Particles dispersed in a dilute gas. II. From the Langevin equation to a more general kinetic approach

Chemical Physics 428 (2014) 144–155

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Particles dispersed in a dilute gas. II. From the Langevin equation to a more general kinetic approach Leonardo Ferrari ⇑ Dipartimento di Fisica e Scienze della Terra, Università di Parma, Parco Area delle Scienze, 7/A, I-43124 Parma, Italy

a r t i c l e

i n f o

Article history: Received 14 June 2013 In final form 31 October 2013 Available online 15 November 2013 Keywords: Test particles in gases Langevin equation Fokker–Planck equation Boltzmann equation Unifying kinetic approach Relaxation processes Fluctuating-force autocorrelation function

a b s t r a c t In the attempt to solve the age-old problem of unifying Langevin, Fokker–Planck and Boltzmann theories for test particles in a dilute gas, the Uhlenbeck and Ornstein’s theory relating Langevin and Fokker–Planck equations is critically analyzed. Agreement and discrepancies between such theory and the results following from the Boltzmann one are also examined. It is concluded that the currently assumed form of the fluctuating-force autocorrelation function, which is extremely successful for Brownian particles in dense fluids, cannot generally guarantee an accurate (or acceptable) relaxation law for the mean square velocity components of generic test particles in dilute gases. This difficulty can be overcome in the framework of a more general kinetic approach which is shown to consistently include Langevin, Fokker–Planck, and Boltzmann theories. The advantages of such approach in interpreting experimental results are particularly evident when the test particles move in a (homogeneous) gas in non-equilibrium conditions and when correlations exist between test- and gas-particle velocities. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction

m

When a test particle (of mass m) moves in a fluid, with velocity ~ v at time t, its equation of motion is given by the Newton’s law

m

d~ v ~ ¼ F int ðtÞ  dt

  d~ p ; dt coll

ð1Þ

where ~ v ] is the force (on the particle p=dtÞcoll , with ~ p  m~ F int [i.e. ðd~ itself) which is due to the interactions (collisions) with the fluid particles. As is well known, in the current Langevin theory [1,2] it is assumed that ~ F int ðtÞ can be decomposed into two parts: one, the drag force mk~ v which represents the average effect of the fluid on the particle, k being the friction coefficient, and the other, the fluctuating force ~ LðtÞ, which is assumed to be independent of the particle position ~ v . In other words one writes r and of velocity ~

~ F int ðtÞ ¼ mk~ v þ ~LðtÞ

ð2Þ

and requires that

0; LðtÞi ¼ ~ h~

ð3Þ

the angular bracketts indicating an appropriate average over a large number of test particles in the fluid. LðtÞ, it has become customOwing to the difficulty in knowing ~ ary to regard the Langevin equation (cf. Eqs. (1) and (2)) ⇑ Tel.: +39 0521 905264. E-mail address: leonardo.ferrari@fis.unipr.it 0301-0104/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemphys.2013.10.024

d~ v ¼ mk~ v þ ~LðtÞ dt

ð4Þ

as a differential stochastic equation in which the fluctuating term ~ LðtÞ is defined through its statistical properties. One of them is given by Eq. (3), while another is assumed to be

LðtÞ  ~ Lðt0 Þi ¼ Cdðt  t0 Þ; h~

ð5Þ

where d is the Dirac delta function, and C is a coefficient which is usually determined by imposing the condition that for t ! 1 the test particles attain the thermal equilibrium with the fluid (see the discussion below). Very often (see, for instance, Refs. [3–5]), assumption (5) is replaced by the more general assumption

hLi ðtÞLj ðt 0 Þi ¼ C0 dij dðt  t 0 Þ

ð6Þ

(with C0  C=3 and dij the Kronecker symbol) which expresses also that no correlation exists (in the isotropic case) between different components of the fluctuating force. Further assumptions on higher moments (multitime higher correlation functions) are sometimes also made [1,2,6]. Of course, in order to make a correct use of the statistical procedures based on the Langevin equation, it is necessary to ascertain if – and to what extent – all the above assumptions are really justified in the physical situations one decides to consider. Indeed, Uhlenbeck and Ornstein (UO) in their famous, fundamental article [6], after having listed the assumptions they made on the fluctuating force, wrote ‘‘The justification, or eventually the criticism, of these assumptions must come from a precise, kinetic, theory. We will not

