KINETIC THEORY OF GASES

CHAPTER I

KINETIC THEORY OF GASES §1. The distribution function T H I S c h a p t e r deals with the kinetic t h e o r y of ordinary gases consisting of electrically neutral a t o m s or molecules. T h e t h e o r y is c o n c e r n e d with non-equilibrium states and p r o c e s s e s in an ideal gas. A n ideal gas, it will b e recalled, is o n e so rarefied that e a c h molecule in it m o v e s freely at almost all t i m e s , interacting with other molecules only during close e_n c31o u n/ t e r s with t h e m . T h a t is to say, the m e a n (where Ν is the n u m b e r of molecules per distance b e t w e e n molecules, f ~ N 3unit volume), is a s s u m e d large in c o m p a r i s o n with their size, or r a t h e r in c o m 3 with the range d of the intermolecular f o r c e s ; the small quantity N d ~ parison ( d / r ) is sometimes called the gaseousness parameter. T h e statistical description of the gas is given b y the distribution function / ( i , q, p) of the gas molecules in their p h a s e s p a c e . It is, in general, a function of the generalized coordinates (chosen in some m a n n e r , and d e n o t e d jointly by q) and the corresponding generalized m o m e n t a (denoted jointly b y p ) , and in a non-steady state also of the time t. L e t άτ - dq dp d e n o t e a volume element in the p h a s e space of the molecule; dq and dp conventionally d e n o t e the p r o d u c t s of the differentials of all the coordinates and all the m o m e n t a respectively. T h e p r o d u c t fdr is the m e a n n u m b e r of molecules in a given e l e m e n t dr which h a v e values of q and ρ in given ranges dq and dp. W e shall return later to this definition of the m e a n . Although the function / will b e e v e r y w h e r e u n d e r s t o o d as the distribution density in p h a s e s p a c e , t h e r e is a d v a n t a g e in expressing it in t e r m s of suitably c h o s e n variables, w h i c h n e e d not b e canonically conjugate c o o r d i n a t e s and m o m e n t a . L e t us first of all decide o n t h e choice t o b e m a d e . T h e translational motion of a molecule is always classical, and is described by the coordinates r = (JC, y, z) of its c e n t r e of m a s s and b y the c o m p o n e n t s of the m o m e n t u m ρ (or the velocity ν = p / m ) of its m o t i o n as a w h o l e . In a m o n a t o m i c gas, the motion of the particles, which are a t o m s , is purely translational. In polyatomic gases, the molecules also h a v e rotational and vibrational degrees of freedom. T h e rotational motion of a molecule in a gas is almost always classical t o o . t It is described in the first place b y the angular m o m e n t u m v e c t o r M of the molecule. F o r a diatomic molecule, this is sufficient. Such a molecule is a rotator turning in a plane perpendicular to M . In actual physical 2 p r o b l e m s , the distribution function t T h e condition for the rotation to be classical is h l2I < T, where I is the m o m e n t of inertia of the molecule and Τ the temperature of the gas. This condition can be violated in ordinary gases only for hydrogen and deuterium at low temperatures.

1

2

Kinetic

Theory of

Gases

may be regarded as independent of the angle φ of rotation of the axis of the molecule in this plane, all orientations of the molecule in the plane being equally probable. This is b e c a u s e the angle φ changes rapidly as the molecule r o t a t e s , and the result m a y be u n d e r s t o o d as follows. The rate of change of φ (the angular velocity of rotation of the molecule) is φ = Ω = MIL Its m e a n value Ω ~ ΰ/d, w h e r e d is the molecular dimension and ϋ the m e a n linear speed. Different molecules h a v e various values of Ω, distributed in some way about Ω. T h u s molecules which initially had the same φ very soon acquire different values; there is a rapid " m i x i n g " with regard to angles. L e t the distribution of molecules in angle φ = φ 0 (in the range from 0 to 2π) and in Ω at the initial instant t = 0 b e given b y a function /(

so that /'(φο, Ω) is a function periodic in
t i n the rotation of a spherical-top molecule, such as C H , the t w o angles remain constant which define 4 the orientation of the molecule relative to M (i.e. the direction of the angular velocity i l ) . In the rotation 2 of an asymmetrical-top molecule, a combination of angles remains constant which represents the rotational energy E t = ΜξΙΙΙχ + Μ / 2 ί + M*j2h, where Μ , Μ , Μ are the c o m p o n e n t s of the ro η 2 ξ η ζ constant vector M along the rotating principal axes of inertia of the molecule.

§1

The Distribution

Function

3

quantized, so that the vibrational state of t h e molecule is specified by the appropriate q u a n t u m n u m b e r s . U n d e r ordinary conditions (at not t o o high temp e r a t u r e s ) , h o w e v e r , t h e vibrations are not excited at all, and the molecule is at its ground vibrational level. In this c h a p t e r w e shall d e n o t e by Γ the set of all variables on which the distribution function d e p e n d s , other t h a n the coordinates of the molecule as a whole (and t h e time t). W e separate from the p h a s e volume element dr the factor dV = dx dy dz, and d e n o t e by d r the remaining factor in t e r m s of the variables used (and integrated o v e r the angles on which / d o e s not d e p e n d ) . T h e quantities Γ h a v e an important c o m m o n p r o p e r t y : they are integrals of the motion, and remain c o n s t a n t for e a c h molecule during its free motion (in t h e a b s e n c e of an external field) b e t w e e n successive collisions; b u t they are in general altered by each collision. T h e coordinates x, y, ζ of the molecule as a whole v a r y , of c o u r s e , during its free motion. F o r a m o n a t o m i c gas, the quantities Γ3 are the three c o m p o n e n t s of the m o m e n t u m ρ = m v of t h e a t o m , so that dT = d p - F o r a diatomic molecule, Γ includes not only the m o m e n t u m ρ b u t also the angular m o m e n t u m M ; accordingly, dT m a y be e x p r e s s e d as

3

(1.1)

dT = 2>rrd pMdMdoM ,

w h e r e d o M is a solid-angle element for the direction of t h e vector M t . F o r a symmetrical-top molecule, the quantities Γ include also the angle θ b e t w e e n M and the axis of the t o p ; t h e n

2 3

2

dr = 4
3 3 d3p 5 ( M . n ) d M d2o „

= d p 5 ( M c o s Θ)Μ dMdoMd

c o s θ d
where do„ = d c o s θ d

4

Kinetic

Theory of

Gases

dV therefore defines its position, at best, only to within distances of the order of its dimensions. This is a very important point. If the coordinates of the gas particles w e r e specified exactly, then the result of a collision b e t w e e n , say, t w o a t o m s of a m o n a t o m i c gas moving in definite classical p a t h s would also b e entirely definite. If, h o w e v e r , the collision is b e t w e e n a t o m s in a given physically small volume (as always in the kinetic theory of gases), the uncertainty in the relative position of the atoms m e a n s that the result of the collision also is uncertain, and only the probability of o n e or another o u t c o m e can b e considered. W e can n o w specify that the m e a n n u m b e r density of particles refers to averaging over the volumes of physically infinitesimal elements thus defined, and correspondingly over times of the order of that t a k e n by the particles to t r a v e r s e such elements. Since the dimensions of the volume elements used in defining the distribution function are large in c o m p a r i s o n with the molecular dimensions d, the distances L over which this function varies considerably m u s t always be large also, in c o m parison with d. T h e ratio b e t w e e n the size of the physically infinitesimal volume elements and the m e a n intermolecular distance r m a y in general h a v e any value. T h e r e is, h o w e v e r , a diiference in the nature of the density Ν determined by the distribution function, according to the value of that ratio. If the element dV is not large c o m p a r e d with r, the density Ν is not a m a c r o s c o p i c quantity: the fluctuations of the n u m b e r of particles p r e s e n t in dV are c o m p a r a b l e with its m e a n value. T h e density Ν b e c o m e s a m a c r o s c o p i c quantity only if it is defined with r e s p e c t to volumes dV containing m a n y particles; the fluctuations in the n u m b e r of particles in these volumes are t h e n relatively small. It is, h o w e v e r , clear that such a definition is possible only if also the characteristic dimensions of the problem L>r.

§ 2. The principle of detailed balancing L e t us consider collisions b e t w e e n t w o molecules, one of which has values of Γ in a given range d T , and the other in a range dTu and which acquire in the collision values in the ranges dT' and dT[ respectively; for brevity, w e shall refer simply to a collision of molecules with Γ and Γι, resulting in Γ and Tj. T h e total n u m b e r of such collisions per unit time and unit volume of the gas m a y be written as a p r o d u c t of the n u m b e r of molecules p e r unit v o l u m e , / ( i , r, O d T , and the probability that any of t h e m has a collision of the t y p e c o n c e r n e d . This probability is always proportional to the n u m b e r of molecules Γι per unit v o l u m e , / ( i , r, T{)dT\9 and to the ranges dV and dT\ of the values of Γ for the t w o molecules after the collision. T h u s the n u m b e r of collisions Γ , Γ ι - » Γ ' , Γ ! per unit time and volume m a y be written as w ( T \ r; ; r, r o / / i d r dvx dr d r ; ;

(2.1)

Ξ h e n c e f o r w a r d , the affixes to / c o r r e s p o n d to those of their a r g u m e n t s here and Γ: f\ fit, r, Γι), / ' — /(f, r, Γ ) , and so on. T h e coefficient w is a function of all its

§2

The Principle

of Detailed

Balancing

5

a r g u m e n t s T.t T h e ratio of w dT' dT[ t o t h e absolute value of t h e relative velocity v - v i of t h e colliding molecules h a s the d i m e n s i o n s of a r e a , and is t h e effective collision cross-section: άσ = ^ Γ ί ΐ Γ , Γ Ο |v-v,|

r d [d

Y

2 ) 2

T h e function w c a n in principle b e d e t e r m i n e d only b y solving the mechanical problem of collision of particles interacting according to s o m e given law. H o w e v e r , certain p r o p e r t i e s of this function c a n b e elucidated from general a r g u m e n t s . ^ T h e collision probability is k n o w n to h a v e an i m p o r t a n t p r o p e r t y which follows τ of m e c h a n i c s (classical or q u a n t u m ) u n d e r time from the s y m m e t r y of the laws r e v e r s a l ; see Q M , § 144. L e t Γ d e n o t e t h e values of t h e quantities obtained from Γ b y time reversal. This operation c h a n gτe s t h e signs of all linear and angular m o m e n t a ; h e n c e , if Γ = (ρ, M ) , then Γ = ( - ρ , - Μ ) . Since time reversal interchanges the states t h a t are " b e f o r e " and " a f t e r " the collision we h a v e

τ

τ

τ

τ

νν(Γ, Γ!; Γ, Γ,) = νν(Γ , Γ , ; Γ , Γ ί ) .

(2.3)

This relation implies, in a state of statistical equilibrium, the principle of detailed τ the τ num Τb e rτ of collisions Γ, Γ ι - > Γ , T[ is equal, in balancing, according to which equilibrium, to the n u m b e r Γ , Τ[ - » Γ , Γ ι . F o r , expressing t h e s e n u m b e r s in t h e form (2.1), w e h a v e

T

w(r,

T

7

T

r;; r, ro/o/oi d r dr! d r dr; = w ( r , i y ; r , r ; ) / ^ d r d i y d r

T

T

dr; ,

w h e r e / 0 is t h e equilibrium (Boltzmann) distribution function. T h e p r o d u c t of p h a s e v o l u m e e l e m e n t s dT dl"i dV dri is unaltered b y time r e v e r s a l ; t h e differentials on the t w o sides of the a b o v e equation m a y thereforeτ b e omitted. N e x t , w h e n t is replaced by -t, t h e energy is u n c h a n g e d : €(Γ) = ε ( Γ ) , w h e r e €(Γ) is the energy of the molecule as a function of the quantities Γ. Since the equilibrium distribution function (in a gas at rest as a whole) d e p e n d s only on the energy,

e(r)/T /ο(Γ) = c o n s t a n t x < T

,

Τ

(2.4)

w h e r e Τ is t h e gas t e m p e r a t u r e , w e h a v e / 0( Γ ) = / 0( Γ ) . L a s t l y , b y the law of c o n s e r v a t i o n of energy in the collision of t w o molecules e + ex = e' + e j . H e n c e /o/oi = /ό/όι,

(2.5)

and t h e a b o v e e q u a t i o n r e d u c e s t o (2.3). This assertion r e m a i n s valid, of c o u r s e , for a gas m o v i n g with a m a c r o s c o p i c t T h e characteristics of the initial (i) and final (/) states in w are written from right to left, w ( / , i) as is customary in quantum mechanics. $It should be emphasized immediately that, although the free motion of m o l e c u l e s is assumed classical, this d o e s not at all mean that their collision cross-section need not be determined quantum-mechanically; in fact, it usually must be so determined. The w h o l e of the derivation of the transport equation given here is independent of the classical or quantum nature of the function w.

PK 10 - Β

6

Kinetic

Theory of

Gases

velocity V. T h e equilibrium distribution function is t h e n

€ /ο(Γ) = c o n s t a n t x e x p ( -

(Ρ Υ Γ ~

'

) ,

) 2 ) 6

and equation (2.5) continues to b e valid b e c a u s e of the c o n s e r v a t i o n of m o m e n t u m in collisions: ρ + pi = p ' + p l . t N o t e t h a t (2.5) d e p e n d s only o n t h e form of t h e distribution (2.4) or (2.6) as a function of Γ ; the p a r a m e t e r s Τ and V m a y vary through the gas v o l u m e . T h e principle of detailed balancing m a y also b e e x p r e s s e d in a s o m e w h a t different form. T o do so, w e apply not only time reversal b u t spatial inversion, changing the sign of all c o o r d i n a t e s . If the molecules are not sufficiently symmetrical, they b e c o m e their stereoisomers on inversion, and they c a n n o t b e m a d e to coincide with these by any rotation of the molecule as a whole.Φ In such c a s e s , inversion would m e a n replacing the gas b y an essentially different s u b s t a n c e , and no new conclusions would be available as to its properties. If, h o w e v e r , the s y m m e t r y of t h e molecule d o e s n o t allow stereoisomerism, the gas remains t h e same on inversion, and t h e quantities which describe the properties of a m a c r o scopicallyτρh o m o g e n e o u s gas m u s t remain unaltered. denote the set of quantities obtained from Γ by simultaneous time Let Γ reversal and inversion. Inversion changes the sign of all ordinary (polar) v e c t o r s , τρ axial v e c t o r s , including the including the m o m e n t u m p , b u t leaves u n c h a n g e d the angular m o m e n t u m M . H e n c e , if Γ = (ρ, M ) , then Γ = (ρ, - Μ ) . A s well as (2.3), w e h a v e the equation§

τρ 1 w ( F , Γ!; Γ, Γ,) = ν ν ( Γ , Γ , * ; Γ

τ ρ τρ , Γ[ ).

(2.7)

Transitions corresponding to the functions w on the t w o sides of (2.3) are said to τ b e mutually time-reversed. T h e y are not strictly direct and reverse transitions, since Γ and Γ are not the s a m e . F o r a m o n a t o m i c gas, h o w e v e r , the principle of detailed balancing can also b e e x p r e s s e d in relation to direct and r e v e r s e τρ the quantities Γ are here just the three m o m e n t u m c o m p o n e n t s of transitions. Since the a t o m , Γ = Γ = ρ, and from (2.7) w ( p \ p i ; P> P i ) = w ( p , p i ; Ρ ' , pi).

(2.8)

This is detailed balancing in the literal s e n s e : each microscopic collision p r o c e s s is balanced b y the r e v e r s e p r o c e s s . T h e function w satisfies o n e further general relation which does not depend o n t h e s y m m e t r y u n d e r time reversal, and which c a n b e m o s t clearly derived in tEquation (2.6) is obtained from (2.4) by transforming the energy of the molecule from the frame of 2 reference Ko in which the gas is at rest to the frame Κ in which it m o v e s with velocity V: € (Γ) = €(Γ) - ρ . V + ïm V ; see Mechanics (3.5). 0 ^Stereoisomers exist for molecules that have no centre of symmetry and n o plane of symmetry. 1 ί Κ §If the quantities Γ include also variables specifying the rotational orientation of the molecule, they too must be transformed in a certain w a y in going to Γ or Γ . For instance, the precession angle of a symmetrical top is given by the product M . n, where η is the direction of the axis of the molecule; this quantity changes sign both under time reversal and under inversion.

§3

The Boltzmann

Transport

Equation

1

q u a n t u m - m e c h a n i c a l t e r m s , the transitions considered being b e t w e e n states forming a discrete series. T h e s e are states of a pair of molecules moving in a given finite volume. T h e probability amplitudes of various collision p r o c e s s e s form +a unitary matrix S, the scattering matrix or S-matrix. T h e unitarity condition is S S = 1, or, in explicit form with the matrix suffixes which label the various states,

η

η

In particular, w h e n i - k,

η

2

T h e s q u a r e | S n | i gives the probability of a collision with the transition i - » n , t and the a b o v e equation is simply the normalization condition for probabilities: the sum of the probabilities for all possible transitions + from a given initial state is unity. T h e + m a y also b e written as SS = 1, with the opposite o r d e r of the unitarity condition S*n = δ*, and w h e n i = k factors S and § . W e t h e n h a v e Σ η Sin

2 Σ

is .nl

=i,

η

so that the sum of the probabilities for all possible transitions to a given final state is unity. Subtracting from e a c h sum the o n e term with η = i (transition without change of state), w e c a n write

2 2 E ' | s n| i = E ' | S i „ l . η

η

This is the required equation. In t e r m s of the functions w, it b e c o m e s

J νν(Γ, Γ!; Γ, Γ,) dV

dY\

= J νν(Γ, Γ,; Γ , Γ!) dT

dY\.

(2.9)

§ 3. The Boltzmann t r a n s p o r t equation L e t us n o w go on to derive the basic equation in the kinetic theory of gases, which is satisfied by the distribution function f(t, r, Γ). If collisions b e t w e e n molecules w e r e entirely negligible, each gas molecule would constitute a closed s u b s y s t e m , and the distribution function of the molecules would o b e y Liouville's t h e o r e m , according to which

2

dfldt=0;

(3.1)

tFor large values of the time t, \S \ is proportional to t, and division by t gives the transition ni probability per unit time; cf QE, §65. If the wave functions of the initial and final particles are normalized to o n e particle per unit volume, this "probability" has the same dimensions (volume/time) as the quantity w dT αΤχ defined by (2.1).

Kinetic

8

Theory of

Gases

see SP 1, § 3 . T h e total derivative here c o r r e s p o n d s to differentiation along the p h a s e path of the molecule, which is determined b y its equations of motion. Liouville's t h e o r e m applies to a distribution function defined as the density in p h a s e space (i.e. in the space of variables that are canonically conjugate generalized coordinates and m o m e n t a ) . This of course does not p r e v e n t / itself from being subsequently expressed in t e r m s of any other variables. In the a b s e n c e of an external field, the quantities Γ for a freely moving molecule remain constant, and only its coordinates r v a r y ; then dfldt = dfldt+

(3.2)

\.Vf.

If, on the other hand, the gas is in, for e x a m p l e , an external field U(r) acting on the coordinates of the centre of m a s s of the molecule (a gravitational field, say), t h e n dfldt = dfldt + ν . V / + F . dfldp,

(3.3)

w h e r e F = - V U is the force exerted on the molecule by the field. W h e n collisions are t a k e n into a c c o u n t , (3.1) is n o longer valid, and the distribution function is no longer c o n s t a n t along the p h a s e p a t h s . Instead of (3.1), w e have (3.4)

dfldt = C(f),

w h e r e C(f) denotes the rate of change of the distribution function b y virtue of collisions: dV dT C(J) is the change due to collisions, per unit time, in the n u m b e r of molecules in the p h a s e v o l u m e dV dT. E q u a t i o n (3.4), in the form d//di = - v . V / + C ( / ) , with dfldt t a k e n from (3.2), gives the total change in the distribution function at a given point in p h a s e s p a c e ; the t e r m dV dT ν . V / is the d e c r e a s e p e r unit time in the n u m b e r of molecules in this p h a s e space element b e c a u s e of their free motion. T h e quantity C ( / ) is called the collision integral, and equations of t h e form (3.4) go by the general n a m e of transport equations. Of c o u r s e , the t r a n s p o r t equation b e c o m e s meaningful only w h e n the form of the collision integral h a s b e e n established. W e shall n o w discuss this topic. W h e n t w o molecules collide, their values of Γ are changed. H e n c e e v e r y collision u n d e r g o n e b y a molecule transfers it out of a particular range d T ; such collisions are referred to as " l o s s e s " . T h e total n u m b e r of collisions Γ , Γ ι - » Γ ' , Γί with all possible values of Γι, Γ', Γί and given Γ, occurring in a volume dV per unit time, is equal to the integral

dVdr

J w ( F , Γί; Γ, Γ ι ) / / ι dri dT'

dT[.

T h e r e are also collisions ("gains") w h i c h bring into the range d r molecules which originally had values outside t h a t range. T h e s e are collisions Γ', Γί -> Γ , Γι, again with

§3

The Boltzmann

Transport

Equation

9

all possible Γι, Γ', Γί and given Γ. T h e total n u m b e r of such collisions in the volume dV per unit time is dV dT j νν(Γ, Γ,; Γ , Γ ί ) / ' / | dTx dT' dT[. Subtracting the losses from the gains, we thus find that as a result of all collisions the relevant n u m b e r of molecules is increased, per unit time, by

dVdr j (w'f'i\-^ff\)dTxdr

dT'u

w h e r e for brevity νν-ννίΓ,Γ^Γ,ΓΟ,

ν ν ' - ν ν ( Γ , Γ ι Γ; , Γ ; ) .

(3.5)

W e therefore h a v e the following expression for the collision integral:

C(f) = j ( w ' / ' / l "

Htffi)

dT, dF dr;.

(3.6)

In the second t e r m in the integrand, the integration over dr' dr; relates only t o w, since / and / i do not d e p e n d on these variables. This part of the integral c a n therefore be transformed b y m e a n s of the unitarity relation (2.9). T h e collision integral then b e c o m e s C ( / ) = j wWi-ffudTidrdT'u

(3.7)

in which b o t h t e r m s h a v e the factor w ' . t Having established the form of the collision integral, w e can write the transport equation as dfidt+ν.

v / = J w ( / ' / ; - / / o dr, d r dr;.

(3.8)

This integro-differential equation is also called the Boltzmann equation ; it w a s first derived b y L u d w i g B o l t z m a n n , the founder of the kinetic t h e o r y , in 1872. T h e equilibrium statistical distribution m u s t satisfy t h e t r a n s p o r t equation identically. This condition is in fact fulfilled. T h e equilibrium distribution is stationary and (in the a b s e n c e of an external field) uniform; the left-hand side of (3.8) is therefore identically z e r o . T h e collision integral also is z e r o , since the integrand vanishes b y virtue of (2.5). T h e equilibrium distribution for a gas in an external field also satisfies the t r a n s p o r t equation, of c o u r s e . W e need only recall that the left-hand side of the t r a n s p o r t equation is the total derivative dfldt, which is t T h e possibility of transforming the collision integral by means of (2.9) w a s noted by E. C. G. Stueckelberg (1952).

10

Kinetic

Theory of

Gases

identically zero for any function / that d e p e n d s only on integrals of the m o t i o n ; and the equilibrium distribution is e x p r e s s e d solely in t e r m s of the total energy e(T) of the molecule, which is an integral of the motion. In the a b o v e derivation of the t r a n s p o r t equation, collisions w e r e regarded as essentially instantaneous and occurring at a particular point in space. It is therefore clear that the equation allows us in principle to follow the variation of the distribution function only over times long c o m p a r e d with the duration of collisions, and over distances large c o m p a r e d with the size of the region in which a collision takes place. T h e s e distances are of the order of the range of action d of the molecular forces (and for neutral molecules, this is equal to their dimensions); the collision time is of the order of d\v. Such values give the lower limit of distances and times that can be dealt with by m e a n s of the t r a n s p o r t equation; the origin of these limitations will be considered in § 16. In practice, h o w e v e r , there is usually n o need (and no possibility) for such a detailed a c c o u n t of the behaviour of the system, which would require, in particular, the specification of the initial conditions (coordinates and velocities of the gas molecules) with the s a m e a c c u r a c y , which is impracticable. In actual physical p r o b l e m s , there are characteristic lengths L and times Τ imposed on the system by the conditions of the problem (characteristic gradient lengths for the m a c r o s c o p i c properties of the gas, wavelengths and periods of sound w a v e s propagated in it, and so on). It is then sufficient to follow the behaviour of the system over distances and times small c o m p a r e d with t h e s e L and T. T h a t is, the physically infinitesimal volume and time elements need b e small only in c o m p a r i s o n with L and T. T h e initial conditions of the problem are also averaged over such elements. F o r a m o n a t o m i c gas, the quantities Γ r e d u c e to the three c o m p o n e n t s of t h e m o m e n t u m p , and from (2.8) the functions w' in the collision integral c a n b e 3 replaced by w = w ( p \ p i ; p , p i ) . T h e n , expressing this function in t e r m s of the da (where t>rei = | v - V | | ; differential collision cross-section du b y w d p' d*p[ = vrel see (2.2)), we find (3.9) T h e function w, and therefore the cross-section da defined by (2.2), contain delta-function factors which e x p r e s s the c o n s e r v a t i o n laws for m o m e n t u m and energy, as a result of which the variables pi, p ' and pi (for a given p) are not in fact independent. H o w e v e r , w h e n the collision integral is expressed in the form (3.9), w e can s u p p o s e that these delta functions h a v e b e e n r e m o v e d b y appropriate integrations; then da will be the ordinary scattering cross-section, depending (for a given vrt \) only on the scattering angle. F o r a qualitative t r e a t m e n t of t r a n s p o r t p h e n o m e n a in g a s e s , the collision integral is roughly estimated by m e a n s of the mean free path I, an average distance traversed by a molecule b e t w e e n t w o successive collisions.t It h a s , of c o u r s e , only qualitative significance; e v e n its definition varies according to which t r a n s p o r t p h e n o m e n o n is under consideration. t T h i s concept is due to R. Clausius (1858).

§4

The H

11

Theorem

T h e m e a n free path c a n b e e x p r e s s e d in t e r m s of the collision cross-section σ and the n u m b e r density Ν of molecules in the gas. If a molecule travels a unit distance in its p a t h , it collides with the molecules p r e s e n t in a volume σ (that of a cylinder with cross-sectional area σ and unit length), the n u m b e r of which is σ Ν . Hence I ~ 1/Νσ.

(3.10)

2 3 T h e collision cross-section σ ~ d , w h e r e d is the dimension of the molecule. With Ν ~ 1/r , f being the m e a n distance b e t w e e n molecules, w e find 2 l~f(fld)

(3.11)

= d(fld)\

Since in a gas r> d, the m e a n free path ί > f. T h e ratio τ ~ IIν is called the mean free time. collision integral, we can put

F o r a rough estimate of the

C ( / ) ~ - (f - /ο)/τ ~ - (β/ïXf - /ο).

(3.12)

By writing the difference / - / 0 in the n u m e r a t o r w e h a v e t a k e n a c c o u n t of t h e fact that the collision integral is zero for the equilibrium distribution function. T h e minus sign in (3.12) e x p r e s s e s the fact that collisions are the m e c h a n i s m for reaching statistical equilibrium, i.e. t h e y tend to r e d u c e the deviation of the distribution function from its equilibrium form. In this s e n s e , τ acts as a relaxation time for the establishment of equilibrium in e a c h volume element of the gas.

§4. T h e Η theorem A gas left to itself, like any closed m a c r o s c o p i c system, will tend to r e a c h a state of equilibrium. Accordingly, the time variation of a non-equilibrium distribution function in a c c o r d a n c e with the t r a n s p o r t equation m u s t b e a c c o m p a n i e d by an increase in the e n t r o p y of the gas. W e shall show that this is in fact so. T h e e n t r o p y of an ideal gas in a non-equilibrium m a c r o s c o p i c state described by a distribution function / is

S = J/log(e//)dVdr;

(4.1)

see SP 1, §40. Differentiating this expression with r e s p e c t to time, w e h a v e

§ = J£(

/log

£) v r d

d

= " / l o g / f dVdr.

(4.2)

Since the establishment of statistical equilibrium in the gas is b r o u g h t a b o u t by collisions of molecules, t h e increase in t h e e n t r o p y m u s t arise from the collisional

Kinetic

12

Theory of

Gases

part of t h e change in t h e distribution function. T h e change in this function d u e t o the free motion of t h e molecules, on t h e other hand, c a n n o t alter t h e e n t r o p y of t h e gas, since this part of t h e change in t h e distribution function is given (for a gas in an external field l / ( r ) ) b y t h e first t w o t e r m s on t h e right-hand side of t h e equation dfldt = - ν . V / - F . dfldp + C ( / ) . Their contribution to t h e derivative dS/dt is - J log / [ - ν . dfldr - F . θ//θρ] dV dT = J [ν . d/dr + F . θ/θρ](/ log fie) dV dT. T h e integral over dV of t h e term involving t h e derivative d/dr is transformed b y G a u s s ' s t h e o r e m into a surface integral; it gives zero on integration through t h e whole volume of t h e g a s , since / = 0 outside t h e region occupied 3 b y the g a s . Similarly, t h e term involving t h e derivative d/dp, on integration over d p , b e c o m e s an integral over a n infinitely distant surface in m o m e n t u m s p a c e , a n d likewise gives zero. T h e change in t h e e n t r o p y is therefore e x p r e s s e d b y

dSldt = - j log / . C(J) dT dV.

