Statistical theory of brownian motion in a moving fluid in the presence of a temperature gradient

Zubarev,

Physica

D. N.

Bashkirov,

39

334-340

A. G.

1968

STATISTICAL THEORY OF BROWNIAN MOTION IN A MOVING FLUID IN THE PRESENCE OF A TEMPERATURE GRADIENT by D. N. ZUBAREV \‘.A. Steklov

Mathematical

Institute

AC. Sci. USSR,

Moscow,

USSR

and A. G. BASHKIROV institute

for Scientific

Information

AC. Sci.

USSR,

Moscow,

USSR

Synopsis The theory of Brownian framework distribution derived “external reflects particles

motion

of the method function

where

force”. with

for a nonequilibrium

the temperature An extra

an additional a

in an inhomogeneous

of one of the authors

diffusion

+ q/kT)/[

is the number

density

system.

The

in the left

fluid is discussed

effect

Fokker-Planck

hand

coefficient

D = kT/[

is also obtained,

of Brownian

The transport

-VT.

where

and

within

for the construction part

equation

thermodiffusion

[ is the friction

the

of the

equation

is considered

term in the right hand part with the kinetic

dissipation

DT = nn/zT(l

gradient

(DNZ)

is

as an

coefficient

~7

for Brownian coefficients

coefficient

and flu

particles.

K irkw ood 1) was the first to analyse in 1946 Brownian motion using the statistical mechanics formalism. Starting from the Liouville equation for the system Brownian particle-fluid, he derived the Fokker-Planck equation for the one-particle distribution function and obtained the relation between the Brownian particle friction coefficient and the time-dependent correlation function of the forces excerted on it. The same results were obtaineds) by means of the Mori local equilibrium distribution method. The Brownian motion in a fluid which a temperature gradient was studied also in ref. 3 and ref. 4, where the diffusion and thermodiffusion coefficients were determined. However, the results of these works are rather contradictory. Besides, the effect of temperature inhomogeneity on the form of the kinetic equation for Brownian particles had not been considered at all. The aim of the present paper is to study the Brownian motion in a fluid with a temperature gradient. We shall obtain the kinetic equation for this case and the transport equation which gives the relation between the Brownian particles and the gradients of the thermodynamical parameters of the system. -

334

-

STATISTICAL

THEORY

OF BROWNIAN

MOTION

IN A MOVING

FLUID

335

We consider in the framework of classical mechanics the system of Brownian particles and fluid with the Hamiltonian #=#1$=@2,

(1)

with

and 3f2

=

C

p”i

i C 2m

+

t4((rt-

X

r,l)

i(
1

where Pa, Ra, M, pi, rg and m are the momentum, position and mass of the cr-th Brownian particle and the i-th fluid particle respectively; V((R, - r6j) is the interaction potential of the Brownian particle with the fluid particle; a((r$ - r,l) is the interaction energy of two fluid particles. The effect of the Brownian particles on the fluid is negligible when their concentration is small. In this case the Brownian particles and the fluid can be considered as two subsystems with Hamiltonians &i and 39’2, respectively. Let us introduce space densities of the energy, momentum and particle number of the Brownian particles and fluid,

1 - 1 fM*) =T +it&4Ix Hi(X)

=

g

5

+

F

v(lx

-

%I)

a@

-

&),

rjl)

W - ri),

[

P(x) = 22P,S(x - Ra), Nl(x) = ; 6(x - RJ.

(2)

p(x) = 22PtW - r0, i N2(4

=

dl

E i

6(x

-

n),

and make use of the equations of motion for them in the form of Poisson brackets with the full Hamiltonian (1). After the necessary calculations, supposing I’ and 21to be of short-range, we obtain the equations of motion in the form of a conservation law Here 9(x) =

Hi(x)

P(x) NM

Hz (4

P(X)

Ndx)

is the matrix of densities (2), Tl@)

h(x)

T2 (4

i2 (4 >

>

(4

336

I>.

is the matrix

N. ZUBAKEV

AND

of the flux densities

A. G. BASHKIKOV

of the subsystems,

S(x)

21(x)