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L. Ferrari / Chemical Physics 428 (2014) 144–155

go into that’’. Their advice, however, seems to have been largely ignored. But, the always more frequent use of the Langevin equation in solving physical problems [1–5] makes particularly urgent to find an adequate answer (at least in some interesting special cases) to the question posed by UO. For this reason in Ref. [7] (hereafter generally referred to as I) we have undertaken an investigation on the limits of validity of the Langevin equation for non-mutually interacting test particles dispersed in a dilute gas, a case which can be treated by the typical methods of the kinetic theory. In such study we have observed that the appropriate average appearing in Eq. (3) must refer to a large number of test particles of velocity ~ v at time t, so that in Eq. (3) the symbol h. . .i should be more conveniently replaced with h. . . ið~v Þ . Moreover, we have established in detail what is the correct averaging procedure able to lead from the Newton’s law (1) to the current Langevin equation (4) (or to a Langevin-like equation). Finally, we have found that, when such appropriate averaging procedure is used, the Langevin equation (4) (with the related condition (3)) can really be written: (i) for heavy test particles dispersed in a background gas of light particles (Rayleigh gas), (ii) in the so-called ‘‘low-velocity limit’’, i.e. for very slow test particles (of any mass) in a gas, (iii) when the test-particle–gas-particle interaction law is supposed to be Maxwellian (Maxwell model). In the first two cases, the result (4) is obtained as a reasonably good approximation, and the (approximate) friction coefficient is given by [7,8]

k ¼ kR 

8 2 M M p 3 m þ M kT

Z

1

F M ðVÞV 5 dV

Z p

rðV; vÞð1  cos vÞ sin v dv;

0

0

ð7Þ where M and V are, respectively, the mass and speed of the gas particles, k is the Boltzmann constant, T is the (constant) gas temperature, and F M ðVÞ is the Maxwellian (equilibrium) distribution of the gas-particle velocities, i.e.,

F M ðVÞ ¼ N



M 2pkT

3=2

exp 

! MV 2 ; 2kT

ð8Þ

N being the gas-particle number density. Moreover, in Eq. (7) v is the scattering angle for test-particle–gas-particle collisions, and rðV; vÞ is the differential cross section for encounters initiated with relative speed g  j~ v j  V. V ~ In the third case, the result (4) is got in an exact way, whatever the test-particle–gas-particle mass ratio may be, and the exact friction coefficient is [7,8]

k ¼ kM  mm

M ; mþM

ð9Þ

where

mm  2pNg

Z p

rðg; vÞð1  cos vÞ sin v dv

ð10Þ

0

is the test-particle–gas-particle momentum-transfer collision frequency, which is g-independent in the Maxwell model. Besides these cases, which refer to very dilute gases (i.e. to large Knudsen numbers), there at least two other (important) physical situations in which the Langevin equation (4) (with the related condition (3)) can be reasonably written. They are: (iv) the well-known case of Brownian particles in a fluid (liquid or dense gas) in which Stokes’ law is expected to hold [9,10] so that one has

k ¼ kS  6pgG Rh =m;

ð11Þ

gG being the viscosity of the fluid, and Rh the hydrodynamic radius of the test particle, and (v) the case of large, heavy particles in a gas in any regime, a case constituting a bridge between Brownian particles (case (iv)) and Rayleigh gases (case (i)). In such case, the friction coefficient is [8,11,12]