(4.3)

This integral c a n b e transformed b y a device which, with a view to later applications, w e shall formulate for t h e general integral /
4

4

= j ( p w ( r , r i r;, r ; ) f / ; d r - | φ * ( Γ ' , Γ ί ; Γ , Γ , ) / / , d T ,

j
4 w h e r e for brevity d T = dT d l \ d F dT[. Since t h e integration here is over all t h e variables Γ , Γι, Γ , Γ!, w e c a n , without altering t h e integral, r e n a m e t h e variables in any manner. Interchanging Γ , Γι and Γ', Γ! in t h e second integral, w e find

4 J 9 ( O C ( f ) d r = J (φ - φ > ( Γ , Γ , ; Γ', Γ ! ) / ' / i d T . Interchanging here Γ , Γ' and Γί9Γ[, taking half t h e sum of t h e resulting integrals, and noting t h e obvious s y m m e t r y of w with respect to t h e t w o colliding particles, w e obtain the transformation rule

, J (p(r)C(/)dr = i J ( 9

4

+ < p 1- ( p ' - 9; ) > v 7 / ; d r .

(4.4)

In particular, / C(f) dT = 0: with C(f) here in t h e form (3.7), w e h a v e

4 j C(S) dT = j w'(f'f[

- / / , ) 4 Γ = 0.

(4.5)

§5

The Change

to Macroscopic

Equations

13

Applied to the integral (4.3), the rule (4.4) gives

4 dSldt = \ J w'f'f[

log ( / ' / Î / / / 0 d r

dV

4 = i | w'ffxx w h e r e χ = f'fllffi. c o n v e r t it to

logxd TdV,

Subtracting from this equation half of the z e r o integral (4.5), we

4 dSldt = i J

log χ - χ + 1) d T dV.

(4.6)

T h e function in the p a r e n t h e s e s in the integrand is non-negative for all χ > 0 ; it is zero w h e n χ = 1, increasing on either side of that point. By definition, the factors w ' , / and / i in the integrand are also positive. W e t h u s obtain the required result, dSldt > 0,

(4.7)

expressing the law of increase of e n t r o p y ; the equality o c c u r s at equilibrium.t N o t e that, since the integrand in (4.6) (and therefore in (4.3)) is non-negative, not only the whole integral (4.3) over dT dV b u t also that over dT alone is positive. T h u s collisions increase the e n t r o p y in e a c h volume element of the gas. This d o e s not, of c o u r s e , imply that the e n t r o p y itself increases in every volume element, since it can b e transferred from o n e region to a n o t h e r b y the free motion of the molecules.

§5. The change to macroscopic equations T h e B o l t z m a n n t r a n s p o r t equation gives a microscopic description of the w a y in which the state of the gas varies with time. W e shall show h o w the t r a n s p o r t equation can b e c o n v e r t e d into t h e usual equations of fluid m e c h a n i c s , which give a less detailed, m a c r o s c o p i c description of this time variation. T h e description is valid w h e n the m a c r o s c o p i c properties ( t e m p e r a t u r e , density, velocity, etc.) of the gas vary sufficiently slowly through its v o l u m e : the distances L over which they change appreciably m u s t b e m u c h greater t h a n the m e a n free p a t h I of the molecules. It has already b e e n mentioned that the integral N(t9r)

= jf(t,r,T)dT

(5.1)

is the spatial distribution density of gas m o l e c u l e s ; the p r o d u c t ρ = mN is correspondingly the m a s s density of the gas. T h e m a c r o s c o p i c velocity of the gas is t T h e proof of the law of increase of entropy by means of the transport equation is due to Boltzmann, and w a s the first microscopic proof of that law. A s applied to g a s e s , the law is often called the Η theorem, since Boltzmann used the symbol Η for the entropy.

14

Kinetic

Theory of

Gases

denoted by V (in contrast to the microscopic velocities ν of the molecules); it is defined as the m e a n V = v = -^|v/dr.

(5.2)

Collisions d o not alter either the n u m b e r of colliding particles or their total energy and m o m e n t u m . It is therefore clear that the collisional part of the change in the distribution function also c a n n o t affect the m a c r o s c o p i c quantities in e a c h volume element of the gas—its density, internal energy, and m a c r o s c o p i c velocity V: the collisional parts of the change in the total n u m b e r , energy and m o m e n t u m of the molecules in unit volume of the gas are given by the zero integrals

J C(f ) d r = 0,

j eC(f) d r = 0,

J p C ( / ) d r = 0.

(5.3)

T h e s e equations are easily derived by applying to the integrals the transformation (4.4) with φ = 1, e and ρ respectively; the first integral is zero identically, the other t w o are z e r o by virtue of the c o n s e r v a t i o n of energy and m o m e n t u m in collisions. L e t us n o w take the t r a n s p o r t equation

ff + £ < * / ) - C ( f ) and integrate over d r after first multiplying by m, ρβ or c. In e v e r y c a s e , the side is z e r o , and w e h a v e t h e equations dpldt + div pV = 0, d(pVa)ldt

+ 3Παβ ΙΘΧβ = 0,

d(Nê)/di + d i v q = 0.

(5.4) right-hand

(5.5) (5.6) (5.7)

T h e first of these is the usual continuity equation of fluid m e c h a n i c s , expressing the conservation of m a s s of the gas. T h e second equation e x p r e s s e s the c o n s e r v a t i o n of m o m e n t u m ; the tensor ΙΙαβ is defined as Π α =β j mVaVpfdT

(5.8)

and is the m o m e n t u m flux t e n s o r ; its c o m p o n e n t Παβ is the α - c o m p o n e n t of the m o m e n t u m transferred in unit time b y molecules across unit area perpendicular to the Χβ-axis. Lastly, (5.7) is t h e equation of conservation of energy; the v e c t o r q is defined as q = | €v/dr, and is the energy flux in the g a s .

(5.9)

§5

The Change to Macroscopic

Equations

15

T o r e d u c e (5.6) a n d (5.7) t o t h e usual equations of fluid m e c h a n i c s , h o w e v e r , w e h a v e still t o e x p r e s s Παβ a n d q in t e r m s of m a c r o s c o p i c quantities. It h a s already b e e n m e n t i o n e d that t h e m a c r o s c o p i c description of t h e gas p r e s u p p o s e s sufficiently small gradients of its m a c r o s c o p i c properties. W e c a n then s u p p o s e , as a first approximation, that in e a c h separate region of t h e gas thermal equilibrium is r e a c h e d , w h e r e a s t h e gas a s a whole is n o t in equilibrium. T h u s t h e distribution function / in e a c h volume element is a s s u m e d t o b e a local equilibrium function, equal t o t h e equilibrium function / 0 for t h e density, t e m p e r a t u r e a n d m a c r o s c o p i c velocity that prevail in that volume element. This approximation implies t h e neglect of all dissipative p r o c e s s e s (viscosity and thermal conduction) in the gas. E q u a t i o n s (5.6) a n d (5.7) t h e n naturally r e d u c e t o those for a n ideal fluid; this m a y b e p r o v e d as follows. T h e equilibrium distribution in a region of t h e gas moving as a whole with velocity V differs from that in a gas at rest only b y a Galilean transformation; on changing t o a frame of reference K' that m o v e s with t h e g a s , w e obtain t h e ordinary B o l t z m a n n distribution. T h e velocities v' of t h e molecules in this frame are related t o t h o s e in t h e original frame X b y ν = ν ' + V. W e write Παβ =

ηιΝ(νανβ)

= m N < ( V a + i O ( V e + t^)>

= ηιΝ(νανβ

+ (ν'αν'β));

the t e r m s νανβ a n d Vpv'a give z e r o o n averaging over t h e directions of ν ' , since all directions of t h e velocity of a molecule in t h e frame K' a r e equally probable. F o r the s a m e r e a s o n ,

t2 (v'av^)^\{v )^\

(5.10)

1

the m e a n square of the thermal velocity is (ν' ) = 3T/m, w h e r e Τ is t h e t e m p e r a t u r e of t h e gas. Finally, since NT is equal t o t h e gas p r e s s u r e P , w e find

ηαβ = ν ν Ρ

α

β

+ δ ρ,

(5.11)

αβ

the familiar expression for t h e m o m e n t u m flux tensor in an ideal fluid; with this tensor, equation (5.6) is equivalent t o E u l e r ' s equation in fluid m e c h a n i c s ( F M , §7). In order t o transform t h e integral (5.9), w e n o t e that the energy e of a molecule in the frame Κ is related t o its energy e' in t h e frame K' b y

2 e = €' + m V . v ' + l m V . Substituting this a n d ν = ν ' + V in q = N e v , w e h a v e

2

2

q = NV[imV + W + e ' ] 2

= V(|pV + P + N Ô ,

16

Kinetic

Theory of

Gases

using (5.10) in averaging the p r o d u c t v ' ( V . v ' ) . But N e ' is the t h e r m o d y n a m i c internal energy of the gas per unit v o l u m e ; the sum N e ' + Ρ is the heat function W of the gas per unit volume. T h u s

2 q = V&>V + W),

(5.12)

in agreement with the k n o w n expression for the energy flux in the d y n a m i c s of an ideal fluid ( F M , §6). Lastly, let us consider the law of conservation of angular m o m e n t u m in the transport equation. This law should apply exactly only to the total angular m o m e n tum of the gas, m a d e up of the orbital angular m o m e n t u m of the molecules in their translational motion and their intrinsic rotational angular m o m e n t a M; the total angular m o m e n t u m density is given by the sum Jrxp/dr + JM/dr.

(5.13)

T h e s e t w o t e r m s , h o w e v e r , have different orders of magnitude. T h e orbital angular m o m e n t u m of the relative motion of t w o molecules at a m e a n distance f apart is of the order of mvr, but the intrinsic angular m o m e n t u m M ~ mvd, which is small in comparison, since w e always h a v e d
(5.14)

T h e r e a s o n for this p r o p e r t y is evident: since, in t h e B o l t z m a n n equation, collisions are regarded as taking place at a point, the sum of the orbital angular m o m e n t a of the colliding molecules is c o n s e r v e d , as well as the sum of their m o m e n t a . In order to derive an equation for the change in the orbital angular m o m e n t u m , it would be n e c e s s a r y to take a c c o u n t of t e r m s of the n e x t higher order in d/r, arising from the fact that the molecules are at a finite distance apart at the time of collision. H o w e v e r , the actual p r o c e s s of angular m o m e n t u m e x c h a n g e b e t w e e n the translational and rotational degrees of freedom c a n b e described in t e r m s of the Boltzmann equation by a relation of the form dWlldt = JMC(f)dT,

(5.15)

w h e r e 3JÎ is the intrinsic angular m o m e n t u m density of the molecules. Since t h e sum of the intrinsic angular m o m e n t a of t w o molecules need not be c o n s e r v e d in a

§6

The Transport

Equation

for a Slightly

Inhomogeneous

Gas

17

collision, the integral on the right of (5.15) is in general not z e r o , and gives the rate of change of 3K. If a non-zero angular m o m e n t u m density is created in the gas by some m e a n s , its s u b s e q u e n t relaxation is described by (5.15).

§ 6. The t r a n s p o r t equation for a slightly inhomogeneous gas In order to t a k e a c c o u n t of dissipative p r o c e s s e s (thermal conduction and viscosity) in a slightly i n h o m o g e n e o u s gas, w e m u s t go to the next approximation b e y o n d that treated in § 5 . Instead of regarding the distribution function in each region of the gas as just the local-equilibrium function / 0, w e shall n o w allow for a slight deviation of / from / 0, putting / = fo + δ/,

δ/ = - (dfolde)x(T)

= foxIT,

(6.1)

w h e r e of is a small correction (
Je/odT,

Jp/odT.

(6.2)

T h e non-equilibrium distribution function (6.1) m u s t yield the s a m e values of these quantities, i.e. the integrals with / and / 0 m u s t be the s a m e . T h e function χ must therefore satisfy the contitions J / o x d r = 0,

ffoXedT

= 0,

J / o X Pd r = 0.

(6.3)

It m u s t be e m p h a s i z e d that even the c o n c e p t of the t e m p e r a t u r e in a nonequilibrium gas b e c o m e s d e t e r m i n a t e only w h e n specific values are assigned to the integrals (6.2). T h e c o n c e p t b e c o m e s entirely rigorous only w h e n t h e gas as a whole is in complete equilibrium; to define t h e t e m p e r a t u r e in a non-equilibrium gas, a further condition is n e c e s s a r y , which m a y b e the specification of these values. L e t us first of all transform the collision integral in the t r a n s p o r t equation (3.8). W h e n the functions (6.1) are substituted, the t e r m s not containing the small correction χ cancel, since the equilibrium distribution function m a k e s the collision integral z e r o . T h e first-order t e r m s give C(f) = foI(x)IT, t T h i s method of solving the transport equation is due to D . E n s k o g (1917).

(6.4)

18 w h e r e I(χ)

Kinetic

Theory

of

Gases

d e n o t e s the linear integral o p e r a t o r (6.5)

H e r e w e h a v e u s e d t h e equation / 0/ o i = /ό/οΓ, t h e factor / 0 can b e t a k e n outside t h e integral, since there is no integration over dT. T h e integral (6.5) is identically z e r o for the functions χ = constant,

χ = c o n s t a n t x €,

χ = ρ . δ V,

(6.6)

w h e r e δ ν is a c o n s t a n t v e c t o r ; this result for t h e second and third functions follows from the c o n s e r v a t i o n of energy and m o m e n t u m in e a c h collision. T h e functions (6.6), which are i n d e p e n d e n t of time and c o o r d i n a t e s , therefore satisfy the t r a n s p o r t equation itself. T h e origin of these solutions is simple. T h e t r a n s p o r t equation is identically satisfied b y the equilibrium distribution function with any (constant) particle density and t e m p e r a t u r e . It is therefore necessarily satisfied also b y the small correction

δ/ = ( θ / 0/ θ Ν ) δ Ν = / 0δ Ν / Ν , which arises w h e n the density changes by 8N; this gives the first solution (6.6). Similarly, the equation is satisfied b y the i n c r e m e n t δ/ =

(df0ldT)ôT,

which arises w h e n Τ c h a n g e s by a small c o n s t a n t a m o u n t δΤ. T h e derivative dfoldT is m a d e u p of a t e r m c o n s t a n t x / 0 (arising from differentiation of t h e normalization factor in / 0) and a term proportional to e / 0; this gives t h e second solution (6.6). T h e third solution e x p r e s s e s Galileo's relativity principle: t h e equilibrium distribution function m u s t satisfy the t r a n s p o r t equation in any other inertial frame. W h e n w e change to a frame moving relative to the original o n e with a small c o n s t a n t velocity δ ν , the velocities ν of the molecules b e c o m e v + δ ν , and the distribution function therefore receives the i n c r e m e n t 8F = ( a / o / a v ) . δΥ = - (/o/T)p . δΥ, corresponding to the third solution (6.6). T h e " e x t r a " solutions (6.6) are excluded b y applying the t h r e e conditions (6.3). W e shall transform the left-hand side of the t r a n s p o r t equation in a general m a n n e r , which c o v e r s b o t h thermal c o n d u c t i o n and viscosity. T h a t is, w e allow t h e p r e s e n c e of gradients of all m a c r o s c o p i c properties of the g a s , including the m a c r o s c o p i c velocity V. T h e equilibrium distribution function in a gas at rest (V = 0) is t h e B o l t z m a n n distribution, which w e write as (6.7)

§6

The Transport

Equation

for a Slightly

Inhomogeneous

Gas

19

w h e r e μ is the chemical potential of the gas. T h e distribution in a moving gas differs from (6.7) only b y a Galilean transformation of the velocity, as already noted in § 5 . In o r d e r to write this function explicitly, w e separate from the total energy €(Γ) of the molecule the kinetic energy of its translational motion:

€(r) = W + € ;

(6.8)

int

the internal energy € i tn includes the energy of rotation of the molecule and the vibrational energy. Replacing ν b y ν - V, w e find t h e B o l t z m a n n distribution in a moving g a s : /„ = e x p ( ^ ) e x p ( ^ ^ ) .

(6.9)

In a slightly i n h o m o g e n e o u s g a s , / 0 d e p e n d s o n the coordinates and the time, as a result of t h e variation through the gas (and in the c o u r s e of time) of its m a c r o s c o p i c p r o p e r t i e s : the velocity V, the t e m p e r a t u r e Τ and t h e p r e s s u r e Ρ (and therefore μ ) . Since t h e gradients of t h e s e quantities are a s s u m e d small, it is sufficient (in this approximation) to replace / b y / 0 o n the left of t h e t r a n s p o r t equation. T h e calculations c a n b e s o m e w h a t simplified b y noting the o b v i o u s fact that the kinetic coefficients, our real subject of interest, d o not d e p e n d o n the velocity V. It is therefore sufficient t o consider a n y o n e point in the gas, and to c h o o s e t h e point w h e r e V (but not, of c o u r s e , its derivatives) is z e r o . Differentiating the e x p r e s s i o n (6.9) with r e s p e c t to time and then putting V = 0, w e obtain

U

μ

ΙΛθΤ/ρ

τ

J

at

Vap /τ at

at '

By the familiar formulae of t h e r m o d y n a m i c s , (3μΙοΤ)ρ

— — S,

(aμ /aP) τ

= 1/N,

μ = W-

Ts,

w h e r e w9 s and 1/N are the h e a t function, e n t r o p y and v o l u m e p e r gas particle. Hence (6.10) Similarly, (6.11) w h e r e for brevity (6.12)

20

Kinetic

Theory of

Gases

in the last term in (6.11), w e have m a d e t h e identical substitution νανβναβ.

νανβ dVpldxa =

T h e left-hand side of t h e transport equation is found by adding t h e expressions (6.10) and (6.11). All derivatives of m a c r o s c o p i c quantities with respect to time c a n be expressed in terms of their spatial gradients b y m e a n s of the equations of an ideal (non-viscous and thermally non-conducting) m e d i u m ; t h e inclusion of dissipative terms here would lead to quantities of a higher order of smallness. At t h e point where V = 0, Euler's equation gives dV/di = - ( l / p ) V P = - ( l / N m ) V P .

(6.13)

At this same point, the equation of continuity gives dN/dt = - Ν div V, or 1 dN

ΝΊΓ

1 dP =

I dT

ρΊΓ-τΊΓ

=

-,

,,

- >

1/ΙΛ

·

άιγΥ

(6 14)

with the equation of state for an ideal g a s , Ν = PIT Lastly, the equation of conservation of entropy, ds/dt + V . Vs = 0, gives ds/dt = 0, or

) Τ dt

Ρ dt

'

with t h e u s e of t h e t h e r m o d y n a m i c formulae (dsldT)p

= c p/ T ,

(dsldP)T

= - 1/P,

c p being t h e specific heat, again p e r molecule; the second of these formulae relates to an ideal gas. E q u a t i o n s (6.14) a n d (6.15) give l £ = -^divV l £ = - l di v V , 1 dt C F dt C v v since for an ideal gas cp - cv = 1. A straightforward calculation leads to t h e result

W T f

ν . V/„ + =*

ν . V T +m , . , , V * . + -

e l ~

(6.16)

)i

r div V } .

. 1 )7

It m u s t be emphasized that n o specific assumption h a s so far b e e n m a d e about the t e m p e r a t u r e d e p e n d e n c e of t h e t h e r m o d y n a m i c quantities; only t h e general equation of state of an ideal gas h a s been used. F o r a gas with a classical rotation of molecules, and vibrations not excited, the specific heat is independent of temp e r a t u r e , and t h e heat function i s t (6.18) t T h e energy €(Γ) of the molecule is assumed to b e measured from its lowest value; accordingly the temperature-independent additive constant in w is omitted.

( 6

§7

Thermal

Conduction

21

in the Gas

T h e last term in (6.17) can then b e simplified; equating (6.17) and (6.4), w e write the t r a n s p o r t equation in the final form

e ( rC r) 7: p v . Vi + [τηυαυβ - δαβ ^ ] ν αβ

= I(χ).

(6.19)

In §§7 and 8, this equation will b e further studied with reference to thermal conduction and viscosity. F r o m the law of increase of e n t r o p y , it follows that a p r e s s u r e gradient (in the a b s e n c e of t e m p e r a t u r e and velocity gradients) does not bring about dissipative p r o c e s s e s ; cf. F M , §49. In the t r a n s p o r t equation, this condition is necessarily satisfied, as is s h o w n by the a b s e n c e of the p r e s s u r e gradient on the left of (6.19).

§7. T h e r m a l conduction in the gas To calculate the thermal conductivity of the gas, we h a v e to solve the transport equation with a t e m p e r a t u r e gradient. Retaining only the first term on the left of (6.19), w e h a v e

e

( cr T) ~ r v . V T = I(x).

(7.1)

T h e solution is to b e sought in the form X = g-VT,

(7.2)

w h e r e the vector g d e p e n d s only on the quantities Γ, since a factor V T results on b o t h sides of (7.1) w h e n this substitution is m a d e . Since the equation m u s t be valid for any vector VT, the coefficients of this on the t w o sides m u s t be equal, and so we obtain for g the equation

€ v

( Cr T )p ~

= I(g),

(7.3)

which does not involve V T (nor therefore any explicit d e p e n d e n c e on the coordinates). T h e function χ m u s t also satisfy the conditions (6.3). With χ in the form (7.2), the first t w o of these are necessarily satisfied, as is evident from the fact that (7.3) contains no vector p a r a m e t e r s which might give the direction of the c o n s t a n t vector integrals / / 0g dT and / / o e g dT. T h e third condition imposes on the solution of (7.3) the further condition / 0v . g d r = 0.

(7.4)

If the t r a n s p o r t equation has b e e n solved and the function χ is k n o w n , the thermal conductivity can b e determined b y calculating the energy flux, or rather its PK 10 - c

Kinetic

22

Theory

of

Gases

dissipative part that is not d u e simply to convective energy transfer, which w e shall d e n o t e b y q'. In the a b s e n c e of m a c r o s c o p i c motion in t h e gas, q' is equal to t h e total energy flux q given b y the integral (5.9). W h e n / = / 0, this integral is z e r o identically, b e c a u s e of the integration over the directions of v. On substitution of / from (6.1), there thus remains q = γ j yfoxe dT = Yj/o€v(g.VT)dr, or in c o m p o n e n t s qa

=

~

καβ 8TIθχβ,

καβ = - γ j

a dr.

/o€V gp

(7.5)

Since a gas in equilibrium is isotropic, there are no preferred directions in it, and the t e n s o r καβ can only b e expressible in t e r m s of the unit t e n s o r δ α , βi.e. it r e d u c e s to a scalar:

=

κ

α

Κδ β,

αβ

Κ

—

3Κ

α. α

T h u s the energy flux is q = - kVT, w h e r e the scalar thermal

conductivity

K

is

= ~Jfj

(7.6)

o€V g ^

m d T

*

77 **^

T h e t r a n s p o r t equation necessarily m a k e s this quantity positive (see §9): the flux q m u s t be in the opposite direction to the t e m p e r a t u r e gradient. In m o n a t o m i c gases, the velocity ν is the only v e c t o r on which the function g d e p e n d s ; it is therefore clear that this function m u s t h a v e t h e form g = Wv)g(v).

(7.8)

In polyatomic g a s e s , g d e p e n d s on t w o v e c t o r s : the velocity ν and t h e angular m o m e n t u m M . If the s y m m e t r y of t h e molecules d o e s not allow stereoisomerism, the collision integral, and therefore equation (7.3), are invariant u n d e r inversion; t h e solution χ m u s t b e similarly invariant. In other w o r d s , χ = g . V T m u s t be a true scalar, and, since t h e gradient V T is a true vector, so m u s t b e t h e function g. F o r instance, in a diatomic gas, w h e r e t h e quantities Γ are just the v e c t o r s ν and M , t h e function g(T) has t h e form g = vgi + M ( v . M ) g 2+ (ν x M ) g 3,

(7.9)

§7

Thermal

Conduction

in the Gas

2

2

2

23

w h e r e gu g2, g 3 are scalar functions of the scalar a r g u m e n t s v , M , ( v . M ) ; this is the m o s t general form of a true vector that can b e c o n s t r u c t e d from the true vector ν and the p s e u d o v e c t o r M . t If, h o w e v e r , the s u b s t a n c e is stereoisomeric, there is no invariance under inversion: as already mentioned in §2, inversion then " t r a n s f o r m s " the gas into w h a t is essentially a different s u b s t a n c e . Accordingly, the function χ m a y also contain p s e u d o s c a l a r t e r m s , and the function g m a y contain p s e u d o v e c t o r t e r m s , e.g. o n e of the form g 4 M . T h e condition for the a b o v e m e t h o d of solving the t r a n s p o r t equation (based on the a s s u m p t i o n that / is close to / 0) to be valid can be ascertained b y estimating the collision integral from (3.12). T h e m e a n energy of a molecule is ë~ T, and so an estimate of t h e t w o sides of (7.3) gives ν ~ g/τ ~ gv/l, w h e n c e g ~ I. T h e condition XJT ~ g | V T | / T <^ 1 (equivalent to 8f
(7.10)

w h e r e c is the specific heat per molecule of the gas. This is a well-known elementary formula in the kinetic t h e o r y of gases (cf. the last footnote to §11). Putting ί ~ l / Ν σ , c ~ 1 and ϋ ~ V ( T / m ) , we h a v e κ~(1/σ)λ/(Τ/,η).

(7.11)

In this estimate, the cross-section σ relates to the m e a n thermal speed of the molecules, and in that sense is to be regarded as a function of t e m p e r a t u r e . As the speed increases, the cross-section in general d e c r e a s e s ; accordingly, σ is a decreasing function of the t e m p e r a t u r e . W h e n the t e m p e r a t u r e is not too low, the gas molecules b e h a v e qualitatively as hard elastic particles which interact only w h e n they actually collide. This type of interaction c o r r e s p o n d s to a collision cross-section varying only slightly with the speed (and therefore with the temp e r a t u r e ) . U n d e r such conditions, κ is approximately proportional to V T . At a given t e m p e r a t u r e , the thermal conductivity is seen from (7.11) to be i n d e p e n d e n t of the gas density, i.e. of the gas p r e s s u r e . It m u s t be e m p h a s i z e d that this important p r o p e r t y is not related to the a s s u m p t i o n s used in making the estimate, b u t is an e x a c t c o n s e q u e n c e of the B o l t z m a n n t r a n s p o r t equation. It arises b e c a u s e this equation takes a c c o u n t only of collisions b e t w e e n pairs of molecules (for which r e a s o n the m e a n free path is inversely proportional to the gas density). t T h e solution of the Boltzmann equation for a gas of rotating molecules w a s first discussed by Yu. M. Kagan and A. M. Afanas'ev (1961).

Kinetic

24

Theory of

Gases

§ 8. Viscosity in the gas T h e viscosity of a gas is calculated b y m e a n s of t h e transport equation in t h same w a y as the thermal conductivity. T h e only difference is that the deviatioi from equilibrium is d u e n o t to t h e t e m p e r a t u r e gradient b u t to t h e non-uniformit; of t h e gas flow as regards t h e m a c r o s c o p i c velocity V. It is again a s s u m e d that th characteristic dimensions of t h e problem L > I. T h e r e a r e , as w e k n o w , t w o kinds of viscosity, the corresponding coefficient being usually denoted b y η and ζ. T h e y are defined as t h e coefficients in th viscous stress tensor σ'αβ which forms part of t h e m o m e n t u m flux tensor: Π αβ = Ρ8αβ + ρ V aν β - σ'αβ9

σ'αβ = 2 η ( ν αβ - \δαβdiv V) + ζδαβ div V,

(8.1

(8.2

w h e r e Vaf}is defined b y (6.12); see F M , § 1 5 . In an incompressible fluid, only th viscosity η o c c u r s . T h e " s e c o n d viscosity" ζ appears in motion such that div V ^ C It is convenient to calculate t h e t w o coefficients separately. Omitting the temperature-gradient term from t h e general transport equatioi (6.19), w e c a n write

2 ηΐΌαΌβ{ναβ

- \δαβdiv V) + [\mv - e(T)lcv]

div V = I(χ),

(8.3

w h e r e the terms containing t h e first and second viscosities have b e e n separated oi the left-hand side. In calculating t h e first viscosity, w e h a v e to a s s u m e tha div V = 0. T h e resulting equation c a n b e identically rewritten as

2 τη(υανβ -\δαβ ν )ναβ

(8.4

= Ι(χ),

w h e r e t h e t w o tensor factors o n the left have zero trace. T h e solution of this equation is sought in the form X = £αβν

α, β

(8.5

w h e r e gapiO is a symmetric tensor; since t h e trace Vaa= 0, b y adding a term in δ0 to g«0 w e c a n always ensure that gaa= 0, without altering χ. T h e equation for g a3/i

12

τη(νανβ - 3δαβ ν ) = I(gap).

(8.6

T h e extra conditions (6.3) are necessarily satisfied. T h e m o m e n t u m flux is calculated from the distribution function as t h e intégra (5.8). T h e required part of this, namely t h e viscous stress tensor, is σ'αβ = - (mlΤ) f

VaVpfoX

η«βγδ = ~ (mIT)

dT = η α βν γ ,δ

foVaVpgys

dT.