0

cP-(x) =(-$1(x) -F(x) is the matrix exchange

of the source densities,

between

while $1(x)

the subsystems,

0>

describes the rate of energy

and

is the density of a force excerted on the Brownian particles by the fluid. And now we turn to the construction of the distribution function for the system. According to the main idea of ref. 5 it may be represented as a functional of the local values (2) 1 p = Q-l exp

I

-

s

x

daXF(x,

t) .9(x)

L

+p(x,t1 -

9-(x, tl)

where 9 ~9 = &k

+

s

dax

I -cc

t) +

dtr e”“‘-t’ x

a*(%, h) at

*9(x,

t1 -

t) II

1

P&Fik,

and the matrix

elements

(6)

are the following:

temperatures of Brownian $11 = @1(x, f), 9-12 = /32(x) are the inverse particles and the fluid respectively, assuming that the fluid heat capacity is so large that Bs does not change with the variation of pr, i.e. it does not depend on the time: .F

21 =

-_y1(x, L) m=z -Bl(X,

where vi(x, t) is the average 922

=

velocity -Y2(%

t)

where 24x, t) is the field of velocities 9731

=

-41(x,

t)

=% -/91(x,

t) 0(x, q,

of the Brownian =

-/92(x)

vz(x,

particles; q>

in the fluid;

t) /Ll(X, t),

932

=

--y2(4

=

-/h(x)

puz(x),

piand ,LL~are the chemical potentials of subsystems. Assume that the deviation of the distribution from the local equilibrium one is small; this corresponds to the thermodynamical forces F@i, 1782, 8.~1, vey2, 17~ 17~2, Pi - P2, y1 - 72, ~1 - vz, 4%/x aylp, ay2iat, wat being small. In so doing we assume that Ofii N O/&T,V * yl N V. yz,Vv~ II Vv2 and that in the difference yi - ys, ,91(x, t) can be replaced by /&4x). In this approximation, the distribution function (6) can be written down in the form of a linear expansion in thermodynamical forces (making use

STATISTICAL

THEORY

OF BROWNIAN

MOTION

IN A.MOVING

337

FLUID

of the equation of motion (3) for Pgk andintegrating by parts thetermswith div Iik). The same distribution function can be obtained by M c Lenn an’s methods)

0

e%%(% t + h) - B~(x))($d%21)- <$I(% h)h)

+

s -co

dtl -

0 P

-co 0

“l(‘;;l+ tl’(H1(x’ tl) - ‘lb

eat1_

+

f --m

’ -j- h) *P(x. tl) -


_

vl.

p)l)

dtl

_

0

s e”l’

-

~~l(~*

t

+

h)

at1

(6”)

(Nl(x, h) - z)dh . '1

--03

Here we made use of the relation aYl

ag

aI%

aY1

-ov,i~-BIP-~P+tHl=-$VBiN+,(H1-vl.p)

(7) 1 where h and n are average densities of the enthalpy and of the number of particles in the system respectively, 1, pl = Q;’ exp{- j dsXY(X) *9(x)} (8) is the local equilibrium distribution and I&)

= Ii(x) +

12(x),

T(x)

=

Tl(x)

+

T2(4,

jN

=

jl(4

+

j2b)

are the full densities of fluxes in the system. Relation (7) follows from the hydrodynamical equations ,for the ideal fluids). We derive the Fokker-Planck equation for the Brownian particle in the fluid. Integrating both parts of the Liouville equation I ap/at = pf,

p>

(9

338

D. N. ZUBAREV

-.

over coordinates

and momenta

~~(~1~1~)

at

Pl

AND

of all the particles

wwlt)

aR1

+M

A. G. BASHKIROV

aP aP~“l

N1

--

but one, we get dt dP dR’ dP’

1

s

(10)

where f(PlRlt)