k ¼ kF 

kR kS : kR þ kS

ð12Þ

Certainly, the cases (iv) and (v) are particularly important, but to check in a simple way the reliability of the assumptions contained in the UO theory, we shall limit ourselves to considering the cases already treated in I. However, since an initial condition compatible with the low-velocity limit (case (ii)) can be conserved during the relaxation process only for a Rayleigh gas (case (i)), we shall actually consider only the cases (i) and (iii). Our attention will be focused on the meaning, correctness, and consequences of the assumption (5) (or (6)), an issue not considered in I. In this connexion, we shall examine in detail the problem of the relaxation of the mean square components of the test-particle velocity in a dilute gas, and we shall compare results and method of the UO theory (including both Langevin and Fokker–Planck equation) with the corresponding ones following from the Boltzmann equation, as well as with results and method of a theory proposed by us some years ago [13]. From such a comparison, it will be clear in what situations the UO theory becomes inaccurate or unreliable. Moreover, it will be evident that our theory of Ref. [13] (a) allows one to avoid the drawbacks introduced by the sole use of the Langevin equation supplemented with the currently assumed statistical properties of the fluctuating force, and (b) constitutes a consistent unification of Langevin, Fokker–Planck, and Boltzmann theories in the test-particle velocity space. To achieve these goals, we shall be forced to recall and discuss (with extension to the three-dimensional case) all the steps of the procedure originally developed by Uhlenbeck and Ornstein [6] in the one-dimensional case. Moreover, we shall also be forced to recall and use results of papers we published in the past with the intention of clarifying specific issues in the theory of the relaxation processes. This should also aid the readers in the complete, direct understanding of the present article. 2. The Uhlenbeck–Ornstein theory of relaxation processes 2.1. The description based on the Langevin equation In their paper [6], Uhlenbeck and Ornstein preliminarily described the particle Brownian motion through the Langevin equation. Extending their treatment to the three-dimensional case, we consider here an ensemble of (independent) test particles all starting at time t ¼ 0 with the same velocity ~ v 0 . At the subsequent time t, such particles will have the velocity ~ v ðtÞ which, formally, can be obtained by integration of the Langevin equation (4). For the generic i-component (i ¼ x; y; z) of ~ v ðtÞ one then gets

v i ðtÞ ¼ v 0i ekt þ

1 kt e m

Z

t

0

Li ðt 0 Þekt dt

0

ð13Þ

0

and

v 2i ðtÞ ¼ v 20i e2kt þ 2 

Z 0

t

v 0i m

e2kt

Z

t

0

0 0

00

0

Li ðt 0 Þekt dt þ 00

Li ðt 0 ÞLi ðt 00 Þekðt þt Þ dt :

1 2kt e m2

Z

t

dt

0

0

ð14Þ

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L. Ferrari / Chemical Physics 428 (2014) 144–155

One can now use the properties of type (3) and (5) (or (6)), UO assumed in their paper [6], i.e.

hLi ðtÞi½~v 0  ¼ 0 and

ð15Þ

1

hLi ðtÞLi ðt 0 Þi½~v 0  ¼ C0 dðt  t 0 Þ;

ð16Þ

the subscript ½~ v 0  recalling that the average refers to test particles having initially velocity ~ v 0 . Averaging then Eqs. (13) and (14), one obtains

hv i ðtÞi½~v 0  ¼ v 0i ekt

time t, probability which is supposed to be independent of the fact that for t ¼ 0 it is ~ v ¼~ v 0. Expanding now in Taylor series about Dt ¼ 0 the left-hand side of Eq. (23), and in Taylor series about ~ v ¼~ v 0 the integrand in the same equation, we obtain

@ Fð~ v 0; ~ v 0 ; tÞ þ Dt Fð~ v0; ~ v 0 ; tÞ þ    @t " Z ¼

D~ v

Fð~ v 0; ~ v 0 ; tÞWðD~ v; ~ v 0 ; tÞ 

X @ Dv i 0 ðF WÞ @ vi i

# 2 X 1X @2 2 @ þ ðDv i Þ ðF WÞ þ Dv i Dv j 0 0 ðF WÞ þ    dðD~ v Þ: 2 i @v i@v j @ v 02 i i
ð17Þ

and

hv

ð24Þ

2 v0 i ðtÞi½~

¼v

2 2kt 0i e

þ

C0 2m2 k

ð1  e

2kt

Þ:

ð18Þ

At this point, since it must be expected that hv 2i ðtÞi½~v 0  reaches, for t ! 1, its equipartition value, i.e. (cf. Eq. (18))

limhv 2i ðtÞi½~v 0  ¼

t!1

C0 2m2 k

¼

kT ; m

ð20Þ

In conclusion, the UO theory leads to the formula

hv 2i ðtÞi½~v 0  ¼ v 20i e2kt þ

kT ð1  e2kt Þ m

3 hðtÞi½~v 0  ¼ 0 e2kt þ kTð1  e2kt Þ 2

WðD~ v; ~ v 0 ; tÞdðD~ vÞ ¼ 1

ð25Þ

ð21Þ

UðD~ v ÞWðD~ v; ~ v 0 ; tÞdðD~ v Þ ¼ hUðD~ v Þið~v 0 Þ ;

 X @  @ v0; ~ v 0 ; tÞ þ O½ðDtÞ2  ¼  i 0 FhDv i ið~v 0 Þ Fð~ @t @v i !  X 1 @2  2 FhðDv i Þ ið~v 0 Þ þ 2 i @ v 02 i  @2  FhDv i Dv j ið~v 0 Þ 0 0 @v i@v j i
þ

ð22Þ

0  mv 20 =2.

ð26Þ

where hUðD~ v Þið~v 0 Þ is the average (evaluated at the velocity ~ v 0 ) of the generic function UðD~ v Þ, Eq. (24) can be rewritten as

Dt

which describes the relaxation of the mean value of v 2i ði ¼ x; y; zÞ. Consequently, the test-particle mean energy hðtÞi½~v 0  relaxes according to the law

X

ð27Þ

On the other hand, from the Langevin equation (4) one has (for a very small Dt, and neglecting higher powers of Dt)

2.2. The Fokker–Planck equation

v 0i  v i ¼ Dv i ¼ kv i Dt þ

Uhlenbeck and Ornstein completed the description of the onedimensional test-particle relaxation process in velocity space with the derivation of the corresponding Fokker–Planck equation. For our purposes we shall give here a brief sketch of the procedure in the three-dimensional case. By the way, we shall also eliminate some inconsistencies appearing in the UO derivation. Thus, if we indicate with Fð~ v 0; ~ v ; tÞ the velocity distribution function (at time t) of the test particles all having velocity ~ v 0 at time t ¼ 0, the test-particle velocity distribution at the subsequent time t0 ¼ t þ Dt will be

Z D~ v

Fð~ v 0; ~ v ; tÞWðD~ v; ~ v ; tÞdðD~ v Þ:

ð23Þ

In writing this equation we have taken into account that at time t 0 the test-particle velocities are ~ v0 ¼ ~ v þ D~ v (with D~ v in general different for every particle) and that WðD~ v; ~ v ; tÞdðD~ v Þ is the probability of a test-particle velocity change between D~ v and D~ v þ dðD~ v Þ at 1

Z D~ v

C0 ¼ 2mkkT:

Fð~ v0; ~ v 0 ; t þ DtÞ ¼

D~ v

and

ð19Þ

one finds that

with

Taking into account that

Z

1 1 Li ðtÞDt  kv 0i Dt þ Li ðtÞDt: m m

ð28Þ

But, if Li ðtÞ rapidly varies in Dt, it is preferable to replace, in Eq. (28), Li ðtÞ with its mean value in Dt, so that we write, in accord with Uhlenbeck and Ornstein,

Dv i ¼ kv 0i Dt þ

1 m

Z

tþDt

0

Li ðt0 Þdt :

ð29Þ

t

Taking the appropriate average over a large ensemble of test particles of velocity ~ v 0 in accordance with what we said in Section 1, we have (cf. Eq. (3) and Paper I)

hDv i ið~v 0 Þ ¼ kv 0i Dt:

ð30Þ

Similarly, from Eq. (29) and using condition (6) (again with h. . . ið~v 0 Þ in place of h. . .i), we get (cf. Eq. (20)) 2 hðDv i Þ2 ið~v 0 Þ ¼ k2 v 02 i ðDtÞ þ