(8.7 (8.8)

§8

Viscosity

in the Gas

25

T h e quantities η α δβ form a t e n s o r of r a n k four, s y m m e t r i c in the suffixes α, β γ and y, δ and giving z e r o o n contraction with r e s p e c t to the pair y, δ. B e c a u s e t h e An gas is isotropic, this t e n s o r c a n only b e e x p r e s s e d in t e r m s of t h e unit t e n s o r δ α . β expression satisfying t h e s e conditions is Ήαβγδ = Τ|[δαγδβδ + δ α δδβγ ~ 3 δ α δβγ ] δ. T h e n σ'αβ = 2 η ν α, β so that η is t h e required scalar viscosity coefficient. It is determined b y contracting the t e n s o r with r e s p e c t t o t h e pairs of suffixes a, y and β, δ : η = - ( m / 1 0 T ) J νανβ8αβ ίο

dr.

(8.9)

In a m o n a t o m i c g a s , gafi is a function only of t h e v e c t o r v. T h e general form of such a s y m m e t r i c t e n s o r with zero trace is

2 g^ = (v^-\b^v )g(v),

(8.10)

with a single scalar function g(v). In polyatomic g a s e s , t h e t e n s o r g a3/is c o m p o s e d of a large n u m b e r of variables, including the t w o v e c t o r s ν a n d M . In t h e a b s e n c e of stereoisomerism, g a3/c a n include only true t e n s o r t e r m s ; in a stereoisomeric g a s , p s e u d o t e n s o r t e r m s also are possible. An estimate of t h e viscosity coefficient, similar t o (7.10) for t h e t h e r m a l c o n d u c tivity, gives a well-known e l e m e n t a r y formula in t h e kinetic t h e o r y of g a s e s , (8.11)

η-mvNl;

see t h e last footnote t o § 1 1 . T h e t h e r m o m e t r i c conductivity a n d t h e kinematic viscosity are found t o b e of t h e s a m e order: KlNcp~vlNm~vL

(8.12)

1/2

Putting in (8.11) I ~ l / Ν σ and ϋ ~ ( T / m ) , w e obtain Tj~V(mT)/a.

(8.13)

T h e description of t h e p r e s s u r e a n d t e m p e r a t u r e d e p e n d e n c e of κ in §7 is entirely valid for t h e viscosity η also. In o r d e r to calculate t h e second viscosity coefficient, w e m u s t t a k e t h e second term o n t h e left of t h e t r a n s p o r t equation (8.3) to b e n o n - z e r o :

2 [\mv

- €(T)lcv]

div V = Ι(χ).

(8.14)

W e shall seek t h e solution in t h e form χ = g div V

(8.15)

26

Kinetic

Theory

of

Gases

and obtain for g the equation

2 \mv -e(Dlcv

= I(g).

(8.16)

Calculation of the stress tensor and c o m p a r i s o n with the expression ζδαβ div V gives t h e viscosity coefficient as (8.17)

2 In m o n a t o m i c gases ε(Γ) = \mv , cv = 3/2, and t h e left-hand side pf (8.16) is z e r o . T h e equation 1(g) = 0 then shows that g = 0, and therefore ζ = 0. W e r e a c h , therefore, t h e interesting conclusion that the s e c o n d viscosity of m o n a t o m i c gases is z e r o . t PROBLEM S h o w that a gas of ultra-relativistic particles has zero second viscosity (I. M. Khalatnikov, 1955). € of a relativistic particle in a frame of reference Κ in which the gas m o v e s with a (non-relativistic) velocity V is related to its energy e' in the frame K' in which the gas is at rest b y €' = € - p . V , where ρ is the momentum of the particle in the frame K ; this is the Lorentz transformation formula with the terms a b o v e the first order in V omitted. T h e distribution function in the frame Κ is /o(c - ρ . V), where /o(c') is the Boltzmann distribution. Considering only the viscosity, w e can immediately assume that the gradients of all macroscopic quantities are zero except that of the velocity V; then dYldt = 0, and the last term in (6.10) vanishes.t In (6.11), the first t w o terms are also absent, and the third b e c o m e s

SOLUTION. The energy

ν . V ( p . V) = ν ρ

αβ ονβΐ3χα

= νρν

αβαβ

;

the directions of ν and ρ are the same, and s o ρ ν = ρ ν . The equations of continuity and entropy αβ βα conservation in the form used in § 6 remain valid in the motion of a relativistic gas (with small velocities V). The formulae (6.16) therefore remain valid also. The transport equation thus b e c o m e s (ν ρ -δ €ΐ€ )ν

αβ αβ ν αβ

= Ι(χ).

In the second-viscosity problem, w e must put ν

αβ= \δαβdiv V, and then

( W - € / c ) d i v V = i(x). In an ultra-relativistic gas, ν « c, e = cp, and the specific heat c„ = 3 (see SP 1, § 4 4 , Problem); the left-hand side of the equation, and therefore χ, are then zero.

§ 9. Symmetry of the kinetic coefficients T h e thermal conductivity a n d t h e viscosity are a m o n g t h e quantities which govern relaxation p r o c e s s e s in systems slightly departing from equilibrium. T h e s e kinetic coefficients satisfy Onsager's symmetry principle, which m a y b e established

3

t i t must be emphasized that these g a s e s are being treated in the approximation with respect to the gaseousness parameter N d which corresponds to the Boltzmann equation (and in which η is independent of the density). In higher approximations (the subsequent terms in the "virial expansion'*; s e e § 18), a non-zero viscosity ζ does appear. Another important point is the quadratic dependence of the particle energy on the momentum; in a relativistic "monatomic" g a s , the second viscosity is not zero (although it vanishes in another limiting case, the ultra-relativistic c a s e ; s e e the Problem). $To avoid misunderstandings, it may b e mentioned that in a relativistic gas the pressure gradient makes a contribution to the thermal-conduction energy flux; see F M , §136.

§9

Symmetry

of the Kinetic

Coefficients

27

in a general form without discussing specific m e c h a n i s m s of relaxation. H o w e v e r , in a specific calculation of kinetic coefficients from t h e t r a n s p o r t equations, t h e s y m m e t r y principle d o e s n o t yield a n y extra conditions t o b e imposed o n t h e solution of t h e equations. In such a calculation the r e q u i r e m e n t s of t h e principle are necessarily satisfied. It is useful to see h o w this o c c u r s . In t h e general formulation of O n s a g e r ' s principle (see S P 1, § 120), t h e r e a p p e a r s a set of quantities xa which describe t h e deviation of t h e system from equilibrium, and a set of quantities " t h e r m o d y n a m i c a l l y c o n j u g a t e " t o t h e s e , Xa = ~ dS/dxa (where S is t h e e n t r o p y of t h e system). T h e relaxation p r o c e s s of a system slightly departing from equilibrium is described b y equations which determine t h e rates of change of t h e xa as linear functions of t h e Xa : *α=-Σ7αΛ,

(9.1)

b

w h e r e t h e y aba r e t h e kinetic coefficients. According t o O n s a g e r ' s principle, if xa and xb b e h a v e in t h e same w a y u n d e r time reversal, then yab

= 7ba-

(9.2)

T h e rate of change of t h e e n t r o p y is given b y t h e quadratic form S = - 2 X e i e = 2 7 e XbeX f .c a

(9.3)

a,b

T h e first of t h e s e expressions is often convenient for establishing t h e corr e s p o n d e n c e b e t w e e n t h e xa and t h e X a. F o r thermal conductivity, w e take as t h e rates xa t h e c o m p o n e n t s q'a of the dissipative heat flux vector (at a n y given point in t h e m e d i u m ) ; t h e suffix a is then the2 same as t h e vector suffix a. T h e corresponding quantities Xa are t h e derivatives T~ dTldxa; cf. S P 2, §88. E q u a t i o n s (9.1) c o r r e2s p o n d t o q'a= -καβ 3ΤΙοχβ9 so that . According t o O n s a g e r ' s printhe kinetic coefficients y ab are t h e quantities Τ καβ ciple, w e should h a v e καβ = κβα . Similarly, for t h e viscosity, w e take as t h e xa t h e c o m p o n e n t s σ'αβ of t h e viscous m o m e n t u m flux tensor; t h e corresponding Xa are - ναβ ΙΤ (the suffix a h e r e answering t o t h e pair of tensor suffixes α β ) . E q u a t i o n s (9.1) c o r r e s p o n d t o σ'αβ= VafiyôVys, and t h e kinetic coefficients are ΤηαβΎδ . According t o O n s a g e r ' s principle, w e m u s t h a v e ηαβΎδ= η γ δ. α β In t h e problems of thermal conduction and viscosity of gases, considered in §§7 and 8, the s y m m e t r y of t h e tensors καβ a n d ηαβΎδw a s a n e c e s s a r y c o n s e q u e n c e of the isotropy of t h e m e d i u m , i n d e p e n d e n t of t h e solution of t h e t r a n s p o r t equation. W e shall show, h o w e v e r , that it would also follow from this solution, independently of t h e isotropy of t h e g a s . T h e p r o c e d u r e for p r o b l e m s of thermal conduction a n d viscosity in a slightly i n h o m o g e n e o u s gas w a s t o seek t h e correction t o t h e equilibrium distribution function in t h e form Χ = Σ8α(ΌΧα,

(9.4)

28

Kinetic

Theory of

Gases

obtaining for the functions ga equations of the form (9.5)

La = I(ga). T h e quantities La are c o m p o n e n t s of the vector T[e(T)-cpT]va for thermal conduction, or the tensor

for viscosity; cf. (6.19). T h e solutions of equations (9.5) m u s t satisfy the further conditions jfogadr

= 0,

jf0gaedT

= 0,

jf0gapdT

= 0.

With these conditions, the kinetic coefficients yab c a n be written as the integrals

2 (9.6)

T yab = -jf0Lagb dr. T h e proof of the s y m m e t r y yab = jba thus r e d u c e s to that of the equation ffoLagbdT

(9.7)

= jfoLbgadT.

It is based on the property that the linearized o p e r a t o r J is "self-conjugate", which m a y be arrived at as follows. L e t us consider the integral J / 0φ ί ( ψ ) dT = J / o / o i W ' φ ( φ ' + ψ{-φ-φι)

dT,

w h e r e φ(Γ) and φ(Γ) are any t w o functions of the variables Γ. Since the integration is over all the variables Γ, Γι, Γ', Γί, w e c a n r e n a m e these in any w a y (as w a s d o n e in §4) without affecting the value of the integral. W e m a k e the change Γ , Γ ' ^ Γ ΐ , Γ ί , and then in each of the t w o resulting forms the further change Γ, Γι <-> Γ', Γί. T h e sum of all four expressions gives

, J /οφί(ψ) dT=\f

/ο/οι[>ν (φ + )Ψι- w(
(9.8)

the notation w and w' is as in (3.5). τL e t us n o w τ consider a similar integral in which ψ(Γ) and <ρ(Γ) are replaced by φ ( Γ ) and ψ ( Γ ) respectively (without changing w

§9

Symmetry

τ

of the Kinetic

Coefficients

29

and νν'). With the change Γ , Γ / , . . . -> Γ, Γ ι , . . . in this integral, and the principle of detailed balancing (2.3), we h a v e

JM I(ç )dr T

T

= \ j /ο/οι[Η>(ψ + φχ) - νν'(ψ' + ψ ! ) ] [ ( φ ' + φ ί ) " ( φ + φι)] d T ,

(9.9)

τ w h e r e the equation / ο ( Γ ) = / 0( Γ ) has also b e e n used. E x p a n d i n g the square brackets in (9.8) and (9.9), and comparing corresponding t e r m s , w e see that the two integrals are equal. In making the c o m p a r i s o n , it is n e c e s s a r y to take a c c o u n t of the unitarity relation (2.9), which gives, for e x a m p l e , j ΜοΜΨ

+ ψ,)(φ + φ,) d T = J /ο/οινν'(ψ + ψ,)(φ + φ,) d T ;

the relation (2.9) is applied here to the integration over the variables Γ' and Γί, on which only w and w' depend in the integrand. T h u s w e r e a c h the equation

T

T

J/o
(9.10)

If the principle of detailed balancing is valid in its simple form (2.8), w = w \ then (9.10) r e d u c e s to a literal self-conjugacy of the operator I: j /οφΐ(ψ) dT = j /οψί(φ) dT,

(9.11)

w h e r e b o t h integrals contain functions φ and ψ of the same variables Γ ; this is immediately evident w h e n w = νν', from the expression (9.8). τ Returning to the kinetic coefficients, w e m a k e in the first integral (9.7) the change Γ -» Γ , and note that

T La(T ) = ±La(T),

(9.12)

τ the u p p e r and lower signs relating to viscosity and thermal conduction respectively. W e n o w u s e the relations (9.5) and (9.10). In the latter, w e c a n integrate o v e r Γ in place of Γ ; this clearly does not affect the value of the integral. W e h a v e T jfogb Ladr

=

T

±jfogbI(ga)dr

T = ± I fog/Hgt)

= ±j

dT

T fogjLb(r)dr .

30

Τ

Kinetic

Theory

of

Gases

N o w , changing Γ - » Γ on the right-hand side, and using (9.12), w e h a v e t h e required result (9.7). T h e kinetic coefficients m u s t also satisfy conditions w h i c h follow from the law of increase of e n t r o p y ; in particular, t h e " d i a g o n a l " coefficients yaa m u s t b e positive. Since the t r a n s p o r t equation g u a r a n t e e s t h e increase of e n t r o p y , t h e s e conditions are of c o u r s e necessarily satisfied w h e n t h e kinetic coefficients are calculated from that equation. T h e increase of e n t r o p y is e x p r e s s e d b y t h e inequality - Jlog/C(/) dT>0; see §4. Substituting / = /o(l + * / T ) ,

C ( / ) = ( / 0/ T ) I ( x ) ,

we have - / l o g / 0C ( F ) D R - £ J / 0l o g ( l + X / T ) I C r ) D R > 0 . T h e first integral is identically z e r o ; in t h e s e c o n d integral, since χ is small, log(l + χΙΤ) « χ/Τ, and so w e find -J/oxKx)dr>0.

(9.13)

This inequality e n s u r e s t h e n e c e s s a r y p r o p e r t i e s of t h e kinetic coefficients. particular, w h e n χ = ga it e x p r e s s e s t h e fact that yaa is positive.

In

§ 10. Approximate solution of the transport equation B e c a u s e of t h e complexity of the law of interaction of molecules (especially polyatomic ones), which d e t e r m i n e s t h e function w in t h e collision integral, the B o l t z m a n n equation c a n n o t really b e e v e n written d o w n in an e x a c t form for specific gases. H o w e v e r , e v e n with linearization and s o m e simple a s s u m p t i o n s a b o u t t h e n a t u r e of t h e molecular interaction, t h e complexity of t h e m a t h e m a t i c a l structure of the t r a n s p o r t equation m a k e s it generally impossible t o solve in an e x a c t analytical form. Fairly efficient m e t h o d s for t h e a p p r o x i m a t e solution of t h e B o l t z m a n n equation are therefore of particular significance in t h e kinetic t h e o r y of gases. T h e principle a s applied to a m o n a t o m i c gas is as follows (S. C h a p m a n 1916). L e t u s first t a k e t h e p r o b l e m of t h e r m a l c o n d u c t i o n . F o r a m o n a t o m i c g a s , t h e specific h e a t c p = 5/2, and t h e linearized equation (7.3) b e c o m e s (10.1)

§10

Approximate

Solution

of the Transport

31

Equation

w h e r e β = m / 2 T ; the linear integral o p e r a t o r 1(g) is defined b y

3 K g ) = | | tWoi(g' + g! - g - gi) d P i da,

(10.2)

c o r r e s p o n d i n g to the collision integral (3.9), and the equilibrium distribution function i s t

3l23 3l2 v (10.3)

fo(v) = (Nfi lm TT )e^ \

A n efficient m e t h o d of approximately solving equation (10.1) is b a s e d on e x p a n d i n g t h e required functions in t e r m s of a c o m p l e t e set of mutually orthogonal functions, w h i c h m a y with especial a d v a n t a g e b e t a k e n as t h e Sonine polynomials (D. B u r n e t t 1935). T h e s e are defined b y t

s

x

Sr(x)

xr s (10.4)

= ±e x-'£;(e- x + ),

w h e r e r is a n y n u m b e r and s is a positive integer or z e r o . In particular,

l S r° = 1,

Sr(x)

(10.5)

= r+l-x.

T h e orthogonality p r o p e r t y of t h e s e polynomials for a given r and different s is

xr s

s

f e- x Sr(x)Sr'(x) Jo

dx = Γ ( γ + s + 1)8 J s !.

(10.6)

W e shall seek the solution of (10.1) as t h e e x p a n s i o n

2 g(v) = ( β / Ν ) ν Σ Α ^ , 2( β ι > ) .

(10.7)

s=\

B y omitting t h e t e r m with s = 0, w e automatically satisfy t h e condition (7.4), t h e 2 integral being z e r o b e c a u s e t h e polynomials with s = 0 and s^0 are orthogonal. T h e e x p r e s s i o n in p a r e n t h e s e s o n t h e left of (10.1) is the polynomial S\^v ), and this equation therefore b e c o m e s

2 - yS\l2 tfv )

= (β/Ν) £

(10.8)

ASI(vSfo).

5=1

2

3

Multiplying b o t h sides scalarly b y ν / ο ( υ ) δ 3/ 2 ( β υ ) and integrating o v e r d p,

we

t T h e distribution function is e v e r y w h e r e taken to be defined in m o m e n t u m space. This, h o w e v e r , d o e s not prevent it from being e x p r e s s e d for c o n v e n i e n c e in terms of the velocity ν = pi m. t T h e y differ only in normalization and affix numbering from the generalized Laguerre polynomials: Sr'W = ( ^ j L ;

+ ( xs) .

32

Kinetic

Theory

of

Gases

obtain a set of algebraic e q u a t i o n s ΣβιΑ=7«ιι»

1 = 1,2,...,

(10.9)

with

2 2

3

j / 0v . Sy(vSîl2 )d p 2 2 = (0 /4N ){vSUvSU

als = - (β ΙΝ )

(10.10)

the notation being

3

3

{F, G} = J /ο(ι>)/ο(ι>ι)|ν - v i | A ( F ) A ( G ) d p d p , Ar, (10.11)

A(F) = F(v') + F(vi) - F(v) - F ( v 0 .

T h e r e is n o equation with I = 0 in (10.9), since a 0s = 0 b e c a u s e of the c o n s e r v a t i o n of m o m e n t u m : Δ ( ν δ 3 / )2 = Δ(ν) = 0. T h e t h e r m a l conductivity is calculated b y sub2 (10.7) in the integral (7.7). T h e condition (7.4) s h o w s that this integral (with stituting e = \mv ) c a n be put in the form

2

3

foS\,2Wv )v.gd p

" = - \ \

and the result is κ=5Α!/4.

(10.12)

T h e advantage of expanding in Sonine polynomials is s h o w n b y the simplicity of the right-hand side of e q u a t i o n s (10.9) and t h e e x p r e s s i o n (10.12). T h e calculations are entirely similar for the viscosity. T h e solution of (8.6) is sought in the form

2 2

2

2

g<* = -(β ΙΝ )(νανβ

-\ν δαβ )

i

B sS ? / (20 u ) .

(10.13)

Substitution in (8.6), multiplication b y

3 and integration over d p leads to the set of e q u a t i o n s 2 5=0

where

bls Bs = 5 δ ί ,0 1 = 0 , 1 , 2 , . . . ,

l

3 2 bis = (β ΙΝ ){(νανβ

-

W8 )S ,2

afi5 9

(10.14)

2 (υανβ - ^ 6 a ) 0S ? / }2.

(10.15)

§10

Approximate

Solution

of the Transport

Equation

33

T h e viscosity is found from (8.9) as η = i m B 0.

(10.16)

T h e infinite set of equations (10.9) or (10.14) is approximately solved b y retaining only the first few t e r m s in the e x p a n s i o n (10.7) or (10.13), i.e. b y artificially terminating the set. T h e approximation c o n v e r g e s extremely rapidly as the n u m b e r of t e r m s i n c r e a s e s : in general, retaining just o n e t e r m gives t h e value of κ or η with an a c c u r a c y of l - 2 % . t W e shall show that the a p p r o x i m a t e solution of the linearized t r a n s p o r t equation for m o n a t o m i c gases b y the a b o v e m e t h o d gives values of the kinetic coefficients that are certainly less than would follow from the e x a c t solution of the equation. T h e t r a n s p o r t equation m a y b e written in the symbolic form I(g) = L,

(10.17)

w h e r e the functions g and L are v e c t o r s in the t h e r m a l c o n d u c t i o n p r o b l e m , and t e n s o r s of r a n k t w o in the viscosity problem. T h e c o r r e s p o n d i n g kinetic coefficient is determined from the function g as a quantity proportional to the integral

3 (10.18)

-jfogI(g)d p;

see §9. T h e a p p r o x i m a t e function g, h o w e v e r , satisfies n o t equation (10.17) itself b u t only t h e integral relation

3

3 (10.19)

ffogI(g)d p=ffoLgd p9

as is evident from the w a y in which the coefficients in the e x p a n s i o n s of g are determined. T h e s t a t e m e n t m a d e a b o v e follows immediately from the "variational p r i n c i p l e " w h e r e b y the solution of (10.17) gives a m a x i m u m of the functional (10.18) within the class of functions that satisfy the condition (10.19). T h e validity of this principle is easily s h o w n b y considering the integral

3 - j

fo(g-
w h e r e g is the solution of (10.17), and φ any trial function that satisfies the condition (10.19). This integral is positive, b y the general p r o p e r t y (9.13) of the o p e r a t o r I. E x p a n d i n g the p a r e n t h e s e s , w e write

3 -Jfo{gKg)

+ φΙ(φ) -
d p.

Î T h e c o n v e r g e n c e is, h o w e v e r , s o m e w h a t less g o o d in problems of diffusion, and especially of thermal diffusion.

34

Kinetic

Theory

of

Gases

Since for a m o n a t o m i c gas the principle of detailed balancing is valid in the form (2.8), the o p e r a t o r I has the self-conjugacy p r o p e r t y (9.11).t H e n c e the integrals of the last t w o t e r m s in the b r a c e s are equal. T h e n substitution of 1(g) = L gives I

- J fo{gKg) + φΚψ) ~ 2
+ φΐ(φ) - 2L 0.

Finally, transforming the integral of t h e last term b y m e a n s of (10.19), w e find - j fogKg) d3p>-f

f
0

d3p

9

as w a s to b e p r o v e d . 4 physical T h e r e is a case that is of formal interest though having n o direct significance, namely a gas of particles interacting according to U = a / r . $ This h a s the p r o p e r t y that the collision cross-section for such particles (determined b y classical m e c h a n i c s ) is inversely proportional t o the relative speed vrelyand so the p r o d u c t Vreida which a p p e a r s in the collision integral d e p e n d s only on t h e scattering angle 0, not on u r i.e T h e p r o p e r t y in question is easily p r o v e d b y dimensional a r g u m e n t s : the cross-section d e p e n d s only on t h r e e p a r a m e t e r s , n a m e l y t h e constant a, the particle m a s s m, and t h e velocity u r i,e and from t h e s e m w e c a n form n o dimensionless combination, and only o n e combination v^\(alm) having the dimensions of area, which m u s t therefore b e proportional to t h e cross-section. This p r o p e r t y of the cross-section greatly simplifies t h e structure of the collision integral, and it b e c o m e s possible to find e x a c t solutions of t h e linearized t r a n s p o r t equations for the thermal c o n d u c t i o n and viscosity p r o b l e m s . T h e s e solutions are found to b e just the first t e r m s in t h e e x p a n s i o n s (10.7) and (10.13).§

PROBLEMS! P R O B L E M 1. Find the thermal conductivity of a m o n a t o m i c g a s , retaining only the first term in the expansion (10.7). S O L U T I O N . With one term of the expansion, equations (10.9) reduce t o Αι = 15/4απ. T o calculate the integral (10.10) with / = s = 1, w e express v, vi, ν', νί in terms of the velocity of the centre of mass and the relative velocities of the t w o atoms: V = !(v + vi) =

k v'

+ vi),

Vre! 2= V-Vl, Vrél 2=l v' - Vi, 2

3v +vi 3 d pd pi

= =

2V +62vli, 3 3 m d Vd v .

rel

t i t must be emphasized that the variational principle as stated a b o v e is dependent on this, and is not valid w h e n the principle of detailed balancing has only its most general form (2.3). i T h e transport properties of this gas model were first discussed by J. C. Maxwell (1866). §A detailed account of the theory for this c a s e is given in § § 3 8 - 4 0 of L . Waldmann's article in Handbuch der Physik 12, 295, 1958. ||Formulae ( l ) - ( 6 ) are due to Chapman and Enskog.

Approximate

§10

Solution

of the Transport

Equation

35

A simple calculation gives A(vSÎ/2) = Δ ( β ΐ Λ ) = β [ ( ν . VielKe. " ( V . V )v ,].

r c rl c

Squaring and averaging over the directions of V, w e obtain

2

2

2 2

2

50 [»?e. " (Vre. . V ^ l V *

= Ιβ Ό*η ΐ V

SIT*

Θ.

Integration over 4TTV dV and over the directions of v i (the latter reducing to a multiplication by 4π) re gives finally

2

12

a

u

= \ β\βΙ2πγ

f* |

o

e x p ( - ϊ β υ ΐ ύ υ ΐ ι sin θ ^

dv

rcidQ\

(1)

the thermal conductivity is K=75/16fln.

(2)

P R O B L E M 2. T h e same as Problem 1, but for the viscosity. S O L U T I O N . W e find in a similar manner Bo = 5/boo, in the integral (10.15) with / = s = 0,

η = 5m/4boo.

2

Δ ( ϋ ϋ β - ϊυ δαβ) = kurel.aurel.p ~ V id. α V ' \, β).

α

rt

The square of this is

2 ïv%\ s i n θ.

3

Integration over d V and over the directions of v i s h o w s that boo= α π , s o that

re

(3)

τ)=4γπκ/15.

For a monatomic g a s , the specific heat c = 5/2; hence the ratio of the kinematic viscosity ν - η / N m to P the thermometric conductivity χ = k/Nc , called the Prandtl number, is, in this approximation,

p

W* = 2/3

(4)

whatever the law of interaction of the a t o m s . t P R O B L E M 3. In the same approximation, find the thermal conductivity and viscosity of a monatomic gas w h e n the atoms are regarded as hard elastic spheres with diameter d. 2 S O L U T I O N . T h e scattering of o n e sphere by another is equivalent to that of a point particle by an impenetrable sphere of radius d\ the cross-section is therefore da = (id) do. Calculation of the integral (1) gives the results^

It

75

K =

o.66

It

...

=

)

64V^Vm "7"Vm'

V ^ T ^

()

4

m0=

T1

8

)

^ -

t F o r a gas with the interaction law 17 = α / r , formulae ( 1 H 4 ) b e c o m e exact, and lead t o the values

,/2

κ = 3.04T(ma)~ ,

, / 2

η = 0.81 T ( m / a ) .

ΦΤο illustrate the rapidity with which s u c c e s s i v e approximations converge, it may b e mentioned that the inclusion of the s e c o n d and third terms in the expansions (10.7) and (10.13) multiplies the expressions (5) and (6) by (1 + 0.015 + 0.001) and (1 + 0.023 + 0.002) respectively.

36

Kinetic

Theory of

Gases

§ 1 1 . Diffusion of a light gas in a heavy gas T h e p h e n o m e n o n of diffusion in a mixture of t w o gases will b e studied here for some particular cases which allow a fairly extensive theoretical analysis. L e t N i and N2 d e n o t e t h e particle n u m b e r densities of the t w o c o m p o n e n t s of the mixture, and let the c o n c e n t r a t i o n of the mixture b e e x p r e s s e d b y c = N J N , w h e r e Ν = Ni + N2. T h e total n u m b e r density of particles is related to the p r e s s u r e and t e m p e r a t u r e by Ν = PIT. T h e gas p r e s s u r e is c o n s t a n t t h r o u g h o u t the v o l u m e ; let the concentration and t h e t e m p e r a t u r e vary along the χ-axis (by allowing a t e m p e r a t u r e variation, w e include thermal diffusion in the problem). L e t us consider diffusion in a mixture of gases of which one (the " h e a v y " gas) consists of molecules w h o s e m a s s is m u c h larger t h a n that of the particles of the other (the "light" gas). T h e latter will be a s s u m e d m o n a t o m i c . Since the m e a n thermal energy of translational motion is the s a m e for all particles (at a given t e m p e r a t u r e ) , t h e m e a n speed of the h e a v y molecules is m u c h less t h a n that of t h e light o n e s , and they can be approximately regarded as being at rest. W h e n a light and a h e a v y particle collide, the latter m a y be a s s u m e d to remain fixed, while the velocity of the light particle changes direction b u t remains unaltered in magnitude. In this section we shall t a k e t h e case w h e r e the c o n c e n t r a t i o n of the light gas (gas 1) in the mixture is small. T h e n collisions b e t w e e n its a t o m s are relatively rare and w e m a y suppose that the light particles collide only with the h e a v y o n e s . t In the general c a s e of an arbitrary gas mixture, a separate t r a n s p o r t equation has to be set u p for the distribution function of the particles of e a c h c o m p o n e n t , the right-hand side containing the sum of t h e collision integrals b e t w e e n the particles of each c o m p o n e n t and t h o s e of that and every other c o m p o n e n t . In the particular case u n d e r discussion, h o w e v e r , it is convenient to derive t h e simplified t r a n s p o r t equation ah initio. T h e required equation is to determine the distribution function for the particles of the light gas, which w e d e n o t e b y / ( p , x). With the a s s u m p t i o n s m a d e , collisions b e t w e e n light and h e a v y particles do not affect the distribution of the latter, and in the diffusion problem this distribution can be t a k e n as given. L e t 0 b e the angle b e t w e e n the direction of the m o m e n t u m p = m\\ of a light particle and the χ-axis. It is evident from the s y m m e t r y of the conditions of the problem that the distribution function will d e p e n d only on 0 (and on the variables ρ and x). L e t da = F ( p , a) do' d e n o t e the cross-section for collisions in which a light particle with m o m e n t u m ρ acquires a m o m e n t u m p ' = m v ' directed into the solidangle element do'; a is the angle b e t w e e n t h e v e c t o r s ρ and p ' (whose magnitudes are equal). T h e probability per unit p a t h length that the particle u n d e r g o e s such a collision is N2da, w h e r e N2 is the n u m b e r density of h e a v y particles; the p r o b ability per unit time is found by multiplying by the speed of the particle: N2vda. L e t us consider particles in a given unit of volume having m o m e n t a in a given 2 the solid-angle element do. T h e range dp of magnitudes and3 directed into n u m b e r of such particles is / d p = / ( ρ , 0, x)p dp do. Of t h e s e ,

2

/ ( ρ , 0, x)p

dp do . N2vF(p,

a) do'

t T h e kinetic theory of this model w a s first d e v e l o p e d by H. A. Lorentz (1905).