= N1 j p dr dp dR’ dP’

and

aL’(lR1 - ril)

sr=-c i

aR1

is the force excerted on the Brownian particle by the fluid. We transform eq. (10) while taking into consideration the form (6”) of the distribution function. We may drop out the correlations between 91 and scalar and tensor quantities, since these correlations vanish due to the Curie principle. We also assume that the thermodynamical forces are weakly changed over the characteristic decay time of the correlation functions of dynamical variables. This makes it possible to put them outside the integral over time. Hence, eq. (10) takes the form af(PlRlt)

af(PlRd) ark

Pl

-at--+M

+
af~dp~~~ (PlRlt)

1

=

where

the

brackets

&-

1

_

1

f(hRlt)

[e(Rl,

I

<.. .>i denote

t) -

averaging

over

Rz, . . . RN,, PI, . . . PN, with the distribution and heat drag kinetic

rl, . . . rN, p1,

function are introduced as

coefficients

5 - -7-i 77, (11)

e(Rl,t)l

. . . pN,

N1plIfl. The friction

(12) The average force
part)

exerted

,

on the Brownian

(13)

where vg = 422 rG( 1 -

e-iiV(r)) dr

0

is the effective system.

volume of the Brownian

particle

and p is the pressure in the

STATISTICAL

THEORY

OF BROWNIAN

To exclude the average velocity

MOTION

of the Brownian

final result we allow for the form of the explicit momentum

I

(10) takes the form Pl

WlRl4

at

particle

FLUID

339

ur(Rrt) from the

dependence

f(PrZ?rt) on the

Pr,

f(PlRlt) - exp +-Then

IN A MOVING

q(PlRlt)

+

.

/hPl I

WPlRl4 ap

+<91>i

aR 1

+T-

;;

=

1-

1

+ kT af(;;R1t)

+ 17 V;

1

Thus, we have shown that the presence of the chemical

f(PlRlt)

potential

* (14

and the

temperature gradients in the system lead to the “external force” <9rii in the left-hand part of the Fokker-Planck equation (nonvanishing only in the inhomogeneous system) and to the dissipation term in the right-hand part, describing the drag of the Brownian particle with the energy flux. There are two kinetic coefficients c and q appearing in the right hand side of eq. (14). We shall obtain the relation between the heat drag coefficient 17 and the thermodiffusion coefficient Dr, similar to Einstein’s relation between c and D. We consider the average force <9r> exerted on the Brownian particle. As was shown earlier 7) s), the average force for stationary processes should be equal to the gradient of the kinetic part of the partial pressure, neglecting the inertial term (of order vs), (9r> Here <. ..> is the averaging

= V(nBkT).

(15)

with the distribution

function

(6*) over all the

coordinates of the system excluding only the coordinate of the Brownian particle while ?&Jris the average density of the Brownian particles. Calculating (*I>, we obtain from (15) the transport equation for the density of the average relative flux of Brownian particles in the fluid (with a constant pressure): IN(R) = NB(U~D is the diffusion

~2) = -DVWB

-

VT DT -. T

coefficient, D = kT/C

and DT, the thermodiffusion

coefficient

DT = nskT

(17)

for Brownian

1 + q/kT

5

Eq. (17) is the well-known

Einstein

relation

particles,

pva/kT . while eq. (18) gives the relation

340

STATISTICAL -____-.-

.~

THEORY _ ~.

OF BROWNIAN

between

the thermodiffusion

friction. From

(17) and (18) we can evaluate

MOTION

coefficient

IN A MOVING

and those

FLUID

of the heat

the order of 17 making

drag and use of the

kinetic theory of gases for the mixture of hard spheres with highly different masses (M > m) and concentrations (1zg Q n), 1~N 0,22kTai/a”

+ % a$zkT,

(19)

assuming that the diameters of the Brownian particle, aB, and the gas molecule, a, are much smaller than the mean free path of gas molecules. It should be pointed out that for gases with small density, the terms of eq. (14) due to inhomogeneity of the system obey the inequality qVT/T > > <9r>i. Thus, (18) can be written in the following form :

(20) This form agrees with that obtained by Nicolisa). It follows from eq. (19) that in the case where CQ > a, eq. (18) can be written in the form DT = nBDg/kT. The authors are grateful to T. N. Khasanovich for a useful discussion of this work, and Miss A. S hu b who kindly took the trouble of helping us with linguistic difficulties Received

9-1-68

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