2kT kDt; m

hDv i Dv j ið~v 0 Þ ¼ k2 v 0i v 0j ðDtÞ2

ðfor i – jÞ:

ð31Þ ð32Þ

In effect, UO assumed that

hLi ðtÞLi ðt 0 Þi½~v 0  ¼ /1 ðt  t0 Þ; where /1 ðfÞ is a function which has a very sharp maximum at f ¼ 0, and whose integral R þ1 /1 ðfÞdf has a value which can be determined through the energy equipartition theo1 rem. Later, however, on the basis of the observation that the duration of a collision is extremely small, so that the collisions themselves can be regarded as instantaneous, it has become customary in the literature to replace this assumption with its particular form (16) (see, for instance, Ref. [2]). We adopt such current choice in this paper.

Moreover, exploiting the further assumption that

hLi ðtÞLj ðt 0 ÞLk ðt 00 Þið~v 0 Þ ¼ 0;

ð33Þ

we find

hDv i Dv j Dv k ið~v 0 Þ ¼ O½ðDtÞ3 :

ð34Þ

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L. Ferrari / Chemical Physics 428 (2014) 144–155

At this point, introducing results (30)–(34) into Eq. (27), neglecting in the same equation all the terms of second and higher order in Dt, and dividing finally the equation by Dt, we obtain (after the inessential replacement of ~ v 0 with ~ v)

X @ @F ¼k @t @v i i



viF þ

 kT @F : m @v i

ð35Þ

On the other hand, since n1 Fð~ v 0; ~ v ; tÞ may be regarded as the transition probability density (n being the (position-independent) testparticle number density), the test-particle velocity distribution f ð~ v ; tÞ, at time t, corresponding to an arbitrary initial distribution f ð~ v ; 0Þ, can be written as

f ð~ v ; tÞ ¼

Z

~ v0

f ð~ v 00 ; 0ÞFð~ v 00 ; ~ v ; tÞd~ v 00 :

ð36Þ

0

So, if we multiply Eq. (35) by f ð~ v 0 ; 0Þ and integrate over the whole ~ v 0 -space, we get

   X @  @f kT @f kT ¼ k div~v ð~ ¼k v if þ v f Þ þ r~2v f @t @v i m @v i m i

ð37Þ

which is the desired Fokker–Planck equation. Of course, its fundamental solution, i.e. the solution satisfying the initial condition

f ð~ v ; 0Þ ¼ ndð~ v ~ v 0Þ

ð38Þ

is just Fð~ v0; ~ v ; tÞ. 2.3. Some remarks 2.3.1. The averaging procedure At this point, also in view of the subsequent discussion, it is necessary to clarify some points relevant to the averaging criteria involved in the Langevin equation and in the related UO theory. As discussed in I, and as already mentioned in Section 1, there is no doubt that the correct form of Eq. (3) is

LðtÞið~v Þ ¼ ~ h~ 0;

ð39Þ

since the average must refer (as the Langevin equation itself) to test particles having velocity ~ v at time t, and subject to a collisional interaction with the fluid particles. Consequently, the average must take into account all the possible collisional events undergone by a large number of test particles of velocity ~ v (at time t). The details of this averaging criterion were given and discussed in Refs. [7,8,13] and will be recalled in Section 5.2 Here we observe that, once the condition (39) is established, if one wants to supplement the Langevin equation (4) with further statistical properties of the fluctuating force, one is naturally led to write conditions again referring to test particles of velocity ~ v . So, it is natural to rewrite Eq. (6) in the form

hLi ðtÞLj ðt 0 Þið~v Þ ¼ C0 dij dðt  t 0 Þ:

ð40Þ

This is the line we have followed in our rederivation of the Fokker– Planck equation (Section 2.2). By contrast, in their paper [6], Uhlenbeck and Ornstein supplemented the Langevin equation (4) with the conditions (15) and (16) (and other multitime conditions) which include an average referring to a large number of test particles having velocity ~ v 0 at time t ¼ 0. Of course, at any time t > 0 such particles have velocities distributed according to Fð~ v0; ~ v ; tÞ, and it must be expected that