§ 11

Diffusion

of a Light Gas in a Heavy

Gas

37

particles per unit time acquire by collisions a m o m e n t u m p' directed into do'. T h u s the total n u m b e r of particles w h o s e m o m e n t u m changes direction is

3 d p j N2vf(p,

e,x)F(p9a)do'.

3

2

C o n v e r s e l y , of the particles in d p ' = p' dp'

do',

2 f(p',

θ', x)p'

dp' do'. N2v'F(p',

a) do

3 acquire a velocity directed into do. Since p' = p, the total n u m b e r of particles that acquire a velocity in d p as a result of collisions is 3 dp j

N2vf(p,e',x)F(p,a)do'.

3 T h u s the change in the n u m b e r of particles in d p is the difference

3 d p . N2v j F(p, a)[f(p,

θ', χ ) - f(p, Θ, x)]do'.

This must equal the total time derivative

3

3 d p(dfldt)

3

= d p v . V/ = d p(dfldx)v

cos θ.

Equating the t w o expressions gives the required transport equation vcosddfldx

= N2v j F(p,a)[f(p,e',x)-f(p,d,x)]do'^C(f).

(11.1)

T h e right-hand side is zero for any function / that does not depend on the direction of p, and not only for the Maxwellian function / 0 as in the case of the Boltzmann equation. This is b e c a u s e of the a s s u m p t i o n that the magnitude of the m o m e n t u m is u n c h a n g e d in the scattering of light particles by h e a v y o n e s : such collisions evidently leave steady any energy distribution of light particles. In reality, equation (11.1) c o r r e s p o n d s only to the zero-order approximation with respect to the small quantity m\lm2, and energy relaxation o c c u r s in the next approximation. If the concentration and t e m p e r a t u r e gradients are not too large (these quantities varying only slightly over distances of the order of the m e a n free path), / m a y be sought as the sum / = / ο ( ρ , χ ) + δ / ( ρ , θ, χ ) , where 8f is a small correction to the local-equilibrium distribution function / 0 and is linear in the gradients of c and T. In turn, we seek 8f in the form

8f = cos θ. g(p, JC), PK 10 - D

(11.2)

38

Kinetic

Theory of

Gases

w h e r e g is a function of ρ and χ only. In substituting in (11.1), it is sufficient to retain the / 0 term o n the left-hand side; in the collision integral, the / 0 term disappears: J F ( p , a ) ( c o s 0' - cos 0) d o ' ;

C(J) = gN2v

the function g, which is i n d e p e n d e n t of the angles, has b e e n t a k e n outside t h e integral. This integral m a y b e simplified as follows. W e take the direction of the m o m e n t u m ρ as the polar axis for the m e a s u r e m e n t of angles. L e t φ and φ' b e t h e azimuths of the χ-axis and the m o m e n t u m p ' relative to this polar axis. T h e n cos 0' = cos 0 cos α -I- sin 0 sin a cos(

C(f) = - N2at(p)vg

(11.3)

where at(p)

= 2π

j F(p,

a ) ( l - cos a) sin a da

= j ( 1 - cos a) da is called the transport cross-section F r o m (11.1), w e n o w find

(11.4)

for collisions.

·

(11 5)

«<ρ·*>-ϊά&

T h e diffusion flux i is, b y definition, the flux of molecules of o n e c o m p o n e n t of the mixture (in this c a s e , the light c o m p o n e n t ) . It is calculated from the distribution function as the integral

3 i= |/vd p,

(11.6)

or, since the vector i is along the χ - a x i s ,

3

2

3

i = j cos 0 . fv d p = j c o s 0 . gv d p ;

(11.7)

§11

Diffusion

of a Light

Gas in a Heavy

39

Gas

t h e fo term d i s a p p e a r s o n integration o v e r angles. Substitution of (11.5) gives

2 • = _ J L J L f foV C O S ' N2dxj at(p) =

p 3

θ

d

,

P

LA (hljn 3N2 dx J at '

This e x p r e s s i o n m a y be written

i

=

{ N i {) }v l ( T

- m 2- k

<

>

w h e r e t h e averaging is o v e r the Maxwellian distribution. L a s t l y , w e u s e the c o n c e n t r a t i o n c = NJN « NJN2 (since b y h y p o t h e s i s N2 > N O , and replace N 2 a p p r o x i m a t e l y b y Ν = P / T . T h e p r e s s u r e being c o n s t a n t , we find the result

i»-JT£{f<«/*>}

-^,>g-JcT^[l<^,>]f.

(11.8)

This is to b e c o m p a r e d with t h e phenomenological e x p r e s s i o n for t h e diffusion flux, (11.9)

i = -ND(vc+^VT^j,

w h i c h defines t h e diffusion coefficient D and the thermal diffusion ratio k T; the p r o d u c t DT = DkT is the thermal diffusion coefficient (see F M , §59).t T h u s w e find ϋ = (Τ/3Ρ)<ϋ/σ ί),

(11.10)

kT= c T ^ l o g ^ .

(11.11)

In diffusion equilibrium in a non-uniformly heated g a s , a c o n c e n t r a t i o n distribution is set u p in which the diffusion flux i = 0. E q u a t i n g to a c o n s t a n t the e x p r e s s i o n in the b r a c e s in (11.8), w e obtain Γ

c = constant X 7 - 1 — r .

(11.12)

1 / the 2 cross-section a is i n d e p e n d e n t of the velocity, and noting that A s s u m i n g that t ( u ) ~ ( T / m i ) , w e find t h a t , in diffusion equilibrium of a mixture with a low t T h e p h e n o m e n o n of thermal diffusion w a s predicted by E n s k o g (1911) for precisely this model of a gas mixture.

40

Kinetic

Theory of

Gases

concentration of the light gas, that c o n c e n t r a t i o n is proportional to V T , i.e. the light gas is c o n c e n t r a t e d in the regions w h e r e the t e m p e r a t u r e is high. T h e diffusion coefficient is, in order of m a g n i t u d e , (11.13)

D~vU

w h e r e ν is t h e m e a n thermal speed of the light-gas molecules and ί ~ l / Ν σ t h e m e a n free path. T h e r e is a well-known elementary derivation of this formula. T h e n u m b e r of molecules of gas 1 passing across unit area perpendicular to the χ-axis from left to right per unit time is equal in order of magnitude to the p r o d u c t Ν{ϋ, w h e r e the density Ni m u s t b e t a k e n at a distance I to the left of the area, i.e. at t h e points from which the molecules r e a c h that area without undergoing collisions. W e similarly find the n u m b e r of molecules crossing the same area from right to left, and the difference b e t w e e n the t w o n u m b e r s gives the diffusion flux: i ~ Nx(x - 1)ϋ - N , ( x 4- 1)ϋ ~ - W

dNJdx,

which gives (11.13).t

§12. Diffusion of a heavy gas in a light gas L e t us n o w consider the opposite limiting c a s e , w h e r e the concentration of the h e a v y gas in the mixture is small. In this c a s e , the diffusion coefficient m a y be calculated indirectly without using the t r a n s p o r t equation, by finding the mobility of the heavy-gas particles, regarding this gas as being in an external field. T h e mobility b is related to the diffusion coefficient of the s a m e particles b y the familiar Einstein's relation D = bT\

(12.1)

see F M , §60. T h e mobility is, b y definition, the proportionality coefficient b e t w e e n the m e a n velocity V acquired b y a gas particle in the external field, and the force f exerted o n the particle b y the field: V = bf.

(12.2)

T h e velocity V is determined from the condition that the force f balances the resistance fr exerted on the moving h e a v y particle by the light particles; collisions b e t w e e n h e a v y particles m a y be neglected, b e c a u s e there are relatively few of tDiffusion, thermal conduction and viscosity are brought about by the same mechanism, namely direct molecular transport. The thermal conduction may be regarded as a "diffusion of energy" and the viscosity as a "diffusion of momentum". W e may therefore assert that the diffusion coefficient D , the thermometric conductivity χ - k/Nc and the kinematic viscosity ν = ηΙΝτη are of the same order of p magnitude; this leads to the formulae (7.10) for the thermal conductivity and (8.11) for the viscosity.

§12

Diffusion

of a Heavy

Gas in a Light

Gas

41

t h e m . T h e distribution function of the light particles is Maxwellian:

/

o

e

x

- ( 2 ^ T F

p

r ^ r j '

w h e r e mx is t h e m a s s of a light particle. L e t u s consider o n e particular h e a v y particle with velocity V, and take coordinates moving with that particle; let ν d e n o t e t h e velocities of t h e light particles in these c o o r d i n a t e s . T h e distribution function of t h e light particles in t h e s e coordinates is /o(v + V ) ; cf. (6.9). Assuming that V is small, w e c a n write /o(v + V) * /o(i>)0 " m i v . V / T ) .

(12.3)

T h e required resistance f rc a n b e calculated as t h e total m o m e n t u m transferred to the h e a v y particle b y light particles colliding with it p e r unit time. T h e frame of reference is u n c h a n g e d in a collision. T h e light particle carries m o m e n t u m mxv\ after t h e collision, in which its m o m e n t u m is t u r n e d through an angle a, it carries a w a y an average m o m e n t u m m i v c o s a. T h e average m o m e n t u m transferred t o the h e a v y particle in such a collision is therefore miv(l - c o s a ) . Multiplying this b y the flux of light particles with velocity ν a n d b y t h e cross-section da for such a collision, and integrating, w e obtain t h e total m o m e n t u m transferred t o t h e h e a v y particle:

3 f r = mi j /o(v + \)v\a

d p,

t

again with the notation (11.4). W h e n / 0( v + V) is substituted in the form (12.3), the first term gives z e r o in t h e integration o v e r directions of v, leaving

3 fr = -

j fo(v)\

. ν Yva

d p,

t

or, averaging o v e r directions of v,

33 î

r

= - j ^ \

=

- N i j ^ \ ( a

jfo(v)a v d p

t

tv

\

w h e r e t h e angle b r a c k e t s again d e n o t e averaging over t h e ordinary Maxwellian distribution. Lastly, since in this case Ni > N 2, w e write Νι « Ν = PIT, so that

Equating to z e r o t h e sum of t h e resistance tr a n d t h e external force f , w e find from

42

Kinetic

Theory of

Gases

(12.2) the mobility b, and t h e n c e the diffusion coefficient

3 2

3

(12.4)

D = bT = 3T lmxP(atv ).

T o calculate the thermal diffusion in this c a s e , it would be n e c e s s a r y to k n o w t h e distribution function of t h e light-gas particles in the p r e s e n c e of a t e m p e r a t u r e gradient. T h e thermal diffusion coefficient therefore c a n n o t b e calculated in a general form h e r e . In order of magnitude D ~ ϋ/Νσ, w h e r e ν ~ V ( T / m O is, as in (11.13), the m e a n thermal speed of the light-gas molecules. T h u s t h e o r d e r of magnitude of t h e diffusion coefficient is the s a m e in e a c h c a s e :

112

(12.5)

D ~ T^laPm, .

PROBLEM Determine the diffusion coefficient in a mixture of t w o gases (one light and o n e h e a v y ) , regarding their 2 particles as hard elastic spheres with diameters d\ and di. 2 S O L U T I O N . The collision cross-section da = ττ(άι + di) do/167r, and s o the transport cross-section a = 47r(di + di) , equal in this case to the total cross-section a. The diffusion coefficient is

t

3/2

2

D = AT /(di + d ) Pm,

2

1/2 ,

where m i is the mass of a light particle and A is a numerical factor. W h e n the concentration of the light gas is small, a calculation from (11.10) gives

32/

A = 5(2/TT)

= 0.68.

W h e n the concentration of the h e a v y gas is small, (12.4) gives

A = 3/2V(2ir) = 0.6. N o t e that the values of A in the t w o limiting c a s e s are almost equal.

§ 13. T r a n s p o r t phenomena in a gas in an external field T h e rotational degrees of freedom of molecules p r o v i d e t h e m e c h a n i s m w h e r e b y an external magnetic or electric field c a n affect t r a n s p o r t p h e n o m e n a in a g a s . t T h e effect is of t h e s a m e n a t u r e in the m a g n e t i c and electric c a s e s ; w e shall first discuss a gas in a magnetic field. A rotating molecule h a s in general a magnetic m o m e n t , w h o s e a v e r a g e value (in the q u a n t u m - m e c h a n i c a l sense) will b e d e n o t e d b y μ. T h e magnetic field will b e a s s u m e d so w e a k that μΒ is small in c o m p a r i s o n with t h e intervals in t h e fine structure of molecular levels.% W e can t h e n neglect t h e influence of t h e field o n t h e t T h i s mechanism w a s pointed out by Y u . M. Kagan and L. A. Maksimov (1961), w h o also derived the results given in this section. $In macroscopic electrodynamics, the mean value (over physically infinitesimal v o l u m e s ) of the magnetic field is called the magnetic induction and denoted b y B. W h e n the density of the medium is l o w , as in a gas, the magnetization is negligible, and the vector Β then coincides with the macroscopic field H.

§13

Transport

Phenomena

in a Gas in an External

Field

43

state of t h e molecule, so that t h e magnetic m o m e n t is calculated for t h e u n p e r t u r b e d state. F o r fairly high t e m p e r a t u r e s , t h e c a s e w e shall consider, μ Β is small in c o m p a r i s o n with Τ also; this enables u s t o neglect the influence of the field o n the equilibrium distribution function of the gas molecules. T h e magnetic m o m e n t is parallel t o t h e rotational angular m o m e n t u m M of t h e molecule, and m a y b e written μ = γΜ.

(13.1)

Classical rotation of t h e molecule c o r r e s p o n d s t o large rotational q u a n t u m n u m b e r s ; w e c a n t h e n neglect in M the difference b e t w e e n the total angular m o m e n t u m (including spin) a n d t h e rotational angular m o m e n t u m . T h e value of t h e c o n s t a n t coefficient y d e p e n d s o n t h e n a t u r e of t h e molecule and t h e n a t u r e of its magnetic m o m e n t . F o r e x a m p l e , with a diatomic molecule having n o n - z e r o spin S, γ « ( 2 σ / Μ ) μ Β,

(13.2)

w h e r e μΒ is t h e B o h r m a g n e t o n , a n d t h e n u m b e r σ = J - Κ is t h e difference b e t w e e n t h e q u a n t u m n u m b e r s J of t h e total angular m o m e n t u m a n d Κ of t h e rotational angular m o m e n t u m ( σ t a k e s t h e values S, S - l , . . . , - S ) ; in t h e d e n o m i n a t o r , t h e difference b e t w e e n J a n d Κ is n o t significant: M « ftJ « fiK. I n (13.2) it is a s s u m e d that t h e spin-axis interaction in t h e molecule is small in c o m p a r i s o n with the intervals in the rotational structure of the levels ( H u n d ' s case b).t In a magnetic field B , the molecule is subjected t o a t o r q u e μ x B. T h e v e c t o r M is t h e n n o longer c o n s t a n t during t h e " f r e e " motion of t h e molecule, b u t varies according t o dMldt

= μΧΒ = -γΒχΜ;

(13.3)

t h e v e c t o r M p r e c e s s e s a b o u t the direction of the field with angular velocity - γ Β . T h e left-hand side of t h e t r a n s p o r t e q u a t i o n t h u s h a s a n a d d e d t e r m ( d / / d M ) . M , and the equation b e c o m e s g + . |v£ +

| L =x CB. ( y . M

/

)

.4)

T h e variables Γ o n which t h e distribution function d e p e n d s m u s t also include t h e discrete variable σ, which determines the value of the magnetic m o m e n t , if there is such a variable, a s in (13.2). In p r o b l e m s of t h e r m a l c o n d u c t i o n a n d viscosity, w e again take a distribution close t o t h e equilibrium o n e , and e x p r e s s it as / = /o(l + * / T ) .

(13.5)

tFormula (13.2) follows from the e x a c t formula for c a s e f>, derived in QM, § 113, Problem 3 , o n taking the limit of large J and Κ with a fixed difference J-K. T h e contribution of the orbital angular momentum Λ is then negligible, being of the next order of smallness in 1 / / .

44

Kinetic

Theory of

Gases

W e shall first show that a term in d / 0/ d M d o e s not o c c u r in the t r a n s p o r t equation. Since / 0 d e p e n d s only on the energy β(Γ) of the molecule, and de/θ M is equal to the angular velocity 11, w e h a v e γ Μ x Β . dfoldM = yM x Β . ft dfolde.

(13.6)

F o r molecules of the rotator and spherical-top t y p e s , M and ft are parallel, and the expression (13.6) is zero identically. In other c a s e s , it b e c o m e s zero after averaging over the rapidly varying p h a s e s , the necessity for which has b e e n explained in § 1. W h e n molecules of the symmetrical-top or asymmetrical-top type r o t a t e , there is a rapid variation b o t h of the direction of the axes of the molecule itself and of that of its angular velocity ft. After the averaging mentioned, ft can retain only the c o m p o n e n t ftM along the c o n s t a n t vector M , and for this c o m p o n e n t the p r o d u c t Μ . Β x ftM = 0. T h e remaining t e r m s in the t r a n s p o r t equation are transformed in the same w a y as in §7 or § 8 . F o r instance, in the thermal c o n d u c t i o n problem w e find the equation

€

( C TF p ) ~

V . V T = - 7 M X B ^ + Ι(χ).

(13.7)

T h e solution of this equation is again to b e sought in the form χ = g . VT, b u t there are now three vectors ν , M , B, not t w o , available to c o n s t r u c t the vector function g(T). T h e external field c r e a t e s a distinctive direction in the gas. T h e p r o c e s s of thermal conduction therefore b e c o m e s anisotropic, and the scalar coefficient κ has , which determines the heat flux to be replaced b y a thermal conductivity tensor καβ by qa = - καβθΤΙοΧβ.

(13.8)

T h e tensor καβ is calculated from the distribution function as the integral καβ = - γ j foevagfi dT;

(13.9)

cf. (7.5). T h e general form of a tensor of rank t w o depending on the vector Β is Καβ = κδαβ + K\babp

+ κ 2β α & β γγ,

(13.10)

w h e r e b = Β / Β , β α γ/ 3is the antisymmetric unit tensor, and κ, κί9 κ2 are scalars depending on the field strength B. T h e tensor (13.10) obviously has the p r o p e r t y t καβ (Β)

= κβα (-Β).

(13.11)

tThis property expresses the symmetry of the kinetic coefficients in the presence of a magnetic field. In the present c a s e , it necessarily follows from the existence of only the o n e vector b from which the tensor κ β can be constructed.

α

§13

Transport

Phenomena

in a Gas in an External

Field

45

T h e e x p r e s s i o n (13.10) c o r r e s p o n d s to the heat flux q = - κ V T - Kib(b . V T ) - K 2V T Χ b .

(13.12)

T h e last term is w h a t is called an odd effect, changing sign with the field. T h e integral term Ι(χ) on the right of (13.7) is given by (6.5). T h e integrand contains the function / 0, which is proportional to the gas density N . Separating this factor and dividing b o t h sides of the equation by it, w e find that Ν a p p e a r s only in the c o m b i n a t i o n s B / N with the field and V T / N with the t e m p e r a t u r e gradient. It is therefore clear that the function / o * = / o g . V T will d e p e n d o n t h e p a r a m e t e r s Ν and Β only t h r o u g h the ratio B / N ; the integrals (13.9) will also d e p e n d only on this quantity, and therefore so will the coefficients κ, K U K2in (13.12). T h e density Ν is proportional (at a given t e m p e r a t u r e ) to the gas p r e s s u r e P . T h u s the thermal conductivity of a gas in a magnetic field d e p e n d s on the field and the p r e s s u r e only through the ratio B / P . t W h e n Β i n c r e a s e s , the first term o n the right of (13.7) i n c r e a s e s , b u t the second term is u n c h a n g e d . It is therefore clear that as Β -> <» t h e solution of the equation m u s t b e a function depending only on the direction (not the magnitude) of the field, and this function m u s t m a k e identically z e r o t h e t e r m M x Β . d*/dM in the e q u a t i o n ; accordingly, the coefficients κ, κ\, κ2 t e n d to c o n s t a n t limits i n d e p e n d e n t of B , as Β ->oo. T h e t r e a t m e n t of the viscosity of a gas in a magnetic field is similar. T h e c o r r e s p o n d i n g t r a n s p o r t e q u a t i o n is (mi>et>, -

δ α ) βναβ = I(χ) - γ Μ x Β ·

(13.13)

cf. (6.19). T h e solution is to be sought in the form χ = gapV^. I n s t e a d of the t w o viscosity coefficients η and £, w e m u s t n o w u s e a t e n s o r η α δβ ofγ r a n k four w h i c h d e t e r m i n e s the viscous stress t e n s o r =

σ'αβ

η « β γ δ ν

γ;

b y definition, the t e n s o r Ύ ) α δ β is sγy m m e t r i c in the pairs of suffixes α, With the k n o w n function χ , its c o m p o n e n t s are calculated as

τ?αβγδ =

~

j

(13.14)

δ β

y8 dT.

mv Vpfog

a

and γ , δ.

(13.15)

T h e viscosity t e n s o r t h u s found will necessarily satisfy the condition η α β( Βγ) δ= η γ δ( -αΒ )β,

(13.16)

which e x p r e s s e s the s y m m e t r y of the kinetic coefficients. With the v e c t o r b = B/B (and the unit t e n s o r s 8αβ and eaPy )9 w e c a n c o n s t r u c t the t T h e change in the thermal conductivity of a gas in a magnetic field is called the Senftleben

effect.

Kinetic

46

Theory

of

Gases

following i n d e p e n d e n t tensor c o m b i n a t i o n s having t h e s y m m e t r y properties of

(1) δαγδβδ + δ δ β ,

α δγ

(2) δ βδ δ» α γ (3) ô ybpb + 8pyb bs + 8 sb b

a

8

a py +

a

(4) 8 b bs + δ ί>αί>β> aPy γδ (5) b bpbyb ,

a

8

(6) ί>αγδβδ + ^βγδαδ + ^αδδβγ + (7) baybpbs + bpyb bs

a

+ b bpb

ab y

8 b by

p8a 9

(13.17)

b^ay ,

+

bp b b ,

8ay

w h e r e ί> αβ = - ί? βα = £αβγί>γ. In all t h e s e c o m b i n a t i o n s e x c e p t (4), t h e p r o p e r t y (13.16) follows automatically from t h e s y m m e t r y with r e s p e c t t o t h e pairs of suffixes α, β and γ , δ ; in (4), t h e t w o t e r m s a r e c o m b i n e d in o r d e r t o satisfy t h e condition (13.16).t In a c c o r d a n c e with t h e n u m b e r of t e n s o r s (13.17), a gas in a magnetic field in general h a s seven i n d e p e n d e n t viscosity coefficients. T h e s e m a y b e defined a s t h e coefficients in t h e following e x p r e s s i o n for t h e viscous stress t e n s o r : <

β = 2η(ναβ - ϊδαβ div V) + ζδαβ div V + η ι ( 2 ν αβ - δαβ div V + δ ανβγί> δ γ{>δ Ι

- 2V ybyb

P

a

+ bb

ap

+ 2 η ( V ybybp

2 a

div V +

a p y8y 8

a —

a

a yp

bbV bb)

+ Vpybyb

+ T ?3( Vaybfiy + Vpyb y

2V yb b

-

2b bfiV byb )

a

~ V b bb

y8ayp 8

+ 2η (V b bpb

y8 8

+ ίι(δ

bp b b ) 4 y 8a y 8 + Vy8 y a8

V bfrb b )

y8 a 8

bb αβ Vy8 y8

a p div V);

+ bb

(13.18)

ναβ is defined in (6.12). This is so c o n s t r u c t e d t h a t 17, TJI, . . . , 174 are coefficients of t e n s o r s which give z e r o o n contraction with r e s p e c t to t h e suffixes α, β ; ζ a n d ζχ are coefficients of t e n s o r s with n o n - z e r o t r a c e , a n d m a y b e called second viscosity coefficients. N o t e that they contain n o t only t h e scalar div V b u t also ν γδ ί> γί>δ. T h e first t w o t e r m s in (13.18) c o r r e s p o n d t o t h e usual e x p r e s s i o n for t h e stress t e n s o r , so that η and ζ a r e t h e ordinary viscosity coefficients. T h e t e n s o r s καβ a n d η αβ γδ m u s t b e t r u e t e n s o r s , since t h e y satisfy t h e condition of s y m m e t r y u n d e r inversion. T h e a b a n d o n m e n t of this condition (for a g a s of stereoisomeric material) would therefore n o t lead t o t h e p r e s e n c e of a n y n e w iV) terms. , ( ) Τ Such a b a n d o n m e n t would, h o w e v e r , bring about n e w effects, with a h e a t flux q d u e to t h e velocity gradients a n d viscous stresses σ due to the temperature gradient. T h e s e cross-effects a r e described b y t h e formulae

( )ν

<Ι γ = ^ , αβ ν αβ ,

Τ)

σϊβ

= -ααβ ,Ύ θΤΙοχΎ9

(13.19)

t i t is unnecessary to write d o w n combinations of terms with t w o factors b : since the product of t w o afi tensors β , such combinations would reduce to those already αβΊreduces to products of tensors δαβ included in (13.17).

§13

Transport

Phenomena

in a Gas in an External

Field

47

w h e r e c % 3a and / α α, β γ are t e n s o r s of r a n k t h r e e s y m m e t r i c in the pair of suffixes separated b y the c o m m a . W i t h 2xa and Xa c h o s e n as in § 9 , the kinetic coefficients yab and yba are Τ ο γ, αβ and Τ α α , γβ. O n s a g e r ' s principle t h u s s h o w s that in the p r e s e n c e of a magnetic field w e m u s t h a v e Τ α α, β γ( Β ) = ο γ >( -αΒ )β.

(13.20)

T h e general form of such t e n s o r s is ααβ ,Ύ = axbabBby

+ αφΎδαβ + α 3(ί> αδ> γ + ί> βδ α)γ+ a4(b^

+ bβyba).

(13.21)

AU the t e r m s here are p s e u d o t e n s o r s , and so the relations (13.19) with t h e s e coefficients are n o t invariant u n d e r inversion. L e t us n o w briefly consider t r a n s p o r t p h e n o m e n a in a gas in an electric field. W e t a k e a gas consisting of polar molecules (i.e. having a dipole m o m e n t d) of t h e symmetrical-top t y p e . In an electric field, a polar molecule is acted o n b y a t o r q u e d x E , so t h a t t h e t r a n s p o r t equation contains a term M.d//dM = dxE.d//dM. T h e direction of d is along the axis of the molecule and is unrelated to that of the rotational anglular m o m e n t u m M . H o w e v e r , as a result of averaging with respect to t h e rapid precession of t h e top's axis a b o u t the direction of the constant vector M , there remains in the above t e r m only the c o m p o n e n t d along M , and it becomes yMxEJ//aM,

(13.22)

w h e r e γ = ad/M ; t h e variable σ (the cosine of t h e angle b e t w e e n d and M ) n o w t a k e s a c o n t i n u o u s series of values from - 1 to + 1 . T h e expression (13.22) differs from t h e c o r r e s p o n d i n g term in the magnetic case only in that Β is replaced b y E. T h u s all the a b o v e t r a n s p o r t equations and the conclusions d r a w n from t h e m remain valid.t T h e r e is, h o w e v e r , a difference arising from the fact that the electric field Ε is a true vector, not a p s e u d o v e c t o r , and is unaffected b y time reversal. F o r this r e a s o n , O n s a g e r ' s principle for t h e t h e r m a l conductivity and viscosity t e n s o r s is here expressed by κ α ( βΕ ) = κ β ( αΕ ) ,

η α β( Εγ) δ= η γ δ( Εα) ,β

(13.23)

instead of (13.11) and (13.16). Correspondingly, κ2 = 0 and η 3 = η4 = 0 in (13.10) and (13.18) (where n o w b = Ε/Ε).Φ O n t h e other h a n d , cross-effects are possible not only in a stereoisomeric g a s , for w h i c h (13.21) is fully valid, b u t also in a gas of nonstereoisomeric molecules: the e x p r e s s i o n (13.21) with a 4 = 0 is t h e n a true tensor. t D i a t o m i c molecules rotate in a plane perpendicular to M; h e n c e σ = 0 for a diatomic polar molecule. In such a c a s e the effect of the electric field o n the motion of the molecules appears in the transport equation only in the quadratic approximation with respect to the field. $In a gas of non-stereoisomeric m o l e c u l e s , the a b s e n c e of the terms in κ , τ)3, Τ\Α in an electric field is 2 also required by the condition of invariance under inversion.

Kinetic

48

Theory of Gases

§ 14. Phenomena in slightly rarefied gases T h e dynamical equations of motion of a gas, including thermal conduction a n d internal friction, contain t h e heat flux q' (the dissipative part of t h e energy flux q) ). and t h e viscous stress tensor σ'αβ(the dissipative part of t h e m o m e n t u m flux ΤΙαβ T h e s e equations acquire real meaning w h e n q' a n d σαβ have b e e n expressed in terms of t h e t e m p e r a t u r e a n d velocity gradients in t h e g a s . H o w e v e r , t h e usual expressions linear in these gradients are just the first t e r m s of expansions in p o w e r s of t h e small ratio IIL of t h e m e a n free path t o t h e characteristic dimensions of t h e problem (called t h e Knudsen number K ) . If this ratio is n o t very small, it m a y b e reasonable t o m a k e corrections b a s e d o n the terms of the next order of smallness in t/L. Such corrections arise b o t h in t h e equations of motion themselves a n d in t h e b o u n d a r y conditions o n these equations at t h e surfaces of bodies in t h e gas flow. T h e successive terms in t h e expansions of t h e fluxes q' a n d σ'αβa r e e x p r e s s e d by m e a n s of t h e spatial derivatives of t e m p e r a t u r e , pressure a n d velocity, of various orders a n d raised t o various p o w e r s . T h e s e t e r m s m u s t in principle b e calculated b y going t o further approximations in t h e solution of t h e transport equation. T h e zero-order approximation c o r r e s p o n d s to t h e local-equilibrium dis(1) T h e first-order tribution function / 0 a n d t h e dynamical equations of a n ideal fluid. approximation c o r r e s p o n d s t o t h e distribution function / = / 0( 1 -4- χ ΙΤ) considered in § § 6 - 8 , a n d t h e N a v i e r - S t o k e s equations of fluid d y n a m i c s , a n d t h e equation of thermal conduction. I n the second-order approximation, t h e distribution function is to b e sought in t h e form

( )1 / =/ο[ΐ+γ*

< 2 )

+γΧ

]

(14.D

{2) and t h e transport equation is t o b e linearized with respect t o t h e second-order correction χ . T h e resulting equation is

/oU -V +V

fo +

fodt

Τ

(

~ψ\

w'foax V~X
dr, dP dT[ = ^ Ι ( Λ

(14.2)

w h e r e ί is again the linear integral operator (6.5). T h e symbol d0ldt signifies that the il) time derivatives of m a c r o s c o p i c quantities which appear as a result of differentiating fox IT are t o b e expressed in terms of spatial derivatives b y m e a n s of t h e zero-order equations of fluid d y n a m i c s (Euler's equations). T h e symbol di/di signifies that t h e time derivatives a r e t o b e eliminated b y m e a n s of t h e first-order terms in t h e N a v i e r - S t o k e s equations a n d t h e equation of thermal conduction (the terms containing η , ζ a n d κ ) . W e shall n o t write o u t all t h e n u m e r o u s t e r m s in q' a n d σ'αβthat arise in t h e second approximation a n d a r e called Burnett terms ( D . Burnett, 1935). I n m a n y cases these terms m a k e a contribution t o t h e solution that is small in c o m p a r i s o n with t h e corrections in t h e b o u n d a r y conditions, t o b e discussed below. I n such

§14

Phenomena

in Slightly

Rarefied

Gases

49

c a s e s , the inclusion of c o r r e c t i o n s in t h e e q u a t i o n s t h e m s e l v e s would b e an unjustifiable exaggeration of t h e attainable a c c u r a c y . W e shall merely consider s o m e typical c o r r e c t i o n t e r m s and m a k e estimates of t h e m for m o t i o n s of various kinds. First of all, let u s n o t e that the small p a r a m e t e r Κ = IIL is related in a certain w a y to t w o p a r a m e t e r s which describe the fluid motion, namely the R e y n o l d s n u m b e r R and t h e M a c h n u m b e r M. T h e R e y n o l d s n u m b e r is defined as R ~ VL/v, w h e r e V is the characteristic scale of velocity of the flow and ν t h e kinematic viscosity; the M a c h n u m b e r M ~ V/α, w h e r e u is t h e speed of sound. In a gas, the order of magnitude of the speed of s o u n d is t h e s a m e as the m e a n t h e r m a l speed ν of the molecules, and t h e kinematic viscosity ν ~ Iv. H e n c e R ~ VL/Ιϋ, M ~ V/ϋ and the Knudsen number K-M/R.

(14.3)

H e n c e it is clear t h a t the condition Κ <^ 1 for the flow to b e g o v e r n e d b y the linear equations of fluid d y n a m i c s imposes a limitation on the relative o r d e r of magnitude of R and M. L e t u s first consider " s l o w " m o t i o n s , with R<1,

M«U.

(14.4)

L e t us take a n y of the B u r n e t t t e r m s in the viscous stress t e n s o r containing the p r o d u c t of t w o first derivatives of the velocity, for instance

1 54 < · >

2 2 2 is an order-of-magnitude estimate. t h e coefficient p i (where ρ is t h( e)2 gas 2density) This term gives a contribution σ ~ p i V / L to σ'αβ . T h e o r d e r of m a g n i t u d e of the principal ( N a v i e r - S t o k e s ) t e r m s in the viscous stresses is il) a ~T)dVldx~plvVIL, and t h e ratio

i2)il)

2 2

a la ~lVILd~l RIL . Since R ^ 1, w e see t h a t the 2t e r m s (14.5) give a correction to the viscous stresses w h o s e relative o r d e r is ^ ( ί / L ) ; t h e c o r r e c t i o n in t h e b o u n d a r y conditions (see below) gives m u c h larger c o r r e c t i o n s ( ~ t / L ) to t h e m o t i o n . T h e c o r r e c t i o n s are e v e n smaller that arise from t e r m s of t h e f o r m t

2 pi dT dT ΈΨΈ^Ί^

6) ·

2 2 this if the t e m p e r a t u r e gradients are t h o s e which result from the m o t i o n itself; follows b e c a u s e t h e characteristic t e m p e r a t u r e differences Δ Τ ~ TV lu . If, t T e r m s of this kind in the v i s c o u s stresses were first discussed by Maxwell (1879).

50

Kinetic

Theory of

Gases

h o w e v e r , t e m p e r a t u r e differences are imposed from outside (e.g. by heated bodies immersed in the gas), the B u r n e t t terms of the form (14.6) m a y cause a steady motion with characteristic velocities determined by the equilibrium equation d

β(σ (l)

/

3χβ «

,

β)dP_ * -οχα·

_(2)λ _

+

σ

A n estimate of the speed of this motion is

2 (14.7)

V-KATflLmdT

(M. N . K o g a n , V. S. Galkin and O. G. Fridlender 1970). In making the estimate, it must be r e m e m b e r e d that the Laplacian of the t e m p e r a t u r e can b e e x p r e s s e d , by m e a n s of the thermal conduction equation div ( K V T ) = 0, in t e r m s of the square of the t e m{p e r a t u r e gradient, and that the motion is c a u s e d only by the non-potential part 8σ αβΙοΧβ of the force; the potential part is balanced by the p r e s s u r e . Similar considerations apply to the correction terms in the heat flux q'. It is impossible to construct a second-order correction term from the derivatives of the t e m p e r a t u r e alone; the first such correction term (after - K V T ) is c o n s t a n t x VAT (where Δ is the Laplacian operator), and thus is of t h e third order. T h e t e r m s which include velocity derivatives as well as t e m p e r a t u r e derivatives, such as

2 ( p / / m ) div V . V T ,

2 2 again give corrections of relative order l IL . L e t us n o w go on to " f a s t " motions, with R>1,

M^l.

(14.8)

In such c a s e s , the gas motion takes place in t w o regions: the main v o l u m e , w h e r e the viscous terms in the equations of motion are unimportant, and a thin b o u n d a r y layer, in which the gas velocity decreases rapidly. L e t us consider, for e x a m p l e , t h e flow of gas p a s t a flat plate, taking t h e direction of flow as the χ-axis. T h e thickness δ of the b o u n d a r y layer on the plate is

ll2

m

ô~(xvlV) ~(xWIV) , w h e r e χ is the distance from the leading edge; see F M , §39. T h e characteristic dimension for the variation of the velocity in the χ-direction is given by the coordinate χ itself, and that in the y-direction, perpendicular to the plate, is given by the thickness δ of the b o u n d a r y layer. H e r e , by the equation of continuity, V y ~ ν χδ/χ. T h e principal term in the N a v i e r - S t o k e s viscous stress tensor is a'xy ~ pv dVJdy

~

ρϋΐν/δ.

2

A m o n g the Burnett t e r m s in a'xy , h o w e v e r , there is n o n e containing (dVJdy) ; it is easily seen that the derivatives dVJdxp do not yield a tensor of r a n k t w o quadratic

§ 14

Phenomena

in t h e m w h o s e xy-component only b e t h o s e of the form

in Slightly

Rarefied

Gases

{

contains that square. T h e largest terms in σ $

51 can

22 pl\dVJdy)div\~pl V lxô.

( 2 )( )1

2

Their ratio to σ $ is σ / σ ~ lV/χϋ ~ (IIδ) , which is again of the s e c o n d order. W e shall n o w show that the correction t e r m s in the conditions at gas-solid b o u n d a r i e s yield effects of the first order in IIL. It follows that appreciable c o n s e q u e n c e s of t h e rarefaction of the gas o c c u r near solid surfaces. In non-rarefied g a s e s , t h e b o u n d a r y condition at the surface of a solid is that the t e m p e r a t u r e s of t h e gas and t h e solid are equal. In reality, h o w e v e r , this is an a p p r o x i m a t e condition, and applies only if t h e m e a n free p a t h m a y b e regarded as infinitesimal. W h e n the finite m e a n free p a t h at the surface of c o n t a c t b e t w e e n a solid and a non-uniformly heated gas is t a k e n into a c c o u n t , there is a difference of t e m p e r a t u r e s , which falls to z e r o , in general, only w h e n there is c o m p l e t e thermal equilibrium a n d t h e gas t e m p e r a t u r e is c o n s t a n t . t N e a r a solid surface (at distances from it that are small, b u t n o t t o o small), the t e m p e r a t u r e gradient of the gas m a y b e a s s u m e d c o n s t a n t , so that t h e t e m p e r a t u r e varies linearly with the distance. In the immediate neighbourhood of the wall, h o w e v e r , at distances —I, t h e t e m p e r a t u r e variation is in general m o r e c o m p l e x and its gradient is n o t c o n s t a n t . T h e c o n t i n u o u s c u r v e in Fig. 1 s h o w s t h e a p p r o x i m a t e form of t h e gas t e m p e r a t u r e near the surface. H o w e v e r , this t r u e form of t h e t e m p e r a t u r e in the vicinity of the wall, which relates to distances c o m p a r a b l e with the m e a n free p a t h , is n o t important w h e n considering the t e m p e r a t u r e distribution t h r o u g h o u t the gas. As regards the temp e r a t u r e distribution near a solid wall, w e are mainly c o n c e r n e d with only the straight p a r t of the c u r v e in Fig. 1, which e x t e n d s to distances large c o m p a r e d with

FIG.

1.

t i n referring to the temperature of a gas in regions w h o s e size is of the order of the mean free path, it is necessary, strictly speaking, to define what is meant by temperature. In the present case it will be defined in terms of the mean energy of the molecules at a given point in the gas, the function which determines the temperature from that mean energy being taken as the same as for large v o l u m e s of gas.

52

Kinetic

Theory of

Gases

the m e a n free path. T h e equation of this straight line is determined b y its slope and b y the intercept on the ordinate axis. W e are thus c o n c e r n e d not with the actual discontinuity of t e m p e r a t u r e at the wall, b u t with the discontinuity that results w h e n the t e m p e r a t u r e gradient is a s s u m e d c o n s t a n t n e a r the wall at all distances d o w n to z e r o , as s h o w n b y the b r o k e n line in Fig. 1. L e t δ Τ d e n o t e this extrapolated t e m p e r a t u r e discontinuity, defined as the gas t e m p e r a t u r e minus the wall t e m p e r a t u r e (the latter being arbitrarily t a k e n as zero in Fig. 1). W h e n the t e m p e r a t u r e gradient is z e r o , so is the discontinuity δΤ. H e n c e , for fairly small t e m p e r a t u r e gradients, 8T = gdTldn;

(14.9)

the derivative is t a k e n along the normal to the surface into the gas. T h e coefficient g m a y be called the temperature discontinuity coefficient If the gas t e m p e r a t u r e increases into the volume ( d T / d n > 0 ) , w e m u s t also h a v e δ Τ > 0 , and so the coefficient g is positive. Similar effects occur at the b o u n d a r y b e t w e e n a solid wall and a moving gas. Instead of " s t i c k i n g " completely to the surface, a rarefied gas maintains a small b u t finite velocity near it, and slips along the surface. As in (14.9), w e h a v e as the speed of slip t > o = i d V f/ a n ,

(14.10)

w h e r e Vt is the tangential c o m p o n e n t of the gas velocity n e a r the wall. Like g, the slip coefficient ξ is positive. T h e s a m e c o m m e n t s apply to vQ as w e r e m a d e regarding the t e m p e r a t u r e discontinuity δ Τ given b y (14.9). This speed is, strictly speaking, not the actual speed of the gas at the wall itself, b u t the speed extrapolated on the assumption of a c o n s t a n t gradient dVJdn in the layer of gas along the wall. T h e coefficients g and ξ h a v e the dimensions of length, and are of the same order of magnitude as the m e a n free p a t h : g-I,

ξ-I

(14.11)

T h e t e m p e r a t u r e discontinuity and the slip speed t h e m s e l v e s are c o n s e q u e n t l y quantities of the first order in i/L. T o calculate the coefficients g and ξ, it would b e n e c e s s a r y to solve the t r a n s p o r t equation for the distribution function of the gas molecules near the surface. This equation would h a v e to take a c c o u n t of collisions b e t w e e n the gas molecules and the wall, and it would therefore b e n e c e s s a r y to k n o w the law governing their scattering in such collisions. If the b r o k e n line in Fig. 1 is continued to intersect the abscissa axis, it m a k e s an intercept of length g. T h u s w e can say that the t e m p e r a t u r e distribution in the p r e s e n c e of a t e m p e r a t u r e discontinuity is the s a m e as if there w e r e no discontinuity but the wall w e r e m o v e d b a c k a distance g. T h e same applies to the slip, with the wall m o v e d b a c k a distance ξ. Of c o u r s e , with these changes only the first-order terms in g or ξ should b e retained in the solutions of problems in fluid m e c h a n i c s . Since taking a c c o u n t of the t e m p e r a t u r e or velocity discontinuities is

§14

Phenomena

in Slightly

Rarefied

Gases

53

equivalent to moving the b o u n d a r i e s by distances of the order of i, the resulting corrections in the solutions are of the order of Idldx ~ IIL, i.e. of the first order in IIL. As well as the a b o v e corrections to the b o u n d a r y conditions, there are other effects of the same o r d e r in IIL, which in m a n y instances are m o r e important, since some qualitatively n e w p h e n o m e n a occur. O n e of t h e s e is a m o v e m e n t of gas near a non-uniformly h e a t e d solid surface, called thermal slip. It bears s o m e analogy to thermal diffusion in a mixture of gases. Just as, in the p r e s e n c e of a t e m p e r a t u r e gradient in a gas mixture, collisions with molecules of the other gas create a flux of particles, so in this case a flux results from collisions with the non-uniformly heated wall by molecules in a thin layer of gas at the wall, w h o s e thickness —I. L e t Vi d e n o t e the tangential velocity acquired by the gas n e a r the wall as a result of thermal slip, and V rT the tangential c o m p o n e n t of the t e m p e r a t u r e gradient. In the first approximation, we can s u p p o s e that Vi is proportional to V,T, i.e. for an isotropic surface V, = V M, T .

(14.12)

T h e coefficient μ m u s t be proportional to the m e a n free p a t h , since it is due to particles in a gas layer of that thickness. T h e n clearly, from dimensional a r g u m e n t s , μ ~ IImv. E x p r e s s i n g the m e a n free path in t e r m s of the collision cross-section and the gas density, w e h a v e I ~ 1/Νσ ~ Τ / σ Ρ , and, finally, (14.13) T h e sign of μ is not determined b y t h e r m o d y n a m i c r e q u i r e m e n t s ; experimental results show that usually μ > 0. O n e further first-order effect is the p r e s e n c e in a moving gas of an additional surface heat flux (i.e. restricted to a layer at the wall with thickness ~ J) q s u, r f proportional to the normal gradient of the tangential velocity: (14.14)

q'surf =
with the dimensions energy/length x time. T h e coefficients μ and φ are c o n n e c t e d b y a relation which follows from O n s a g e r ' s principle. T o derive this, let us consider the " s u r f a c e " part of the rate of increase of e n t r o p y S s u, due r f to the motion of the gas at the wall and t a k e n per unit 2 quantity consists of t w o p a r t s . T h e p r e s e n c e of the area of the wall surface. This heat flux q s fucontributes r~ q S rf. r U V T ; cf. the c o r r e s p o n d i n g expression for the rate of increase of e n t r o p y due to a bulk heat flux ( F M , §49; SP 2, §88). Secondly, the wall past which the gas is flowing is subject to a frictional force -j]d\tldn per unit area. T h e energy dissipated per unit time is equal to the w o r k d o n e by this force, -r)d\t/dn. \ t, and division b y Τ gives the contribution to the rate of increase of e n t r o p y . T h u s w e have Ssurf

PK 10 - Ε

= - ψ

Qsurf

. VT -

J

T>

V, '

(14.15)

54

Kinetic

Theory of

Gases

W e n o w take as the X e, in the general s t a t e m e n t of O n s a g e r ' s principle (§ 9), the vectors

X

l

V

"T^

i

T

X

'

2

~ T ~ ^ '

A c o m p a r i s o n of (14.15) with the e x p r e s s i o n s (9.3) s h o w s that the c o r r e s p o n d i n g quantities xa are the vectors Xl =

qsurf,

À2 =

η

V,.

T h e " e q u a t i o n s of m o t i o n " (9.1) are the relations (14.12) and (14.14); writing t h e s e as

2 Χι = Τ φ Χ 2,

χ2= η μ Τ Χ , ,

w e obtain the required relation φ = Τημ

(14.16)

(L. W a l d m a n n 1967).

PROBLEMS P R O B L E M 1. T w o v e s s e l s containing a gas at different temperatures T\ and Τι are connected by a long tube. A s a result of thermal slip, a pressure difference is established b e t w e e n the g a s e s in the t w o v e s s e l s (the thermo-mechanical effect). Determine this difference. S O L U T I O N . The boundary condition at the surface of the tube for Poiseuille flow under the influence of the pressure and temperature gradients, with allowance for thermal slip, is ν - μ dTldx at r = R (where R is the tube radius and the χ - a x i s is along the length of the tube). W e find in the usual w a y (see F M , § 17) the velocity distribution over the tube cross-section:

The mass of gas flowing through a cross-section of the tube per unit time is pirR'dP^

Λ

2 dT

Ό

where ρ is the gas density. In mechanical equilibrium Q = 0, w h e n c e dP dx

$ημ = ~W

dT dx '

Integration over the w h o l e length of the tube gives the pressure difference:

2

Ρ2-Ρ.

= (8ημ/Κ )(Τ -Τ,)

2

(if T2 - Ti is fairly small, η and μ may be taken as constants). A n estimate of the order of magnitude of the effect by means of (14.13) and (8.11) gives

22

δΡ/Ρ ~

(l IR )8TIT.

§14

Phenomena

in Slightly

Rarefied

Gases

55

T h e velocity distribution over the tube cross-section w h e n Q = 0 is

2

AdT

(2r

The gas flows along the walls in the direction of the temperature gradient (v > 0 ) , and near the axis of the tube it flows in the opposite direction (v < 0 ) . P R O B L E M 2. T w o tubes of length L and different radii (JRi Ti), the difference being small. A s a result of thermal slip, a circulatory motion of gas is established in the tubes. Find the total gas flow through the 4 tube cross-sections. S O L U T I O N . Dividing (1) in Problem 1 by R and integrating along a closed contour formed by the two tubes, w e have

2

Q =

P^ T -T,)(R

( 2

2

2

-R

1)^r^.

The flow takes place in the direction s h o w n in Fig. 2 .

τ

"2/?,

2/?,

FIG.

2.

P R O B L E M 3. Determine the force F acting on a sphere of radius R immersed in a gas where a constant temperature gradient V T = A is maintained. S O L U T I O N . The temperature distribution within the sphere is given by

T =

3K

Kl +

2

2K2

Ar c o s 0 ,

where κι and κι are the thermal conductivities of the sphere and the gas; r and 0 are spherical polar coordinates with the origin at the centre of the sphere and the polar axis along A (see F M , § 5 0 , Problem 2). H e n c e w e find for the temperature gradient along the surface of the sphere

\_dT_ R se '

3K

Kl +

2

2K2

A sin 0.

The laminar flow of the gas resulting from the thermal slip is determined only by the o n e vector A . The corresponding solution of the N a v i e r - S t o k e s equation may therefore be sought in the same form as in the problem of liquid flow past a sphere moving in it (see F M , § 2 0 ) :

1

ν = - a

A + n(A.

r

1

η) , , 3 n ( A . n) - A - + b — —τ, r

where η = r/r; the additive constant in ν is omitted, since w e must have ν = 0 as r-><». The constants a and b are found from the conditions

vr = 0, ve = falR)dTlde at r = R;

56

Kinetic

their values are

Theory of

Gases

2 a = bIR

= - 3κ Κμ/2(κι + 2 κ ) .

2

2

The force on the sphere is F = 8τταη A = -

12ττημ1?κ2νΤ/(κι +

2κ ).

2

For the surface effects considered in these Problems to be in fact small compared with the volume effects, the temperature must vary only slightly over the radius of the tube in Problems 1 and 2, and over the radius of the sphere in Problem 3. P R O B L E M 4. T w o v e s s e l s joined by a long tube contain gas at the same temperature and at pressures Pi and P . Determine the heat flux b e t w e e n the v e s s e l s which accompanies Poiseuille flow in the tube 2 (the mechano-caloric effect). S O L U T I O N . According to (14.14) and (14.16), the heat flux along the walls of the tube is q' = 2vRqL

= 2ττΚΤημ

r{

dV/dr.

From the condition of mechanical equilibrium of the liquid in a steady flow, w e have

2

2TTRT)

Hence,

dVldr

=

TTR

dP/dx

=

ITR\P

2

-

POIL.

finally, q' =

TTR^{P -Px)IL.

2

§15. Phenomena in highly rarefied gases T h e p h e n o m e n a discussed in § 14 are n o m o r e than correction effects associated with higher p o w e r s of the ratio of the m e a n free path I to the characteristic dimensions L of the p r o b l e m ; this ratio w a s supposed still small. If the gas is so rarefied, or the dimensions L are so small, that i / L a : l , the equations of fluid dynamics b e c o m e completely inapplicable, even with corrected b o u n d a r y conditions. In the general case of a n y i/L, it is in principle n e c e s s a r y to solve the t r a n s p o r t equation with specified b o u n d a r y conditions on solid surfaces in c o n t a c t with the gas. T h e s e conditions d e p e n d on the interaction b e t w e e n the gas molecules and the surface, and relate the distribution function for particles incident on the surface to that for particles leaving it. If this interaction a m o u n t s to scattering of molecules without chemical transformation, ionization, or absorption by the surface, it is described by the probability w ( F , T)dT' that a molecule with given values of Γ strikes the surface and is reflected into a given range d T ' ; the function w is normalized b y the condition

(15.1)

fw(r,T)dr=l. With this function, the b o u n d a r y condition for the distribution function becomes

f Jn.v<0

w(F,r)n.v/(r)dr =

-n.v7(r)

with

n.v>0.

/(Γ)

(15.2)

§15

Phenomena

in Highly Rarefied

Gases

57

T h e integral on the left multiplied by dT' is the n u m b e r of molecules incident on unit area of the surface per unit time a n d scattered into a given range dT'; the integration is t a k e n over the range of values of Γ that c o r r e s p o n d s to molecules moving t o w a r d s the surface (n being a unit v e c t o r along the o u t w a r d normal to the surface of t h e b o d y ) . T h e e x p r e s s i o n o n t h e right of (15.2) is t h e n u m b e r of molecules leaving unit area of the surface p e r unit time. T h e values of Γ' on each side of the equation m u s t c o r r e s p o n d to molecules moving a w a y from the surface. In equilibrium, w h e n the t e m p e r a t u r e of the gas is the s a m e as that of the b o d y , the distribution function m u s t h a v e the B o l t z m a n n form for b o t h the incident and the reflected particles. H e n c e it follows that the function w m u s t satisfy identically the equation

€ l/ T f

νν(Γ, Γ ) η . ν
7Τ

d T = - η . ν ' e~< »,

(15.3)

Jn. ν<0

w h i c h is obtained b y substituting in (15.2) / ( Γ ) = c o n s t a n t x e x p ( - e/TO, with Ti t h e t e m p e r a t u r e of the b o d y . In the general formulation described, the solution of the problem of highly rarefied gas flow is of c o u r s e very difficult. T h e p r o b l e m c a n , h o w e v e r , b e m o r e simply stated in the limiting case w h e r e the gas is so highly rarefied that IIL> 1. A large class of such p r o b l e m s relate to situations w h e r e a considerable m a s s of gas occupies a v o l u m e large c o m p a r e d with the dimensions L of solid bodies i m m e r s e d in the g a s , and also c o m p a r e d with the m e a n free p a t h i. T h e n collisions of molecules with solid surfaces are comparatively r a r e , and are u n i m p o r t a n t relative to collisions b e t w e e n molecules. If the gas itself is in equilibrium, with t e m p e r a t u r e T 2, w e c a n a s s u m e u n d e r t h e s e conditions t h a t the equilibrium is not d e s t r o y e d b y the i m m e r s e d b o d y . T h e r e m a y b e any t e m p e r a t u r e difference b e t w e e n the gas and the b o d y . T h e same is true of the m a c r o s c o p i c velocities. L e t τ = T 2- T i b e the difference b e t w e e n the t e m p e r a t u r e of the gas and that of s o m e part df of the surface of the b o d y , and V t h e velocity of the gas relative to the b o d y . F o r n o n - z e r o τ and V there is heat e x c h a n g e b e t w e e n the gas and the b o d y , and a force is exerted on the b o d y by the gas. L e t q be the dissipative heat flux from the gas to the body, and let F - F n denote the force per unit area acting at each point on the surface of the body; η is the outward normal to the surface. The second term here is the ordinary gas p r e s s u r e ; F is the additional force u n d e r consideration, due to τ and V. T h e quantities q and F are functions of τ and V, and are z e r o w h e n t h e s e are z e r o . If τ and V are sufficiently small (τ with r e s p e c t to the t e m p e r a t u r e s t h e m s e l v e s of the gas and t h e solid, V with r e s p e c t to the thermal velocity of the gas molecules), t h e n q and F c a n b e e x p a n d e d in p o w e r s of τ and V as far as the linear t e r m s . L e t F„ and Vn d e n o t e the c o m p o n e n t s of F and V along the normal n ; ¥ t and u, their tangential p a r t s , which are vectors having t w o i n d e p e n d e n t c o m p o n e n t s . T h e n the e x p a n s i o n s mentioned are q = ar + fiVn, Fn = yr + 8Vn9 F, = 0V„

(15.4)

w h e r e α, β, γ, δ, θ are c o n s t a n t s (or rather functions of t e m p e r a t u r e and p r e s s u r e ) , characteristic of any given gas and solid material. T h e " s c a l a r " quantities q and F n

58

Kinetic

Theory of

Gases

cannot, b y s y m m e t r y , contain t e r m s linear in t h e vector \ t. F o r t h e same r e a s o n , the expansion of t h e vector F t d o e s n o t contain t e r m s linear in t h e " s e a l a r s " τ a n d T h e quantities a, δ and θ are positive. F o r e x a m p l e , if t h e gas t e m p e r a t u r e e x c e e d s t h e b o d y t e m p e r a t u r e ( r > 0), heat will pass from t h e gas to t h e b o d y , a n d the corresponding part of the flux q will b e positive; h e n c e a > 0 . N e x t , t h e forces Fn and F, due to t h e gas flow relative to t h e b o d y m u s t b e in t h e same direction as V„ and V, ; h e n c e δ > 0 and θ > 0. T h e sign of t h e coefficients β and γ d o e s n o t follow from general t h e r m o d y n a m i c considerations, although in practice they seem to b e usually positive. T h e r e is a simple relation b e t w e e n t h e m which is a c o n s e q u e n c e of t h e s y m m e t r y of t h e kinetic coefficients. T o derive this relation, w e calculate t h e time derivative of t h e total e n t r o p y of the system comprising t h e gas and t h e b o d y in it. A quantity of heat q df is gained by the b o d y from t h e gas in unit time through each surface element df. T h e increment in t h e entropy S i of t h e b o d y is

w h e r e the integration is over t h e whole surface of t h e b o d y . T o calculate t h e increase in t h e e n t r o p y of the g a s , w e t a k e coordinates such that the gas is at rest at t h e position of t h e b o d y ; then t h e velocity of e a c h point o n t h e surface is - V . In order to d e m o n s t r a t e t h e required relation, w e shall s u p p o s e that the shape of t h e b o d y m a y vary during its m o t i o n ; then t h e velocities V of various points on its surface are arbitrary i n d e p e n d e n t variables. F r o m the t h e r m o d y n a m i c relation dE = TdS -P dT, t h e change in t h e e n t r o p y of t h e gas p e r unit time is s2 =

(Ê2+p2f2)iT2,

quantities with t h e suffix 2 relating to the g a s . T h e derivative È2 is, b y t h e conservation of t h e total energy of t h e system, minus t h e change in t h e energy of the b o d y . This change is m a d e u p of t h e quantity of heat φ q df a n d t h e w o r k φ - V . (F - Ρ η) df d o n e o n t h e b o d y . T h u s w e find as t h e change in t h e energy of t h e gas Ê2 = é(-q

+ FnVn +

Ft-Yi -PiVn)df.

T h e change in t h e gas volume is equal to minus t h e change in t h e volume of t h e body:

T h e change in the entropy of t h e gas is therefore

§15

Phenomena

in Highly Rarefied

Gases

59

Adding t h e derivatives of S i a n d S 2, a n d t h e n putting (for small τ ) T\ « T 2= T, w e finally have as t h e rate of c h a n g e of t h e total e n t r o p y of t h e s y s t e m S =

J [ f ^ +^ ] d / .

(15.5)

W e t a k e a s t h e quantities X I , x 2, x 3, * 4 in t h e general formulation of O n s a g e r ' s principle ( § 9 ) respectively q, Fn a n d t h e t w o c o m p o n e n t s of t h e v e c t o r Fi at a n y given point o n t h e surface of t h e b o d y . T o find t h e c o r r e s p o n d i n g quantities Xa, w e 2 c o m p a r e (15.5) with t h e general e x p r e s s i o n (9.3) for t h e rate of c h a n g e of t h e e n t r o p y , a n d see that X i , X 2, X 3, X 4a r e respectively - τ / Τ , -VJT and the two at t h e s a m e point. T h e kinetic coefficients (i.e. c o m p o n e n t s of t h e vector -\tIT t h o s e in t h e relations (9.1)) a r e

2 γιι = α Τ ,

γ

2 = 2δ Τ ,

Jn = β Τ ,

γ

33= 2yu= ΘΤ,

y2\ = yT .

T h e s y m m e t r y yxl = yi\ t h u s gives t h e required relation: β = γΤ.

(15.6)

M o r e o v e r , from t h e condition that t h e q u a d r a t i c form ( 9 . 3 ) is positive ( S > 0 ) , w e h a v e t h e inequalities α , β , θ > 0 already m e n t i o n e d , a n d also t h e inequality

2 Ταδ > β .

T o calculate t h e coefficients in (15.4), w e need t o k n o w t h e specific form of t h e law of scattering of gas molecules b y t h e surface of t h e b o d y , e x p r e s s e d b y t h e function w(T', Γ) defined a b o v e . A s a n e x a m p l e , let u s derive a formula w h i c h in principle allows α t o b e calculated. T h e energy flux from t h e gas t o t h e b o d y is given b y t h e integral

f q = J ( e - e')\v \w(T ,

Γ ) / ( Γ ) dT dT',

x

(15.7)

t a k e n o v e r t h e ranges vx <0, v'x>0, since a n a m o u n t of energy € - e' is transferred to t h e wall at e a c h collision of a molecule with t h e wall. L e t u s transform this e x p r e s s i o n b y m e a n s of t h e principle of detailed balancing, τ according t o w h i c h , in equilibrium, t h e n u m b e r of transitions Γ - ^ Γ ' in/Τ t h e scattering of molecules b y t h e wall is equal t o t h e n u m b e r of transitions Γ -> Γ . This m e a n s that

w ( R , nkl

= w ( r T

'

ρ Τ ) | ϋ ; | 6Χρ

( ΤΓ ) ίί

:

;

in equilibrium, t h e t e m p e r a t u r e s of t h e gas and t h e wall a r e equal.

( 1 5

·

8 )

Kinetic

60

Theory of

Gases

Τ

Τ

In (15.7) w e r e n a m e t h e variables of integration: Γ - » Γ , Γ - * Γ . Half t h e sum of the t w o resulting expressions gives

Τ 2 q=±j

(6-€')^

τ

IT

[νν(Γ\

T

Ίτ

τ

T)\vx\e -* >-w(T ,

Υ' )\ν'χ^ ηάΤάΓ.

τ

Lastly, substituting νν(Γ , Γ ) from (15.8) and then expanding t h e integrand in p o w e r s of t h e small difference τ = T 2- Tu w e find that q = ar, w h e r e μ-*(Γ)

a =

dTdT'

(vx<0,

v'x>0);

(15.9)

the subscript is omitted from t h e t e m p e r a t u r e T{ ~ T 2. T h e distribution function for molecules scattered from t h e wall d e p e n d s o n t h e specific nature of their interaction with t h e wall. T h e r e is said to b e complete accommodation if t h e molecules reflected from each surface element of t h e b o d y have (whatever t h e magnitude a n d direction of their velocity before t h e impact) t h e same distribution as in a b e a m leaving a small a p e r t u r e in a vessel containing gas at a t e m p e r a t u r e equal to that of t h e body. T h u s , with c o m p l e t e a c c o m m o d a t i o n , t h e gas scattered by t h e wall r e a c h e s thermal equilibrium with it. T h e values of t h e coefficients in (15.4) m a y reasonably be c o m p a r e d with those for complete a c c o m m o d a t i o n . In particular, energy e x c h a n g e b e t w e e n t h e gas molecules a n d t h e solid wall is usually described b y t h e a c c o m m o d a t i o n coefficient, defined as t h e ratio α / α 0, w h e r e a0 c o r r e s p o n d s to complete a c c o m m o d a t i o n . In actual c a s e s , complete a c c o m m o d a t i o n is n o t usually achieved, a n d t h e a c c o m m o d a t i o n coefficient is less than unity. T h e fact that a 0 is in fact the greatest possible value is easily s h o w n as follows. L e t us view the entropy S in (15.5) s o m e w h a t differently: n o t as t h e total e n t r o p y of t h e b o d y and t h e gas together, b u t as t h e entropy of t h e b o d y together with just the gas molecules that reach t h e surface of t h e b o d y in a time Δί. F o r this system, reflection of t h e molecules with complete a c c o m m o d a t i o n d e n o t e s a transition to a state of complete equilibrium, a n d its entropy therefore takes t h e m a x i m u m possible value. Accordingly, t h e change of entropy AS = S A i a c c o m p a n y i n g this transition will also b e a m a x i m u m . t T h a t is, for c o m p l e t e a c c o m m o d a t i o n t h e quadratic form (9.3) m u s t b e a m a x i m u m for a n y given values of t h e Xa (i.e. of τ, Vn and V r) . Denoting t h e corresponding values of t h e coefficients yab b y t h e suffix zero, w e c a n write this condition as

τ

v„+

80> δ,

θ0>0,

2

flo-e

τ

V?>0.

F r o m this, there follow t h e inequalities a0>

a,

2

Τ ( α 0- α ) ( δ 0- δ ) > ( β ο - β ) .

(15.10)

tlmportant points in this argument are that the body (which acts as a "heat reservoir") may be regarded as in equilibrium throughout the process, and that the entropy of an ideal gas depends only on the distribution law for its molecules, not on the law of interaction b e t w e e n them.

§15

Phenomena

in Highly Rarefied

Gases

61

L e t us consider the outflow of a highly rarefied gas from a small orifice with linear dimensions L. In the limit l/L>l, this p r o c e s s is a very simple one. T h e molecules will leave the vessel independently, forming a molecular b e a m in which each molecule m o v e s at the speed with which it reached the orifice. T h e n u m b e r of molecules leaving the orifice per unit time is equal to the n u m b e r of collisions per 1 / to 2 that of the orifice. unit time b e t w e e n molecules and a surface with area s equal T h e n u m b e r of collisions per unit wall area is P/(27rmT) , w h e r e Ρ is the gas pressure and m the m a s s of a molecule; see SP 1, §39. T h u s the m a s s of gas leaving per unit time is Q = sPV(m/27rT).

(15.11)

If t w o vessels containing gas are joined by an orifice, for KL in mechanical equilibrium the p r e s s u r e s P i and P 2 of the gases in the t w o vessels are equal, w h a t e v e r their t e m p e r a t u r e s Ti and T 2. If l> L, the condition of mechanical equilibrium is that the n u m b e r s of molecules passing through the orifice in each direction are equal. By (15.11), this gives P i / V T i = P 2/ V T 2.

(15.12)

T h u s the p r e s s u r e s of rarefied gases in two communicating vessels will b e different, and proportional to the square roots of the t e m p e r a t u r e s (the Knudsen effect). So far w e h a v e discussed p h e n o m e n a in a large m a s s of highly rarefied gas in equilibrium by itself. L e t us n o w briefly consider p h e n o m e n a of another t y p e , w h e r e the gas itself is not in equilibrium, for instance in heat transfer b e t w e e n two solid plates h e a t e d to different t e m p e r a t u r e s and immersed in a rarefied gas, the distance b e t w e e n them being small c o m p a r e d with the m e a n free path. Molecules moving in the space b e t w e e n the plates u n d e r g o almost n o collisions with o n e another; after reflection from one plate, they m o v e freely until they strike the other. W h e n scattered by the hotter plate, the molecules gain some energy from it, and t h e n transfer s o m e of their energy to the cooler plate w h e n they r e a c h it. T h e heat transfer m e c h a n i s m in this case thus differs essentially from that of ordinary conduction in a non-rarefied gas. It m a y be described by a heat transfer coefficient κ, defined (by analogy with the ordinary conductivity) so that q = K(T2-Tx)IL,

(15.13)

w h e r e q is the a m o u n t of heat transferred per unit area of the plates per unit time, Ti and T 2 the t e m p e r a t u r e s of the plates and L the distance b e t w e e n t h e m . T h e value of κ m a y be estimated in order of magnitude b y m e a n s of (7.10). Since collisions b e t w e e n molecules are now replaced by collisions of molecules with the plates, the m e a n free path I m u s t be replaced by the distance L b e t w e e n the plates. Thus κ ~ LvN

~ PLN(mT).

(15.14)

T h e heat transfer coefficient in a highly rarefied gas is proportional to the p r e s s u r e , in contrast to the conductivity of a non-rarefied gas, which is i n d e p e n d e n t of the

62

Kinetic

Theory

of

Gases

p r e s s u r e . It should b e e m p h a s i z e d , h o w e v e r , that κ here is n o t a p r o p e r t y of t h e gas alone: it d e p e n d s also on the specific conditions of the p r o b l e m , namely the distance L b e t w e e n the plates. A similar effect is the " v i s c o s i t y " of a highly rarefied gas, which o c c u r s , for e x a m p l e , in the relative motion of t w o plates in it (again with L < I). T h e viscosity coefficient η m u s t here b e defined so that F = TJV/L,

(15.15)

w h e r e F is t h e friction force per unit area o n t h e moving plate and V t h e relative speed of the plates. Replacing the m e a n free p a t h / in ( 8 . 1 1 ) b y the distance L , w e have η ~ mvNL

(15.16)

~ LP\/(mlT)9

i.e. the viscosity of a rarefied gas is likewise proportional to the p r e s s u r e .

PROBLEMS P R O B L E M 1. At the initial instant t = 0, a gas occupies the half-space χ < 0 . Neglecting collisions, determine the density distribution at subsequent instants. S O L U T I O N . If collisions are neglected, the transport equation reduces to d//ar + v . d / / a r = 0, the general solution being / = / ( r - vr, v). With the given initial condition, w e have /ο = /ο(υ)

for

/ = 0

v >x\U

x

for

v
x

where /o is the Maxwellian distribution. The gas density is

3 N(r, x) = J

J ^j

Œ

h(v)m

dv dv

x y

dv

z

where

and No is the initial density. Since collisions have b e e n neglected, these formulae are actually valid only in the range \x\
2

F=-(4TT/3)V1* (5 +

20).

P R O B L E M 3. Determine the speed of m o v e m e n t , in a rarefied gas, of a light plane disc w h o s e sides are heated to different temperatures Ti and T i . S O L U T I O N . The speed V of the disc (in the direction perpendicular to its plane) is found from the condition that the total forces acting o n the t w o sides of zero. It m o v e s with the cooler side forwards at a speed given (when Ίι > Ti) by ν = γ(Τ -Τι)/2δ.

2

Phenomena

§15

in Highly Rarefied

63

Gases

P R O B L E M 4. Calculate the value ao of the coefficient a corresponding to c o m p l e t e accommodation. S O L U T I O N . The amount of energy contributed per unit time by molecules colliding with unit area of the surface of a b o d y is / fiv e dT, where f i is the Boltzmann distribution function with the temperature x Ti of the g a s , € is the energy of a molecule and the χ - a x i s is perpendicular to the surface. The amount of energy carried a w a y b y the same m o l e c u l e s is found (in the c a s e of c o m p l e t e accommodation) simply by replacing Ti b y the temperature Ti of the body. The heat flux is J(f2-fl)€V dT,


x

2

the integration over v being from 0 to » . The energy of the molecule is written as e = €i„t + l m u , where x tint is the internal energy. The value given by calculation for each integral is j fev

x dT

= y(€i„ + 2 T ) = v(i + 2-T) = vT(c

t

+I),

v

where € = c«T is the mean energy of a molecule and ν = P/V(27rmT) the number of molecules striking unit area of the surface per unit time. The heat q is equal to the difference b e t w e e n the energies of the molecules arriving and leaving in equal numbers, i.e. for the same v. The value obtained for the coefficient in q - a(Ti~ T\) is

ρ

the difference Ti - T\ is assumed small, and so w e put Ti « Ti = T. P R O B L E M 5. T h e s a m e as Problem 4, but for the coefficients β and γ. S O L U T I O N . The normal c o m p o n e n t of the m o m e n t u m contributed per unit time by the molecules striking unit area of the surface of the body is half the gas pressure. Expressing the pressure in terms of v, w e have

5 P = vV(kmT). The difference b e t w e e n the values of this quantity at the temperature Ti and Ti for the same ν gives the additional force F caused by the temperature difference. If Ti - T\ is small, w e find

n

γο = Ρ/4Τ. For β, in accordance with (15.6), βο = P/4. P R O B L E M 6. The same as Problem 4, but for the coefficients δ and 0. S O L U T I O N . W e take coordinates in which the b o d y is at rest and the gas m o v e s with velocity V, the x-axis being normal to the surface and the xy-plane containing V. The distribution function in these coordinates is

2

/ = constant χ e x p | - ψ

-

[(v - V f

x

x

+ (v - V )

y

y

2

+ υ ] J.

ζ

With c o m p l e t e a c c o m m o d a t i o n , the reflected m o l e c u l e s h a v e a distribution function with V = 0; τ is assumed to be zero. T o calculate the tangential force F , w e put V = 0. The total y - c o m p o n e n t of m o m e n t u m contributed y x by molecules reaching the surface of the body is J mvyVxf

dV = mVy

j Vxf dT =

mV v,

y

the integration over v being always from 0 to » . The y - c o m p o n e n t of m o m e n t u m carried a w a y by these x molecules is zero. Thus F = mvV , and so

y

y

0o= vm =PV(m/27rT).

N o w let Vx ^ 0, V = 0. A s far as the first order in V w e have

y

x

/ = /o+

V (mvJT)fo,

x

Kinetic

64

Theory of

Gases

where /o is the distribution function with V = 0. The number of molecules colliding with unit area of the surface per unit time is

„ =jMdr =

V ( 2 —7. r m T ) +

The χ - c o m p o n e n t of momentum contributed by these molecules is

2

j mv f dT = \P +

PV V(2mlTrT).

x

x

The molecules reflected from the bounding surface have the distribution function with V* = 0, normalized so that the integral / fv dT is equal to the number ν of incident molecules determined above. x The χ-component of momentum carried away by these molecules is -\v V(27rmT) = -\P-

lPV V(vml2T).

x

The normal force additional to the pressure is F = 8oV , where

x

x (2+27R)=260(4+7R)

δο=p

V^F

-

P R O B L E M 7. Assuming complete accommodation, determine the temperature of a plate moving in its 2 o w n plane with speed V in a rarefied gas. S O L U T I O N . Proceeding as in Problem 4, w e have for the energy contributed v(c Ti + 2T2 + \m V ) , and v for that carried away vT\(c + 5). Equating these gives

v

2

Ti-T

2=

mV /(2c„ + l).

P R O B L E M 8. Determine the quantity of gas flowing per unit time through the cross-section of a cylindrical tube of radius R as a result of pressure and temperature gradients. The gas is so rarefied that the mean free path I > RA There is complete accommodation in collisions of molecules with the tube walls. 3 S O L U T I O N . The speed distribution of the molecules reflected from the wall with complete accommodation is v fd p, where / is the Maxwellian distribution function and the χ-axis is perpendicular to x the surface. If ϋ is the angle b e t w e e n the velocity of a molecule and the χ - a x i s , w e find that the distribution of the reflected molecules with respect to their directions of motion (whatever their speed) is

(ν/ττ) cos udo, this function being normalized s o as to give ν o n integration over all solid angles o n o n e side of the plane. W e take the ζ-axis along the axis of the tube, and the origin in the cross-section considered. Molecules last reflected from various parts of the tube surface pass through this cross-section. Of those scattered by an element df of the wall surface at a distance z, the ones that pass through the cross-section concerned are those reflected in directions lying in the solid angle subtended by this cross-section at the relevant point o n the surface of the tube; their number is thus df. ν / c o s ^dojir, with integration over the angle range mentioned. This integral is evidently the same for all points lying at the same distance from the cross-section concerned. The total number of molecules passing through this cross-section per unit time is therefore obtained by replacing df by the annular surface element lirR dz and integrating along the whole length of the tube; multiplying also by the mass m of a molecule, w e get the mass flow rate of the gas through a cross-section of the tube: Q = 2mR

j v(^j

c o s d d o ^ dz.

The number i>, being a function of pressure and temperature, varies along the tube. If the lengthwise gradients of pressure and temperature are not too great, w e can write v(z)=v(0) t G a s flow of this type is called free-molecular

+ flow.

z[dvldz] =o.

2

§15

Phenomena

in Highly Rarefied

Gases

65

The integral containing i>(0) is evidently zero, and s o

Q = 27rR[dv/dz]=o J J ζ cos ΰ do dz. z

T o carry out the integration, w e take coordinates r and φ in the plane of the cross-section considered, r being the distance of a variable point A' from a fixed point Ο o n the circumference of the cross-section, and φ the angle b e t w e e n OA' and the radius of the cross-section (Fig. 3). A molecule

/

"A I \ I \ I

\

-TV!

! /

r\\\

! / S£

FIG.

3.

reflected from the wall at a point A o n the same generator as Ο and then passing through A' must have a velocity at an angle ϋ to the normal to the tube surface at A such that _

r cos φ

The solid-angle element may b e written

d, r

07 = 2ζ rdrdç + z* V(r +

zV

the area rdrdç is projected o n the plane perpendicular to the line A A \ and the result is divided by the square of the length of that line. T h e integration is carried over the region -ιπ ^φ^ϊπ, 0^r=^ 2R c o s
(SirR'lDdvldz.

Finally, putting ν ••• P / V ( 2 7 r m T ) , w e obtain

3

4TTR

3L

V(2nm)(^r-^r) f

where the difference in parentheses is b e t w e e n the values of P / V T over a length L of the tube; the replacement of the derivative b y the difference is allowable b e c a u s e Q, and therefore this derivative, are constant along the tube. P R O B L E M 9. Assuming complete accommodation, find the frictional force b e t w e e n t w o solid planes at a distance apart L 0 and t; < 0 y y are reflected from planes 1 and 2 respectively; with complete accommodation, their distribution functions are

Kinetic

66

Theory

of

Gases

where N i and N2 are the corresponding number densities of particles; the total density Ν = N i + N2. The condition of zero total flux in the y-direction gives N i V T , = N2VT2. A pressure Ρ = N i Γι + N T

22

acts o n each plane, and the frictional force per unit area is F = - F , = mvi

2

vyfd'p

Jv >0

y

= VN V(2mT /7r)

2

2

, / 2T12

m

= V N V(2m/7r)(T T2) /( i ' + T ).

1

If Ti = T ^ T , then

2

2

F = - F i = VPV(m/2irT),

2

in agreement with (15.15) and (15.16). P R O B L E M 10. Assuming complete accommodation, determine the heat transfer coefficient κ b e t w e e n t w o plates with almost equal temperatures Ti and T2. S O L U T I O N . With complete accommodation, the molecules incident o n plate 1 have an equilibrium distribution with temperature T2. T h e energy flux from plate 1 to plate 2 is therefore q ao(T -T\). 2 Taking ao from Problem 4 and determining κ from (15.13), w e find

L K = «oL =

£

Vm (( cT„ + )i ) ,

in accordance with the estimate (15.14). P R O B L E M 11. Determine the g a s density o n the axis behind a circular disc of radius R<1, moving in a gas with a velocity - V much greater than the mean thermal speed υτ of the atoms. S O L U T I O N . When V > VT, the particles reflected from the rear surface of the disc are unimportant (except for a narrow region near that surface; s e e below). T h e problem is a matter of the "shadow" of the disc in the incident flow. In coordinates for which the disc is at rest (and the gas is moving with velocity V), in the absence of the disc the distribution function would be

2

/1 \

No

ί

m(v-V) l

In the presence of the disc, the number density of gas particles o n the z-axis (Fig. 4) is

2 N ( z ) = 2ττ f JO

f /o(v)p sin d dti dp, J*o

where d is the angle b e t w e e n ν and the z-axis, and do the angle subtended by the radius of the disc at the point of observation o n the z-axis (tan do = Λ / ζ ; particles with d < # o are cut off by the disc). Integration, with the condition V > υτ, gives

y f {"2if * ° ° 2

ψ

n(z)=

exp

f

ttt V

[{v

2

vc

sd

}

) 2 + y 2 s i n 2 βο]

υdv

1

--2jr-sin doj =No

l~ ΤΓ W+z y

exp

1

where No is the gas density far from the disc. T h e integration over dp is carried out with the assumption that c o s ϋο> vtIV; it can be s h o w n that this inequality is also the condition for particles reflected from the rear face to be negligible.

§ 16

Dynamical

Derivation

of the Transport

67

Equation

V R ζ V

FIG. 4.

§ 16. Dynamical derivation of the t r a n s p o r t equation Although the derivation of the t r a n s p o r t equation given in § 3 is satisfactory from the physical point of view, there is considerable interest in ascertaining h o w the e q u a t i o n c a n b e derived analytically from t h e m a t h e m a t i c a l formalism of t h e t h e o r y , i.e. from t h e equations of motion of the gas particles. Such a derivation has b e e n given b y Ν . N . Bogolyubov (1946). T h e value of the m e t h o d lies also in the fact that it affords a regular p r o c e d u r e for deriving in principle not only the B o l t z m a n n equation b u t also the corrections to it, i.e. 3the t e r m s of higher orders in the small " g a s e o u s n e s s p a r a m e t e r " — t h e ratio ( d / r ) , w h e r e d is t h e molecular dimension (range of action of molecular forces) and r the m e a n distance b e t w e e n the molecules. T h e derivation given below relates to a m o n a t o m i c gas in purely classical t e r m s , i.e. on the a s s u m p t i o n that not only free motion b u t also collisions of the gas particles are describable b y classical m e c h a n i c s . W e start from Liouville's t h e o r e m regarding t h e distribution function for t h e gas m of Ν particles. This function, in 6 ^ - d i m e n s i o n a l p h a s e as a w h o l e , as a system s p a c e , is d e n o t e d b y f (U Ή , τ 2, . . . , i > ) , w h e r e τ α is the set of c o o r d i n a t e s and m o m e n t u m c o m p o n e n t s for the a t h particle: τ α = ( r e, ρ α) . T h e function is a s s u m e d normalized to unity:

iJ f °(t,

3 τ „ T 2, . . . , τ*) d r , . . . dr* = 1,

dra = d xa

3 d pa.

iJf) function which a p p e a r s in the B o l t z m a n n equation T h e " o n e - p a r t i c l e " distribution is obtained b y integrating f over all dra b u t o n e : (16.1) the function also is normalized to unity, and w e shall retain t h e notation / ( 1 ) for the distribution function normalized to the total n u m b e r of (without superscript) particles: / = JV/ . It has b e e n noted in SP 1, § 3 , that Liouville's t h e o r e m arises as a c o n s e q u e n c e of the equation of continuity in p h a s e space which m u s t b e satisfied b y the distribution function for a closed system: (16.2)

68

Kinetic

Theory of

Gases

With Hamilton's equations ra = dHldpa,

this gives

a/

w ±

dt

Aii/

w

(16.3)

va = -dHldra,

. ι

. ^df^

£ί\ I dra

dpa

df

w Λ

η

.

Λ Λ

J

w h e r e t h e ra = v fl and p fl a r e a s s u m e d to be expressed in t e r m s of ru τ 2, . . . b y m e a n s of equations (16.3). E q u a t i o n (16.4) e x p r e s s e s t h e c o n t e n t of Liouville's theorem. W e write t h e Hamiltonian function for a m o n a t o m i c gas in t h e form Η = Σ„f^+.

Σ

(16.5)

^ ( k a - r . l ) . b
H e r e it is a s s u m e d that there is n o external field, and that t h e interaction b e t w e e n the gas particles r e d u c e s to t h e s u m of their pair interactions.t E q u a t i o n (16.4) then becomes f=\ I d r a

dt

b
dpa

w h e r e Uab( a ^ b) d e n o t e s U ( | r a - r ^ l ) . L e t u s n o w integrate this equation over dr2... άτ#. T h e n , of all t h e terms in the sum in (16.6), only those remain which involve differentiation with r e s p e c t to pi or Γι; t h e integrals of t h e other t e r m s are transformed into integrals over infinite surfaces in m o m e n t u m or coordinate s p a c e , a n d are z e r o . T h u s w e h a v e

(1) (1) a / ( t , T i ) , V| a / ( t , T , ) _ ,r (dUl2 dt

( )2 where / integral

' "

a ri

J

2 df \UrUT2) dp

a ri

is t h e two-particle distribution function normalized t o unity, i.e. t h e

(

( 2 ) /

(ί,τ„τ ) = | / ^ ί ί τ 3 . . . ^ ;

(16.8)

2

the factor Λ Γ in (16.7) t a k e s a c c o u n t of t e r m s that differ only in t h e n o m e n c l a t u r e of the variables of integration; strictly speaking, t h e n u m b e r of such t e r m s is J{—\, b u t this is very large a n d m a y b e replaced b y Jf. Similarly, integrating (16.6) over d r 3. . . dr^ w e obtain

{2)

df dt

.a/^ | 'ar

1

(3)

V 2al / + y

( )2

ar

2

V dUn df™ a ri ' a pi

2)

dUn ar

=

df

2 ap2

wrf^ f^i J Lapi

+ ari

ap

2

d T 3 ar J

,

.

(16 9)

2

w h e r e / ( ί , τι, τ 2, τ 3) is t h é three-particle distribution function. t T h e latter assumption constitutes a model, but it d o e s not affect the result in the first approximation (which corresponds to the Boltzmann equation): in this approximation, only pair collisions of particles occur, in which other (non-pair) interactions play no part.

§16

Dynamical

Derivation

of the Transport

Equation

69

( )Ran almost infinite ( n (Jf + 1being ) Continuing in this w a y , w e should obtain v e r y large) s e q u e n c e of e q u a t i o n s , e a c h expressing / in t e r m s of / . All t h e s e equations are e x a c t in the sense that no a s s u m p t i o n has b e e n m a d e in t h e m as to the rarefaction of the gas. T o obtain a closed set of e q u a t i o n s , the series has to b e terminated in s o m e w a y by making u s e of the condition that the gas is rarefied. In 2 particular, the first a p p r o x i m a t i o n in this m e t h o d c o r r e s p o n d s to terminating ( )the series already at t h e first equation, (16.7), in w h i c h the two-particle function / is e x p r e s s e d approximately in t e r m s of This is d o n e b y using the rarefaction of the g a s , b y m e a n s of equation (16.9). Returning to this equation, w e shall first of all show that the integral on the right-hand side is small. T h e function 17 (r) is noticeably different from zero only within the range of action of the forces, i.e. w h e n r^d. H e n c e , in b o t h p a r t s of the 3 practice only over the integral in (16.9), the integrations over c o o r d i n a t e s are in 3 ) integration ( 2 ) v o l u m e ~ d . Since in ( an region | r 3- rx\ ^ d or | r 3- r 2| ^ d, i.e. over a 3 over t h e whole v o l u m e of the gas, T ~ ^ V r , w e should h a v e / / d T 3 = / , we obtain t h e estimate

3 F r o m this w e see t h a t t h e right-hand side of (16.9) is small in t h e ratio ( d / r ) relative to t h e t e r m s containing dUldr on the left-hand side, and m(a2y ) therefore b e neglected. T h e t e r m s on t h e left constitute the total derivative d / / d i , in which r i , Γ2, Pi, P2 are regarded as functions of time w h i c h satisfy t h e e q u a t i o n s of motion (16.3) with the t w o - b o d y Hamiltonian

Thus we have

2

df \t,

T „ T 2) / d i = 0 .

(16.10)

So far, all t h e transformations of the e q u a t i o n s h a v e b e e n purely mechanical o n e s . T o derive t h e t r a n s p o r t equation, of c o u r s e , s o m e statistical a s s u m p t i o n is also n e c e s s a r y . This m a y b e formulated as t h e statistical i n d e p e n d e n c e of e a c h pair of colliding particles, w h i c h has essentially b e e n a s s u m e d in deriving the t r a n s p o r t equation in § 3 (where t h e collision probability w a s written in t h e form (2.1), proportional to t h e p r o d u c t / / i ) . In the m e t h o d u n d e r consideration, this s t a t e m e n t acts as t h e initial condition for t h e differential equation (16.10). It c r e a t e s t h e a s y m m e t r y in relation to t h e t w o directions of time, and as a result t h e irreversible t r a n s p o r t equation is derived from the e q u a t i o n s of m e c h a n i c s invariant u n d e r time reversal. T h e correlation b e t w e e n t h e positions and the m o m e n t a of the gas particles arises only as a result of their collisions and e x t e n d s to distances ~ d. T h u s the a s s u m p t i o n of t h e statistical i n d e p e n d e n c e of colliding particles is also t h e s o u r c e of t h e f u n d a m e n t a l limitations as r e g a r d s t h e distances and time intervals allowed b y t h e t r a n s p o r t equation, already discussed in § 3 . PK 10 - F

70

Kinetic

Theory

of

Gases

L e t i 0 b e s o m e instant before t h e collision, w h e n the t w o particles are still far a p a r t (|ri 0- r 2o| > d, w h e r e t h e suffix z e r o d e n o t e s t h e values of quantities at t h a t instant). T h e statistical i n d e p e n d e n c e of colliding particles m e a n s that at such an instant i 0 t h e two-particle distribution function is the p r o d u c t of t w o one-particle functions H e n c e t h e integration of (16.10) from i 0 to t gives

(2

f \t,

(1)

(1)

ru τ 2) = / ( ί ο , τ 1 ) 0/ ( ί ο , r 2 ) 0.

(16.11)

H e r e τ !0 = (ΙΊ0, pio) and τ 2ο = (r 2o, P20) are to b e u n d e r s t o o d as t h o s e values of t h e coordinates and m o m e n t a w h i c h the particles m u s t h a v e at t h e instant i 0 in o r d e r t o acquire t h e n e c e s s a r y values Ti = ( r i , p i ) and Τ2= (Γ2, P2) at the instant t; in this s e n s e , τ ί0 and τ 2ο are functions of τι, τ 2 and t - i 0 (only Γ ϊ0 and r 2o d e p e n d on t - i 0; t h e values of pio and p 20relate to particles moving freely before the collision, and do not d e p e n d on t h e choice of t - i 0) . L e t u s n o w r e t u r n to (16.7), which is to b e c o m e the t r a n s p o r t equation. T h e left-hand side already h a s t h e required form; w e shall n o w b e c o n c e r n e d with t h e integral o n the right, w h i c h is ultimately to b e c o(m)2 e t h e collision integral in t h e il B o l t z m a n n equation. Substituting in this integral / from (16.11) and changing on b o t h sides from to / = Jff \ we write

where (16.12) ~ d, i.e. t h e region in which the collision o c c u r s , is i m p o r t a n t Only t h e range | r 2in the integral (16.12). In this r a n g e , h o w e v e r , w e c a n neglect (in t h e first approximation, w h i c h is being considered here) the c o o r d i n a t e d e p e n d e n c e of / , w h i c h varies appreciably only o v e r distances L , the characteristic dimensions of t h e p r o b l e m , which are certainly large in c o m p a r i s o n with d. T h e final form of t h e collision integral will therefore b e unaltered if, in o r d e r to simplify s o m e w h a t the analysis and t h e formulae, w e t a k e t h e c a s e of spatial h o m o g e n e i t y , i.e. a s s u m e that / is i n d e p e n d e n t of t h e c o o r d i n a t e s . It m a y b e n o t e d immediately that t h e explicit time d e p e n d e n c e through r i 0( i ) and r 2( 0i ) t h e n d i s a p p e a r s from t h e functions / ( i 0, pio) a n d /(ίθ, Ρ2θ). W e c a n transform t h e integrand in (16.12) b y using the fact that the e x p r e s s i o n in the b r a c e s is an integral of t h e m o t i o n (and a p p e a r e d as such in (16.11)); ind e p e n d e n t l y of this, it is o b v i o u s that p î0 and p 2 , 0the values of t h e m o m e n t a at a fixed instant i 0, are b y definition integrals of t h e motion. U s i n g also t h e fact m e n t i o n e d a b o v e that t h e y contain no explicit d e p e n d e n c e on t h e time i, w e h a v e ^ / ( ί ο , Pio)/(to, P20)

§16

Dynamical

Derivation

of the Transport

Equation

71

F r o m this, w e e x p r e s s the derivative with r e s p e c t to pi in t e r m s of t h o s e with r e s p e c t to ri, r2 and p2, and substitute in (16.12). T h e t e r m containing the derivative 2 d i s a p p e a r s w h e n t h e integral is t r a n s f o r m e d to a surface integral in m o m e n tum s p a c e . W e t h e n find

d/dp

3

3

C ( / ( t , pO) = J v r le· £ { / ( t 0, Ριο)/(ίο, P20)} d x d p2,

(16.14)

with t h e relative velocity of t h e particles v r ie= vi - v 2, taking into a c c o u n t the fact t h a t pio and p2o (and t h e r e f o r e t h e w h o l e e x p r e s s i o n in t h e b r a c e s ) d e p e n d on Γι and r2 only t h r o u g h t h e difference r = ri - r 2. Replacing r = (x, y, z) b y cylindrical polar and t h e c o o r d i n a t e s ζ, ρ, φ with t h e z-axis along v r i,e w e h a v e v r ie. d/dr = u rid/dz, e integration o v e r dz c o n v e r t s (16.14) i n t o t

3 C ( / ( i , pO) = J [/(to, Pio)/(io, P2o)]z=-oou r pe ldp d

(16.15)

W e n o w u s e t h e fact that ρ ΐ0 and p2o are t h e initial (at time i 0) m o m e n t a of particles w h i c h at t h e final instant t h a v e m o m e n t a pi and p2. If at the final instant - z2 = - oo, it is clear t h a t at the initial instant the particles w e r e " e v e n z = Ζχ f u r t h e r " a p a r t , i.e. t h e r e h a s b e e n n o collision. In this c a s e , t h e r e f o r e , the initial and final m o m e n t a are t h e s a m e : Pio = P i ,

P2o = P2

for

ζ = -oo.

If =z -h oo, p10 and p20 act as the initial m o m e n t a for t h e collision which gives t h e particles m o m e n t a pi and p2; in this c a s e , w e write Ριο = ρί(ρ),

P2o = pKp)

for

z = +oo.

T h e s e are functions of t h e c o o r d i n a t e p, which acts as the i m p a c t p a r a m e t e r for t h e collision. T h e p r o d u c t ρ dp d

d/v t o b e satisfied: at the instant i 0, t h e distance b e t w e e n t h e particles m u s t b e large in c o m p a r i s o n with the range d of t h e f o r c e s . T h e difference ί - ί 0, h o w e v e r , m a y b e so c h o s e n as to satisfy also t h e condition t-to
72

Kinetic

Theory

of

Gases

periods of possible time variation of the distribution function. T h e change in this function during the time t - i 0 will t h e n b e relatively small and m a y b e neglected. F r o m these considerations, w e obtain the final form of the integral (16.15):

3 C ( / ( i , p,)) = j {f(t, Ρί)/(ί, pD " fit, p , ) / ( t , p 2) b r leda d p2,

(16.16)

which agrees with the B o l t z m a n n collision integral (3.9).

§ 17. The t r a n s p o r t equation including three-particle collisions T o find t h e first correction t e r m s to the B o l t z m a n n equation, w e m u s t go b a c k to the points in § 16 w h e r e t e r m s w e r e neglected, and increase the a c c u r a c y of t h e calculations by one further order of magnitude relative to( )3the g a s e o u s n e s s w e r e omitted in p a r a m e t e r . First of all, t e r m s containing the triple correlation / (16.9), and three-atom collisions w e r e t h e r e b y left out of consideration. M o r e o v e r , in converting the collision integral (16.12) to t h e final form (16.16), w e neglected t h e variation of the distribution function over distances ~ d and times ~ div ; the pair collisions w e r e t h e r e b y regarded as " l o c a l " events occurring at a single point. W e m u s t n o w t a k e b o t h t h e s e corrections into a c c o u n t : three-particle collisions, and the " n o n - l o c a l n e s s " of pair collisions. ( )2s e q u e(n c3e ) of equations w a s te rm ina ted at the In the first approximation, the ( )4 second equation, which relates / and (/)3 . In (t h4e )s e c o n d approximation, we must ( ) 3 go t o the third equation, which relates / and / , omitting t h e / t e r m s in t h e s a m e w a y as t h e / t e r m s w e r e omitted in (16.9) in t h e first approximation. T h e e q u a t i o n then becomes df°\t,

T i , r 2, T 3) / d i = 0,

(17.1)

( 2 )

corresponding to the earlier equation (16.10) for / ; the variables τχ, τ 2, τ 3 in (17.1) are a s s u m e d to vary with time according to t h e equations of motion in t h e three-body p r o b l e m ; a pair interaction b e t w e e n particles is again a s s u m e d . t With the statistical i n d e p e n d e n c e of t h e particles before t h e collision, t h e solution of (17.1) is

(3)

( 1 )

(1)

(1)

/ ( t , tu T2, τ 3) = / ( i 0, τ 1)0/ ( ί ο , τ 2)0/ ( ί ο , r 3 ) 0.

(17.2)

T h e quantities ί ο 5τ ΰ0 (a = 1,2,3) h e r e h a v e t h e s a m e sense as in (16.11); T Û0 = Tao(t, t0, τι, τ 2, τ 3) are t h e values of t h e c o o r d i n a t e s and m o m e n t a which the particles m u s t h a v e at the instant i 0 in o r d e r to r e a c h the specified points τ χ, τ 2, τ 3 in p h a s e s p a c e at the instant i. T h e only difference from (16.11) is that τ α0 = ( r a , 0p eo) are n o w t i n contrast to the first approximation (cf. the first footnote to § 16), this assumption n o w places s o m e limitations o n the generality of the treatment, since in three-body collisions there could be an effect of three-body interactions, i.e. terms of the form ϋ(ηη , r - η ) in the Hamiltonian, which do not reduce 3 to pair interactions.

§17

The Transport

Equation

Including

Three-particle

Collisions

73

t h e initial c o o r d i n a t e s and m o m e n t a in a three-body p r o b l e m , w h i c h will b e s u p p o s e d solved in principle.t T o write d o w n and t r a n s f o r m t h e s u b s e q u e n t f o r m u l a e , it is c o n v e n i e n t t o define an o p e r a t o r §m w h o s e effect on a function of t h e variables τι, τ 2, τ 3 (pertaining to t h e t h r e e particles in t h e t h r e e - b o d y problem) is to c h a n g e t h e s e variables according to r a -» fa = r a0 + (Pao/m)(i - ί 0) , 1 Pa

^

Pa

=

(17.3)

J

PaO-

Similarly, t h e o p e r a t o r S !2 will m a k e this c h a n g e in functions of t h e variables τι and τ 2 w h i c h pertain t o t h e t w o particles in the t w o - b o d y p r o b l e m . A n i m p o r t a n t p r o p e r t y of t h e t r a n s f o r m a t i o n (17.3) is t h a t for times i - i 0> d / t J it is n o longer t i m e - d e p e n d e n t : for such t - i 0, the particles are far a p a r t and m o v e freely with c o n s t a n t velocities v a0 = p ao / w , t h e values of t h e r ao v a r y with time as c o n s t a n t -νβ(0 ί - ίο), and t h e time d e p e n d e n c e in (17.3) d i s a p p e a r s . M o r e o v e r , if t h e r e w e r e n o interaction b e t w e e n t h e particles, t h e t r a n s f o r m a t i o n (17.3) would r e d u c e to an identity: in m o t i o n t h a t is free at all t i m e s , the right-hand sides are identically equal s t h e particles, say particle 1, to t h e left-hand sides. F o r t h e s a m e r e a s o n , if o n e of d o e s n o t interact with particles 2 a n d 3, t h e n $ 1 2 3 -§23; t h e o p e r a t o r s § i2 and $13 t h e n r e d u c e to unity. It is therefore e v i d e n t that t h e o p e r a t o r G123 = $123 - S 12- S,3 - S 23+ 2

(17.4)

is z e r o if any o n e of t h e t h r e e particles d o e s n o t interact with t h e o t h e r t w o , i.e. this o p e r a t o r s e p a r a t e s from t h e functions t h e p a r t that is d u e to t h e interaction of all t h r e e particles ( w h e r e a s t h e t h r e e - b o d y p r o b l e m also includes, as particular c a s e s , pair interactions, with t h e third particle in free motion). With t h e o p e r a t o r S123, (17.2) b e c o m e s

(1) f°\t,

(Ι)

(1)

ru T2, τ 3) = S 1 /2 3( i , ίο, τ , ) / ( ί , ίο, τ 2) / ( ί , ίο, τ 3) ,

(17.5)

where

( 1 ) / ( ί , ίο, τ ) = /

( 1 ) ( ί 0, r - ρ ( ί - i 0) / m , p ) ;

(17.6)

(c)o2m p e n s a t e s t h a t d u e to t h e o p e r a t o r S i 2. (3 )3 t h e shift of the a r g u m e n t r in T h e two-particle distribution / is obtained b y integrating t h e function / with ( 1 τ) , and integration with r e s p e c t to τ and τ gives the r e s p e c t to t h e variables 3 2 3 distribution function / : (2)

( 3 )

/ ( t , τι, r 2) = j / ( ί , η , T2, r 3) d r 3,

( 1 )

(17.7)

( 3 )

/ ( ί , T,) = j / ( ί , Τ„ T2, r 3) d r 2d r 3.

(17.8)

tin practice, of course, an analytical solution of the three-body problem can be given only in a few cases such as that of hard spheres.

Kinetic

74

Theory of

Gases

( )3 s u b s e q u e n t calculation is to ( )2eliminate T h e object of the from t h e s e t w o equations, with / from (17.5), and so e x p r e s s / with t h e n e c e s s a r y a c c u r a c y in t e r m s of T h e n , substituting this e x p r e s s i o n in (16.7), which is itself e x a c t , w e arrive at the t r a n s p o r t equation sought. T o carry out this p r o g r a m m e , w e first of all transform t h e integral (17.8), expressing the o p e r a t o r S i 23 in (17.5) in t e r m s of G i 23 b y (17.4). With t h e e q u a t i o n s ( 1 ) J > > ( i , ίο, τ) dr = J /

(1)

( t 0, τ) d r = l ,

(1)

J S 1/ 2 ( i , ίο, τι)/ (ί, ίο, τ 2) dn dr2 = 1, which are obvious from t h e c o n s e r v a t i o n of the total n u m b e r of molecules, w e obtain

(Ι)

(1)

(1)

(Ι)

/ ( ί , τΟ = / ( ί , ίο, τΟ + 2 I { ( S 12- 1)/ (ί, ίο, τ,)/ (ί, ίο, r 2)}dr 2

0)

(1)

(1)

+ J { G i 2 3 / ( i , ίο, T,)/ (t, ίο, τ 2) / ( ί , ίο, τ 3)} dr 2dr 3.

(17.9)

This equation can be solved for b y successive approximation, bearing in mind that Si2 - 1 is of t h e first order of smallness, and G i 23 of t h e s e c o n d o r d e r ; c o m p a r e ( 1 )estimate (of1 the ) the right-hand side of (16.9). In the zero-order approximation, / ( i , i 0, Ή) = / ( ί , τι). In the n e x t t w o a p p r o x i m a t i o n s ,

(1)

(1)

(1)

( 1 )

/ ( ί , ίο, τ,) = / ( ί , η ) - 2 J { ( S 12- 1)/ (ί, )T/ l ( i , r 2)}dr 2 - j

{Gi23

- 4 ( S 12-

ΐχ5 1 3 + S 23-

(1)

(1)

(1)

2)/ (ί, τ,)/ (ί, τ 2) / ( i , r 3)} d r 2 dr 3.

2 It now remains to substitute this e x p r e s s i o n in (17.5) and t h e n in (17.7), retaining only the t e r m s that are not a b o v e t h e second order of smallness, ~ ( 5 i 2- l ) or ~ G i 2 . 3T h e final result is (2)

(1)

(1)

/ ( ί , τ,, r 2) = S 1/ 2 ( i , τ,)/ (ί, r 2) + J

( 1 )

{lW (i,

(1)

(1)

τ,)/ (ί, τ 2) / ( ί , r 3)} dr 3,

(17.10)

where -R123 = S i 23 — S\2Si3 — S i 2S 23 + S i 2.

(17.11)

It should be e m p h a s i z e d that the order of the S o p e r a t o r s in their p r o d u c t s is significant. T h e o p e r a t o r S Î S22 , 3for e x a m p l e , first c h a n g e s t h e variables τι, τ 2, τ 3- + τι, ί 2( τ 2, τ 3), τ 3(τ 2, τ 3), the functions τ 2, 3( τ 2, τ 3) being determined from t h e e q u a t i o n s of motion of the interacting particles 2 and 3; the variables τι, τ 2, τ 3 are t h e n subjected t o t h e transformation r u τ 2, τ 3- > ί ι ( τ ι , τ 2), τ 2(τι, τ 2), τ 3, w h e r e n o w t h e functions ίι, 2(τι, τ 2) are d e t e r m i n e d b y the problem of t h e motion of a pair of interacting particles 1 and 2.

§ 17

The Transport

Equation

Including

Three-particle

75

Collisions

(1) N e x t , substituting (17.10) in (16.7) a n d changing e v e r y w h e r e from t h e functions t o / = JV/ , w e h a v e t h e t r a n s p o r t e q u a t i o n in t h e f o r m t < 2 )c =)( /+« , c) () / )

È i l k i û + . VÈiikià l 01 OT\

. 1 )2

where

C<(/(t, τ,)) = f

-^ /(i, rO/(t, r)} dr,

2)

C (/(i, τ ι » - I J ^ . J L (3)

12

2

(17.13)

2

)/(i, )/(i, r)} dr dr.

K { 1 /2( ,3iTl

T2

3

2

(17.14)

3

T h e first of t h e s e is t h e pair collision integral, a n d t h e second is t h e t h r e e - b o d y collision integral. L e t u s c o n s i d e r their s t r u c t u r e in m o r e detail. In b o t h integrals, t h e integrands involve functions / t a k e n at different points in s p a c e . I n t h e pair collision integral, t h e effect of this " n o n - l o c a l n e s s " is t o b e separated a s a c o r r e c t i o n t o t h e ordinary (Boltzmann) integral. T o d o s o , w e e x p a n d t h e functions / , w h i c h v a r y only slowly (over d i s t a n c e s ~ d ) , in p o w e r s of 2 Since t h e s e functions in t h e integrand a r e p r e c e d e d b y t h e o p e r a t o r § i 92let u s first c o n s i d e r t h e quantities Ê\2r\ a n d S i 2r 2 into w h i c h t h a t o p e r a t o r t r a n s f o r m s t h e variables Γι a n d r2 . T h e c e n t r e of m a s s Kri + r2 ) of t h e t w o particles m o v e s uniformly in t h e t w o - b o d y p r o b l e m ; t h e o p e r a t o r § l2 therefore leaves this s u m u n c h a n g e d . W e c a n t h u s write

r -ri.

S n r ^ ^ r i + r ^ + Kri-r,))

l

= r i + \(r2- r i ) - 2Sl2 (r2 - r O , S i 2 r = r , + \(τ - r i ) + | S i ( r - Γι).

2

2 2

2

N o w , e x p a n d i n g t h e functions SX2 f(U

ΓΙ, P I ) = / ( * , S , 2r b pio),

S /(i, r 2, p 2) = /(t, S i 2r 2, p 2 ) 0 î2

in p o w e r s of r 2- r i as far as t h e first-order t e r m s , w e obtain

(2)

( 2 )

( 2 )

C (/) = C0 (/)+C,

(/),

(17.15)

where

C (/(i, r „ p , ) ) = Jf ^d r i ·d/-{/(i, r „ p )/(i, r i , p 2) }0 dr, pi (2)

0

I0

2

(17.16)

(2) C, (/(i, r „ p , ) ) = \ j + [fit, r i , p 1 ) 0

·

{ ( r 2- r . ) · £ / ( t , r „ p 1 ) 0/ ( i , n , P20)

fit, r , , P20) - fit, n, p2o)

t h e differentiation with r e s p e c t t o

fit, r , , p,o)l. & 2( r 2 - r , ) } dr2;

' is t a k e n at c o n s t a n t pio o r p 2 . 0

(17.17)

t T h e w a y t o derive the correction terms t o the Boltzmann equation w a s pointed o u t b y Ν . N . B o g o l y u b o v (1946). T h e s e terms were first brought to their final form by M. S . Green (1956).

Kinetic

76

Theory

of

Gases

T h e integral (17.16) is the same as (16.12);t it has b e e n s h o w n in §16 h o w this integral is reducible to the ordinary B o l t z m a n n form by carrying out one of the three integrations with r e s p e c t to spatial c o o r d i n a t e s . Let us now consider the t h r e e - b o d y collision integral (17.14). T o include " n o n l o c a l n e s s " in this integral would be t o go b e y o n d the a s s u m e d a c c u r a c y , since the integral itself is a small correction. H e n c e , in the a r g u m e n t s of the three functions /, all the radius vectors r i , r 2, r 3 are to be taken as the s a m e r j , and m o r e o v e r we must assume that the o p e r a t o r R m does not act on these variables at all:Φ

r

(3) C ( / ( i , r,, p,)) =

J

*

r

â^tfWC' *> P i ) / ( ' > " P2)/(i> ^

p 3)} dr2 dr3.

(17.18)

Let us next examine in s o m e w h a t m o r e detail the structure of the operator R 1 ,2 in 3 order to elucidate the n a t u r e of the collision p r o c e s s e s c o v e r e d b y the integral (17.18). First of all, the o p e r a t o r R 1 ,2 like 3 G i 23 in (17.4), is zero if any one of the t h r e e particles does not interact with the o t h e r s . H o w e v e r , the p r o c e s s e s for which R i 2 3 ^ 0 include not only three-body collisions in the literal s e n s e , but also combinations of several pair collisions. In genuine three-body collisions, three particles c o m e simultaneously into the " s p h e r e of interaction", as s h o w n diagrammatically in Fig. 5a. But the o p e r a t o r j R m is also different from zero for " t h r e e - b o d y i n t e r a c t i o n s " which consist of three successive pair collisions, one pair colliding t w i c e ; Fig. 5b shows diagrammatically an example of such a p r o c e s s , for which S i 3= l , so that fl123= S ! 2 - S 3i 2S 2 . §3 M o r e o v e r , the o p e r a t o r R m also t a k e s a c c o u n t of c a s e s w h e r e one (or more) of the three collisions is " i m a g i n a r y " , i.e. o c c u r s only if the influence of one of the real collisions on the path of the particles is ignored. An example is s h o w n in Fig. 5c, 2

(a)

3

1

2

(b)

3

1

2

3

(c)

FIG. 5.

tit differs in that to is replaced by t in the arguments of the functions / , but the right-hand equation (16.13) is then still valid, since the dependences on η , Γ2, pi, Ρ2 enter only through ρ and p o, which are !0 2 integrals of the motion. tit should be stressed, to avoid misunderstanding, that these simplifications do not imply that the integrand no longer depends on η and n ; a dependence on these variables still occurs through the S operators, which transform the momenta p into functions ρ ( η , Γ2, r , pi, Ρ2, Ρ3). fl β 3 §The operator R123, unlike G123, is zero for a sequence of t w o collisions. For instance, in a process consisting of collisions 2 - 3 and 1-2, w e should have S123 = S12S23, S13 = 1, and s o R123 = 0.

The Virial Expansion

§18

of the Kinetic

Coefficients

77

w h e r e t h e collision 1-3 would o c c u r only if the p a t h of particle 3 w e r e unaffected by its =collision with particle 2;t for this p r o c e s s , S i 23 = S i 2S 23 b u t S I 3t * 1, so that #123

~ Si Si3 +

2

( )2

Si .

2

In the same kind of w a y as the integral C 0 w a s transformed in § 16, one of the six integrations with r e s p e c t to coordinates in the three-body collision integral can b e carried o u t ; the interaction potential Ui2 t h e n no longer a p p e a r s explicitly in the formulae.Φ § 18. The virial expansion of the kinetic coefficients It w a s s h o w n in §§7 and 8 that the thermal conductivity and the viscosity w e r e i n d e p e n d e n t of the gas density (or pressure) b e c a u s e only pair collisions of molecules w e r e t a k e n into a c c o u n t . F o r such collisions, the collision frequency, i.e. the n u m b e r of collisions u n d e r g o n e b y a given molecule per unit time, is proportional to t h e density N , the m e a n free p a t h ! α 1/N, and since η and κ are proportional to Nl they are i n d e p e n d e n t of N . T h e values TJ 0 and κ 0 t h u s obtained are, of c o u r s e , only t h e first t e r m s in e x p a n s i o n s of t h e s e quantities in p o w e r s of the density, called virial expansions. In the n e x t approximation, t h e r e is already a density d e p e n d e n c e in the form

3

κ = κο(1 + a N d ) ,

3

η = η 0( 1 + P N d ) ,

(18.1)

w h e r e d is a p a r a m e t e r of the order of molecular dimensions, and a and β are ( )2 corrections h a v e a twofold origin reflected in dimensionless c o n s t a n t s(. )3T h e s e first the correction t e r m s C and C i in the2 t r a n s p o r t equation. T h r e e - b o d y collisions (whose frequency is proportional to N ) d e c r e a s e the m e a n free p a t h . T h e nonlocalness of the pair collisions m a k e s possible a transfer of m o m e n t u m and energy across a certain surface without its actually being crossed b y the colliding particles: the particles a p p r o a c h to a distance ~ d and t h e n s e p a r a t e , remaining on opposite sides of the surface. This effect increases the m o m e n t u m and energy fluxes. T h e solution of the problem of thermal c o n d u c t i o n or viscosity with the m o r e a c c u r a t e t r a n s p o r t equation (17.12) is to b e b a s e d on the p r o c e d u r e as already described in § § 6 - 8 . W e seek the distribution function in the form / = / 0( 1 + χ / Γ ) , ) ( and 2 ) χΙΤ ~ IIL is a small correction. T h e w h e r e / 0 is the local-equilibrium( 3 function, )3 t h r e e - b o d y collision integral C , like C 0 , is z e r o for the (function / 0. W e m u s t (2) therefore retain the (χ 2 t e) r m in it, and so the integral C 3 is, relative to the B o l t z m a n n integral C , a correction of relative order ~ ( d / f ) . In the integral C i , 2) h o w e v e r , which contains spatial derivatives of the distribution function, it is sufficient to take / = / 0, and in this sense the term C / should b e t a k e n to the 3 side of t h e equation, w h e r e it gives ( )3 a correction 2) left-hand of the same relative order ~ ( d / f ) . T h u s the t w o additional t e r m s C and C\ in the t r a n s p o r t equation give contributions of the same order. § t H a v i n g regard to the sense of action of the S operators, w e must follow the paths of the particles backwards in time. ( )2 ÎThe transformation is carried out in a paper by M. S. Green (Physical Review 1 3 6 , A905,1964). ( 3 ) §This argument clears up any misapprehension which might arise b e c a u s e the integral C i contains derivatives d//dr ~ fIL, which are not found in C , as a result of which the t w o terms might appear to give corrections of different orders of magnitude.

78

Kinetic

Theory

of

Gases

F o r r e f e r e n c e , t h e results of solving the m o r e a c c u r a t e t r a n s p o r t e q u a t i o n for t h e thermal conductivity and the viscosity of t h e g a s , with t h e m o d e l of hard s p h e r e s (diameter d ) are

3 K = K 0( l + 1.2Nd ),

3 η = η 0( 1 + 0 . 3 5 Ν 4 ) ,

(18.2)

w h e r e κ 0 and TJ0 are t h e values obtained in § 10, P r o b l e m 3 (J. V . S e n g e r s 1966).t By making further c o r r e c t i o n s to t h e t r a n s p o r t equation (arising from four-body collisions, etc.), it would in principle b e possible to d e t e r m i n e also t h e s u b s e q u e n t t e r m s in t h e virial e x p a n s i o n of t h e kinetic coefficients. It is i m p o r t a n t to n o t e , h o w e v e r , that t h e s e t e r m s will involve non-integral p o w e r s of N ; the functions K ( N ) and T J ( N ) are found to b e non-analytic at the point Ν = 0 . T o elucidate t h e origin of this b e h a v i o u r , let u s consider t h e c o n v e r g e n c e of the integrals occurring in the t h e o r y ( E . G. D . C o h e n , J. R. D o r f m a n and J. W e i n s t o c k 1963). W e t a k e first the integral in (17.10), w h i c h d e t e r m i n e s the contribution of t h r e e - b o d y collisions t o t h e two-particle distribution function. T h e t y p e of c o n v e r g e n c e of t h e integral is different for t h e different kinds of collision p r o c e s s c o v e r e d b y t h e o p e r a t o r J?i 2. 3L e t u s u s e as an e x a m p l e t h e p r o c e s s as in Fig. 5b. T h e integration is o v e r t h e p h a s e v o l u m e d r 3 with given p h a s e points Ή a n d τ 2. A s the variable in t h e last integration w e leave t h e distance r 3 of particle 3 (at time 2 t) from the point w h e r e t h e collision 2-3 o c c u r r e d . Before this last integration, the integrand will contain the following factors: (1) t h e v o l u m e element r 3 d r 3 for t h e variable r 3; (2) if w e follow the m o t i o n of particle 3 b a c k w a r d s in time, it will b e clear that the direction of its m o m e n t u m p 3 m u s t lie in a certain solid-angle e l e m e n t for the collision 3-2 to o c c u r , n a m e l y 2the2 angle s u b t e n d e d b y t h e region of collision at the distance r 3, giving a factor d / r 3 ; (3) a n o t h e r such factor arises from t h e further limitation on t h e possible directions of t h e m o m e n t u m p 3 i m p o s e d b y t h e condition that the " r e c o i l i n g " particle 2 enters2 t h e s p h e r e of collision with particle 1. T h u s w e get an integral of t h e form / d r 3/ r 3, w h i c h is t o b e t a k e n from r 3~ d t o oo, and w e see that it c o n v e r g e s . Similarly, it c a n b e s h o w n that for collision p r o c e s s e s of other t y p e s t h e c o n v e r g e n c e of t h e integral is e v e n m o r e rapid. T h e contribution of four-body collisions w o u l d b e e x p r e s s e d in (17.10) b y a n integral of similar form, t a k e n o v e r t h e p h a s e s p a c e of particles 3 and 4, again for given τι and r 2. Let us consider a four-body collision of the kind shown in Fig. 6. W e again leave as the last variable of integration the distance r 3. T h e difference from t h e preceding estimate arises from the presence of the integral over dx4 in t h e 2 integrand. This integration clearly gives a contribution proportional to the region of collisions 1-4, that is oc rf . (The second collision 1-2 can again b e ensured by restricting the range of integration over angles p 3. ) It is therefore evident from 2 integral over dx dimensional arguments that 3the 4 m a k e s an additional contribution of t h e o r d e r of p r3d . T h e integral over dr3 t h e n has t h e form / dr3/r3, and so diverges logarithmically at the u p p e r limit. Cutting off the integral at some distance A~\t-t0\/v, we obtain a contribution to the function f& which t T h e calculations, which are e x c e e d i n g l y laborious, are given by Sengers in Lectures in Theoretical Physics, Vol. I X C , Kinetic Theory (ed. b y W. E. Brittin), Gordon & Breach, N e w York, 1967.

§19

Fluctuations

of the Distribution

Function

FIG.

in an Equilibrium

Gas

6.

32 in the contains t h e large logarithm l o g ( A / d ) . This logarithm appears correspondingly 32 correction to the transport coefficients, which is proportional not to (Nd ) but to (Nd ) log(A/d). T h e presence of divergent terms signifies that the four-body collisions cannot be treated separately from those of all higher orders (five-body, etc.). F o r the divergence shows that large r 3 are important, but even when r 3 ~ / particle 3 can collide ) so on. T h e w a y to r e m o v e t h e divergence is t h u s clear: in with s o m e particle 5,( 2and t h e e x p r e s s i o n for / ( ί , T bτ 2) w e m u s t t a k e a c c o u n t of t e r m s relating t o collisions of all o r d e r s , retaining in e a c h order t h e m o s t rapidly divergent integrals. Such a s u m m a t i o n c a n b e carried out, and h a s a foreseeable result: t h e arbitrarily large p a r a m e t e r Λ in t h e 2logarithm is replaced b y the order of magnitude of the m e a n free p a t h , J ~ 1 / N d . t T h u s t h e e x p a n s i o n of t h e t r a n s p o r t coefficients h a s t h e form 3 κ = κ 0[1 + axNd

3

32 + a2(Nd )

log(l/Nd ) + . . . ] ,

(18.3)

and similarly for η .

§ 19. Fluctuations of the distribution function in an equilibrium gas T h e distribution function d e t e r m i n e d b y t h e t r a n s p o r t equation, d e n o t e d in §§ 19 3 20 b y / , gives t h e m e a n n u m b e r s of molecules in the p h a s e v o l u m e element and d x d r ; for a gas in statistical equilibrium, /(Γ) is t h e B o l t z m a n n distribution function / 0 (6.7), i n d e p e n d e n t of time and (if t h e r e is n o external field) of the c o o r d i n a t e s r. It is natural to consider t h e fluctuations of t h e e x a c t microscopic distribution function / ( t , r, Γ) as it varies with time in the motion of the gas particles u n d e r their e x a c t e q u a t i o n s of m o t i o n . t W e define t h e correlation function of t h e fluctuations as («/(ίι,Γ,,ΓΟδΜ,^,Π)), t S e e K. Kawasaki and I. Oppenheim, Physical Review 139, A 1 7 6 3 , 1 9 6 5 . t T h i s topic w a s first discussed by Β. B. K a d o m t s e v ( 1 9 5 7 ) .

(19.1)

80

Kinetic

Theory of

Gases

w h e r e 8f = f - / . In an equilibrium gas, this function d e p e n d s only o n the time difference t = t x- i 2; the averaging is t a k e n with r e s p e c t to o n e of the times tx and i 2, with a fixed value of their difference. Since t h e gas is h o m o g e n e o u s , t h e coordinates ri and r 2 also o c c u r in the correlation function as the difference r = ri - r 2. W e can therefore arbitrarily t a k e i 2 and r 2 as z e r o , and write the correlation function as <δ/(ί,Γ,ΓΟδ/(0,0,Γ 2)>.

(19.2)

Since the gas is isotropic, the d e p e n d e n c e of this function on r in fact r e d u c e s to a d e p e n d e n c e on the magnitude r. If the function (19.2) is k n o w n , integration of it gives the correlation function of the particle n u m b e r density: <δΝ(ί, Γ)δΝ(0,0)) = J <δ/(ί, r, ΓΟδ/ίΟ, 0, Γ2)> dr, dT2.

(19.3)

F o r distances r that are large c o m p a r e d with the m e a n free p a t h J, the density correlation function m a y b e calculated b y the h y d r o d y n a m i c t h e o r y of fluctuations (see SP 2, §88), b u t at distances a kinetic t r e a t m e n t is n e e d e d . It is immediately evident from the definition (19.1) that <δ/(ί, r, Γ 0 δ / ( 0 , 0 , Γ2)> = < δ / ( - ί , - r , Γ 2) δ / ( 0 , 0 , Γ ι » .

(19.4)

T h e correlation function also h a s a m o r e profound s y m m e t r y which c o r r e s p o n d s to that of the equilibrium state of the system u n d e r time reversal. T h e latter p r o c e s s replaces a later time tτ b y an earlier o n e - i , and also replaces t h e values of Γ b y the time-reversed ones Γ . T h e s y m m e t r y in question is therefore e x p r e s s e d b y

τ

Τ

<δ/(ί, r, Γ , ) δ / ( 0 , 0 , Γ2)> = < δ / ( - ί , r, Γ , ) δ / ( 0 , 0 , Γ 2)>.

(19.5)

W h e n t = 0, the function (19.2) relates the fluctuations at different points in p h a s e space at the s a m e instant. B u t the correlations b e t w e e n simultaneous fluctuations are propagated only to distances of the o r d e r of the range of molecular forces, w h e r e a s in the t h e o r y u n d e r consideration such distances are regarded as z e r o , so that the simultaneous-correlation function vanishes. It should b e e m p h a s i z e d that this result is due to the equilibrium n a t u r e of the state relative to which the fluctuations are considered. W e shall see in § 20 that simultaneous fluctuations also are correlated in the non-equilibrium c a s e . In the a b s e n c e of correlation at n o n - z e r o distances, the simultaneous-correlation function r e d u c e s to delta functions, w h o s e coefficient is t h e m e a n s q u a r e fluctuation at o n e point in p h a s e s p a c e ; cf. SP 2, §88. In an ideal equilibrium gas, the m e a n square fluctuation of the distribution function is equal to the m e a n value of the function itself (see SP 1, §113); t h u s <δ/(0, r, Γ , ) δ / ( 0 , 0 , Γ 2)) = /(ΓΟδΟΟδίΓ, - Γ 2).

(19.6)

§19

Fluctuations

of the Distribution

Function

in an Equilibrium

Gas

81

T h e n o n - s i m u l t a n e o u s correlation b e t w e e n fluctuations at different points o c c u r s e v e n in t h e t h e o r y which neglects molecular d i m e n s i o n s . T h a t this correlation necessarily arises is evident from t h e fact that particles which participate at a certain instant in fluctuations at s o m e point in p h a s e s p a c e will already b e at other points at a n y s u b s e q u e n t instant. T h e p r o b l e m of calculating t h e correlation function for t -φ 0 c a n n o t b e solved in a general f o r m , b u t c a n b e r e d u c e d t o t h e solution of particular e q u a t i o n s . T o d o s o , a proposition is n e e d e d from t h e general t h e o r y of quasi-steady fluctuations; see SP 1, §§118 a n d 119. L e t xa(t) b e fluctuating quantities (with z e r o m e a n values). It is a s s u m e d that, if the s y s t ç m is in a non-equilibrium state with values of xa b e y o n d t h e limits of their m e a n fluctuations (but still small), t h e p r o c e s s of relaxation of t h e system to equilibrium is d e s c r i b e d b y linear " e q u a t i o n s of m o t i o n " xa =

(19.7)

- ^ K b X b

>. T h e n w e c a n say that t h e correlation functions of t h e with c o n s t a n t coefficients Afli xa satisfy similar e q u a t i o n s ft (Xa(t)Xc(O))

= - 2*ab(x (t)xM),

b

t > 0,

(19.8)

with c a free suffix. Solving t h e s e e q u a t i o n s for t > 0, w e t h e n find t h e values of t h e functions for t < 0 from t h e s y m m e t r y p r o p e r t y (xa(t)xb(0))

= - <*b(-0*.(0)>,

(19.9)

w h i c h follows from t h e definition of t h e correlation functions. In t h e p r e s e n t c a s e , t h e e q u a t i o n s of m o t i o n (19.7) are r e p r e s e n t e d b y t h e linearized B o l t z m a n n e q u a t i o n for t h e small addition 8f t o t h e equilibrium distribution function / . T h u s t h e correlation function of t h e distribution function m u s t satisfy t h e integro-differential e q u a t i o n ( ^ + v , ~ - ί 1) < δ / ( ί , Γ , Γ 1) δ / ( 0 , 0 , Γ 2) ) = 0

for

t>0,

(19.10)

w h e r e I\ is a linear integral o p e r a t o r acting o n t h e variables Γ, in the function following it: i i g ( r , ) = f w(Tu Γ ; Γί, r)[f[g\

+ f'g'-

flgl - fg] dT d T , d F .

(19.11)

T h e variables Γ 2in (19.10) are free variables. T h e initial condition for t h e equation is t h e value (19.6) of t h e correlation function for t = 0; that for t < 0 is then given b y (19.4), t h e condition (19.5) being automatically satisfied b y t h e result. T h e formulae (19.10), (19.11) a n d (19.4) constitute a set of e q u a t i o n s sufficient in principle for a c o m p l e t e d e t e r m i n a t i o n of t h e correlation function.

82

Kinetic

Theory

of

Gases

W h a t is usually of interest is n o t t h e correlation function itself b u t its F o u r i e r transform with r e s p e c t t o c o o r d i n a t e s a n d t i m e , d e n o t e d b y ( 6 / i 6 / 2) w, kw h e r e t h e suffixes 1 a n d 2 refer t o t h e a r g u m e n t s Γι a n d Γ 2:

I (W)K3I R

at J < δ / ( ί , r , Γ 0 δ / ( 0 , 0 , r 2) ) 6 -

(δ/ιδ/ΛΛ =

-

d x,

(19.12)

t h e spectral function o r spectral correlation function of t h e fluctuations. If a fluctuating function is e x p a n d e d a s a F o u r i e r integral with r e s p e c t t o time a n d c o o r d i n a t e s , t h e m e a n value of t h e p r o d u c t s of its F o u r i e r c o m p o n e n t s is related t o the spectral correlation function b y

4

,

< δ / ^ ( Γ , ) δ / β ((Γν2) > = (2ττ) δ(ω + û> )e(k + ^ ) ( δ / , δ / 2) ω ; | ι

(19.13)

cf. SP 1, §122. It is e a s y t o derive an equation which in principle allows a determination of t h e spectral function of t h e fluctuations without p r e v i o u s calculation of t h e s p a c e - t i m e correlation function. Dividing t h e range of integration with r e s p e c t t o t in (19.12) into t w o p a r t s , from - » t o 0 a n d from 0 t o » , a n d using (19.4), w e h a v e

where

( δ / ι δ / Λ * = (δ/ιδ/ΎΚ + ( e / 2e / , ) Î 2 - k,

(19.14)

| ( w k ) ri 3 (δ/ιβ/ζ)£2 = £ ° dt f ( δ / ( ί , r , Γ 0 δ / ( 0 , 0 , r 2) > e -

-

d x.

(19.15)

T o t h e equation (19.10) w e apply t h e one-sided F o u r i e r transformation (19.15). T h e t e r m s containing derivatives with r e s p e c t t o t a n d r a r e integrated b y p a r t s , using t h e facts t h a t the correlation function m u s t tend t o z e r o a s r -> oo and a s t » , and m u s t b e given b y (19.6) w h e n t = 0. T h e required e q u a t i o n is t h e n found t o b e [i(k. V l - ω ) - ί ι ] ( β / , δ / 2 ) ί 2 = / ( Γ ι ) δ ( Γ ! - Γ 2).

(19.16)

If w e a r e interested in t h e fluctuations of t h e g a s density, a n d n o t in t h o s e of t h e distribution function itself, it is ap p r o p r i at e t o integrate e q u a t i o n (19.16) o v e r d r 2: [i(k. ν - ω ) - ί ] ( δ / ( Π β Ν ) £ 2 = /(Γ).

(19.17)

2 T h e spectral furîction ( δ Ν ) ^ sought is found from t h e solution of this e q u a t i o n b y a single integration, n o t a double 2o n e a s in (19.3). A n o t h e r m e t h o d of finding (8Ν )ω)ί is b a s e d o n t h e relation b e t w e e n t h e density correlation function a n d t h e generalized susceptibility with r e s p e c t t o a w e a k external field of t h e form , ( w k ri ) l / ( i , r ) = l / we k -

;

(19.18)

see SP 2, §86.t If this field c a u s e s a density change ô N û =) | k ~|a(W , k ) U a ,) k t T h i s relation exists only in the equilibrium c a s e .

(19.19)

§ 19

Fluctuations

of the Distribution

Function

in an Equilibrium

83

Gas

t h e n from SP 2 (86.20) t h e spectral correlation function of t h e density is, in the classical limit,

2 ( S N U = (2Τ/ω) im α ( ω , k).

(19.20)

L e t δ / ( ί , τ ) b e the change in the distribution function due to the s a m e field; it satisfies the t r a n s p o r t equation

T h e F o u r i e r c o m p o n e n t s of δ/(ί, r, Γ) are written /•k(0

= -X*k(OU.k,

* is in which the external field is separated as a factor. T h e n the equation for χω [ i ( k . ν - ω) -

îyr) = -

ik.

dfldp.

(19.21)

T h e solution of this equation gives the required spectral correlation function by a single integration:

2 (8N U

= (2Τ/ω) im j

(19.22)


PROBLEMS P R O B L E M 1. Determine the density correlation function in a monatomic gas in equilibrium, neglecting collisions. S O L U T I O N . For a monatomic gas, the quantities Γ are the three c o m p o n e n t s of the momentum p. The s o l u t i o n of ( 1 9 . 1 0 ) for

ii = 0 is

<δ/(ί, r, ρ , ) δ / ( 0 , 0 , ρ )> = / (ρι)δ(Γ- ν , ί ) δ ( ρ , - ρ ),

2

2

and its Fourier c o m p o n e n t is ( δ / , δ / ) κ = 2 π / ( ρ ι ) δ ( ρ ι - ρ ) δ ( ω - k. Vi).

2ω

2

Integration of these expressions with the Maxwellian function / gives as the density correlation function

Ι23

<δΝ(ί, Γ)δΝ(0,0)> = Ν(ηι12πΤγ Γ (8N U

2 2

(1)

e x p ( - rruo l2Tk ).

(2)

, 2/

2

= (N/k)(27rm/T)

2

e x p ( - mr l2Tt\

P R O B L E M 2. The same as Problem 1, but for a collision integral in the form hg = -gh with a ( constant relaxation time τ. S O L U T I O N . Equation ( 1 9 . 1 6 ) reduces to an algebraic equation, from which w e determine (δ/ιδ/ ) οΓιί, 2 and then find from ( 1 9 . 1 4 ) (3)

Kinetic

84

Theory

of

Gases

- 1

The presence of even a small number of collisions changes the asymptotic behaviour of the spectral correlation function of the density at high frequencies,

ku, τ , i.e. for fluctuations with a phase velocity much greater than the thermal speed of the molecules: in this limit,

2

2

(δΝ )

= 2Ν/τω ,

(4)

ω[ 1

i.e. the correlation function decreases with increasing frequency according to a power law, instead of exponentially as in (2).

§ 20. Fluctuations of the distribution function in a non-equilibrium gas L e t a gas be in a steady b u t non-equilibrium state with some distribution function fir, Γ) which satisfies the t r a n s p o r t equation v.a//ar=C(/);

(20.1)

the function / m a y deviate greatly from t h e equilibrium distribution function / 0, and so t h e c o l l i s i o n integral C(f) is not a s s u m e d linearized with r e s p e c t to t h e difference / - / 0. T h e steady non-equilibrium state has to b e maintained in the gas by external interactions: the gas m a y , for e x a m p l e , contain a t e m p e r a t u r e gradient supported b y external s o u r c e s , or it m a y e x e c u t e a steady motion (which does not consist in a motion of the gas as a whole). L e t us seek to calculate the fluctuations of t h e distribution fit, r, Γ) relative to fir, Γ). T h e s e fluctuations will again b e described b y a correlation function (19.1), in which the averaging is carried out in t h e usual w a y with r e s p e c t to time for a given difference t = t\-t2, and t h e correlation function d e p e n d s only on t. Since t h e distribution fir, Γ) is not uniform, h o w e v e r , the correlation function n o w d e p e n d s on the coordinates r, and r 2 separately, and not only on their difference. T h e p r o p e r t y (19.4) b e c o m e s <δ/ 1(ί)δ/ 2(0)) = <δ/ 2(-ί)δ/ι(0)>,

(20.2)

w h e r e / , ( t ) = fit, r„ Γ,), / 2( 0 ) » / ( 0 , r 2, Γ 2). T h e relation (19.5) involving time reversal d o e s not apply in general in the non-equilibrium c a s é . T h e correlation function of t h e distribution function again satisfies the equation (19.10):

l {Ît

+V ' ok

<ί "

)

ιδ

/ >ΐ

(

Ο =

δ

°'

/

2

(

0 3)) '

w h e r e I 4 is t h e linear integral o p e r a t o r (19.11), which acts on the variables I Y t T h e problem of the initial condition on this equation, i.e. the form of the single-time correlation function, is considerably m o r e c o m p l e x t h a n in the equilibrium c a s e , w h e r e it was given simply by (19.6). In a non-equilibrium gas, the single-time correlation function is itself determined from a t r a n s p o r t equation w h o s e form can t T h e use of this equation in the non-equilibrium case is due to M. L a x (1966).

§20

Fluctuations

of the Distribution

Function

in a Non-equilibrium

Gas

85

(2) b e t w e e n the correlation function and the be established b y using the relation ί 2 ) two-particle distribution function f defined in § 16. In a steady state the function / Ο ι , Γι; r 2, Γ 2) , like / ( r , Γ ) , d o e s n o t d e p e n d explicitly on the time. T o derive this relation, we note that, since the volume dr = d^xdT is infinitesimal, it c a n n o t contain m o r e t h a n o n e particle at a time.t H e n c e the m e a n n u m b e r / dr is also the probability that a particle is in the element dr (the probability that there are t w o particles in it at o n c e being a quantity of a higher order of smallness). It follows that t h e m e a n value of the p r o d u c t of the n u m b e r s of particles in the t w o elements dr\ and dr2 is equal to the probability of simultaneously finding one particle in each of t h e m . F o r a given pair of particles this is the p r o d u c t fn dr\ dr2, by the definition of the two-particle distribution function. Since a pair of particles2 can be c h o s e n from the (very large) total n u m b e r of particles in j V ( j V - 1) ~ j V w a y s , we h a v e

2

2

{fxdTX .f2dr2)

= ^ f^dTX

dr2.

T h e equation (f\f2) = N fn thus obtained relates, h o w e v e r , only to different points in p h a s e s p a c e . T h e passage to the limit R I , Γ Ι - * Γ 2, Γ 2 m a k e s it n e c e s s a r y to take into a c c o u n t that, if drx and dr2 coincide, an atom in dri is also in dr2. A relation which allows for this is

2

= # f?l

+ / ι δ ( Γ , - Γ )δ(Γ, - Γ ) :

2

2

(20.4)

2 Δ τ, the first w h e n it is multiplied b y drxdr2 and integrated o v e r some small volume term on t h e right gives a small quantity of t h e s e c o n d order, α ( Δ τ ) , and the term containing the delta functions gives / Δ τ , a first-order quantity. T h u s we h a v e

<({.>«*)>'-• as it should b e , since as far as first-order quantities there c a n only be either no particle or o n e particle in t h e small v o l u m e Δτ. Substituting (20.4) in t h e definition of the single-time correlation function <β/ι(0)δ/ (0)> = < / , ( 0 ) / 2 ( 0 ) > - / , / ,

2

2

w e obtain t h e required relation b e t w e e n it and t h e two-particle distribution function:

2

<δ/,(0)δ/ 2(0)> = Jf f®

- fj2 + / , 8 ( R , - Γ 2) δ ( Γ , - Γ 2) .

(20.5)

In an ideal gas in 2equilibrium, the two-particle distribution function r e d u c e s to the p r o d u c t f?2 = fJ2IN , and (20.5) r e d u c e s t o (19.6). In a n y c a s e , tends to this p r o d u c t as t h e distance b e t w e e n t h e points 1 and 2 i n c r e a s e s , so that < δ / , ( 0 ) δ / 2( 0 ) > ^ 0

as

| r , - r 2| ^ o c .

t T h e derivation which follows is a paraphrase of the argument in SP 1, § 116. PK

10 - G

(20.6)

Kinetic

86

Theory

of

Gases

( )2 T h e two-particle distribution function satisfies a t r a n s p o r t equation analogous to the B o l t z m a n n equation, which could b e derived from equation (16.9) for / in t h e ( )2 s a m e w a y as the e q u a t i o n for t h e single-particle function w a s derived from (16.7).t H e r e , h o w e v e r , w e shall give a derivation of the equation for / analogous to that 2 of t h e B o l t z m a n n equation in § 3 , b a s e d on( )intuitive physical a r g u m e n t s . itself b u t t h e difference W e t a k e as t h e u n k n o w n function n o t / 2
(20.7)

which t e n d s to z e r o as | r i - r 2| - > ° o ; it is the correlation function (20.5) w i t h o u t the last term. This quantity is small in the^usual sense of fluctuation t h e o r y , n a m e l y of the o r d e r of \IN in c o m p a r i s o n with / i / 2. ( )2 In t h e a b s e n c e of collisions, the function φ satisfies an e q u a t i o n which simply e x p r e s s e s Liouville's t h e o r e m — t h e c o n s t a n c y of / along t h e p h a s e trajectory of a pair of particles:

df^_dç_ at

~ dt "

Vl

, β φ +2 V dç_ a

8)

é>r -°-

ri

·

2

T h e c h a n g e in φ as a result of collisions is d u e to p r o c e s s e s of t w o kinds. Collisions of particles 1 and 2 with any other particles, b u t not with e a c h other, c a u s e t h e a p p e a r a n c e , on t h e right of (20.8), of t e r m s 1\φ + ί 2φ , w h e r e î\ and ί 2 are the linear integral o p e r a t o r s (19.11) acting on the variables Γι and Γ 2 respectively. Collisions b e t w e e n the particles 1 and 2 play a special role, causing a simult a n e o u s " j u m p " of b o t h particles from o n e pair of points in p h a s e s p a c e to a n o t h e r pair. E x a c t l y t h e s a m e a r g u m e n t s as w e r e u s e d in t h e derivation of (3.7) give on t h e right of (20.8) a t e r m δ ^ - r 2) C 1 ( 2 / ) , where C 1( 2 / ) = j w(Tu Γ 2; Γί, m

m

- fj2)

dT[ dH;

(20.9)

in this integral, fluctuations m a y b e neglected. T h e factor δ ( ΐ Ί - Γ 2) e x p r e s s e s t h e fact that particles undergoing collisions are at the s a m e point in s p a c e . t T h u s w e h a v e finally t h e e q u a t i o n v, ·

σ Γι

/). + v 2 - 1 * . - f up - Ι2φ = β ( η - r 2) C 1 ( 2 o r 2

(20.10)

Solution of this equation gives, in a c c o r d a n c e with (20.5), the function which acts as t h e initial condition for e q u a t i o n (20.3) at t = 0.§

( )2 t i n § 17 equation (16.9) w a s used only for a specific purpose, namely to eliminate /

from the equation for

I $A further integration of (20.9) over dYi yields the ordinar Boltzmann collision integral. §This result is due to S. V. Gantsevich, V. L. Gurevich and R. Katilyus (1969) and t o Sh. M. Kogan and A. Ya. Shul'man (1969).

§20

Fluctuations

of the Distribution

Function

in a Non-equilibrium

Gas

87

Without the right-hand side, the h o m o g e n e o u s equation (20.10) has the solution Ψ = / θ ΐ Δ / θ 2 + /θ2Δ/οΐ, Δ

>°=!^ # +

Δ Τ +

(20.11)

£ · '. 0

Δ ν

corresponding to arbitrary small changes in the n u m b e r of particles, the temp e r a t u r e , and the m a c r o s c o p i c velocity in the equilibrium distribution / 0. This " s p u r i o u s " solution is, h o w e v e r , excluded by the condition that
3 /(t,r) = ^ | / ( i , r , r ) d x ,

(20.12)

which w e d e n o t e b y the s a m e letter / but without the argument r. T h e c o r r e s p o n d ing correlation function satisfies an equation that differs from (20.3) in not having a t e r m containing t h e derivative with r e s p e c t t o t h e c o o r d i n a t e s : ( ^ + Γ 1. ^ - ί 1) < δ / ( ί , Γ 1) δ / ( 0 , Γ 2) ) = 0

for

i>0;

(20.13)

o n t h e left-hand side, a t e r m h a s b e e n a d d e d which arises from t h e force F acting on the particles in t h e external field. T h e single-time correlation function

1 <δ/(ο, Γ θ δ / ( ο , r2)> = ΛΓ 7 ® ( Γ , , r 2) - 7 ( Γ Ο 7 ( Γ 2) + ϊψ

β(τ, - Γ 2)

^ φ ( Γ „ Γ 2) + Î f f l î δ(Γ, - Γ 2)

(20.14)

satisfies the equation

[Fi - ^

+ F 2· ^

- (f, + Ϊύ]φ(Γι,

Γ 2) = C n M r , , Γ 2)).

(20.15)

88

Kinetic

Theory of

Gases

If the gas is in a closed vessel, this equation is to b e solved with the additional condition that e x p r e s s e s a fixed value (without fluctuations) of the total n u m b e r of particles in the gas: J <δ/(0, Γ,)δ/(0, Γ2)> dr, = J <δ/(0, Γ,)δ/(0, Γ2)> d r 2 = 0.

(20.16)

This condition m u s t b e satisfied in the equilibrium case also, but it is not satisfied by the expression [/(Γ,)/ΊΓ|δ(Γι--Γ 2) which c o r r e s p o n d s to the correlation function (19.6). T h e c o r r e c t expression is obtained b y making u s e of the arbitrary choice (20.11); with the appropriate value of the p a r a m e t e r <δ/(0, Γ,)δ/(0, Γ2)> = ψ / ( Γ , ) δ ( Γ , - Γ 2) - ± / ( Γ , ) / ( Γ 2) .

(20.17)

This correlation function includes a term which d o e s not contain a delta function.