Z 1 h~ h~ v 0; ~ v ; tÞd~ v; LðtÞi½~v 0  ¼ LðtÞið~v Þ Fð~ n ~v

ð41Þ

2 Other averaging criteria followed in the past [6,14–16] have been critically examined by us in I.

hLi ðtÞLj ðt 0 Þi½~v 0  ¼

Z

1 hLi ðtÞLj ðt0 Þið~v Þ Fð~ v 0; ~ v ; tÞd~ v: n ~v

ð42Þ

Obviously, when the averages h. . . ið~v Þ have – or are assumed to have – a ~ v -independent value, as it happens in Eqs. (39) and (40), then the same value is taken also by the corresponding averages h. . . i½~v 0  . So, from Eqs. (39) and (40) it follows also that

LðtÞi½~v 0  ¼ ~ h~ 0;

ð43Þ

hLi ðtÞLj ðt 0 Þi½~v 0  ¼ C0 dij dðt  t 0 Þ;

ð44Þ

which correspond to (or include) Eqs. (15) and (16). But, for the reasons already said, these conditions, in spite of their apparent formal identity with Eqs. (39) and (40), should not be directly associated to the Langevin equation (4). Note parenthetically that, while conditions (39) and (40) necessarily imply Eqs. (43) and (44), the vice versa in general does not hold (cf. Eqs. (41) and (42)). We observe moreover that a correct derivation of the Fokker–Planck equation (see Section 2.2) involves the averages h. . . ið~v Þ at ~ v ¼~ v 0 (and not h. . . i½~v 0  ) and that it is conceptually incorrect to evaluate such averages (as Uhlenbeck and Ornstein [6] seem to do) through conditions of type (43) and (44). Also on this point the UO paper [6] appears to be questionable or unclear. 2.3.2. The approximations Another important point, which has to be discussed, is the validity of the approximations made in the derivation of the Fokker–Planck equation (37). We observe, in fact, that the Taylor expansion of the product F W is equivalent to the product of the Taylor expansions of F and W (which is the procedure followed by Chandrasekhar [10]), and that the truncation after the secondorder terms of the expansion of F W corresponds, obviously, to the product of the truncated expansions (after the second-order terms) of F and W, once, in this product, all the higher-order contributions have been dropped. Thus we can pass to analyze, separately, the validity of the truncated expansions of F and W around ~ v ¼~ v 0. First of all, we notice that truncation of F after the second-order terms requires that Fð~ v 0; ~ v ; tÞ varies slowly with ~ v , so that it is possible to approximate F, around ~ v , with a second-degree polynomial form in the Dv i ’s. In other words, F cannot be a ‘‘too peaked’’ function of ~ v when ~ v varies along any (chosen) direction in velocity space. But at t = 0 F coincides with the Dirac delta function (38), a circumstance which excludes that, in principle, F itself can really constitute an accurate representation of the ‘‘true’’ test-particle distribution function at the early times t > 0. Observations of this type (even if much more detailed) were already made by us [17–21] in relation to the derivation of the Fokker–Planck equation (for a Rayleigh gas) from the Boltzmann equation (see Section 3.1). Earlier considerations on the inadequacy of the Fokker–Planck equation in describing the initial stages of the relaxation process undergone by a group of test particles of assigned velocity ~ v 0 were made by Keilson and Storer [22] and Berman [23]. A more subtle problem concerns the truncated expansion of WðD~ v; ~ v ; tÞ. Again the truncation requires that W slowly varies with ~ v , around the chosen value of ~ v . But the Taylor expansion of W around ~ v ¼~ v 0 introduces a further D~ v -dependence of W through ~ v itself. Of course, in order that the expansion can be reliable, the D~ v involved (which, we remember, is due to the collisional interaction with the fluid particles) is expected to be small in magnitude. Since, on the other hand, the integral in Eqs. (23) and (24) is extended over all the conceptually possible D~ v ’s, i.e. for Dv i ði ¼ x; y; zÞ ranging over the interval ð1; þ1Þ, the only possibility to meet both these exigences is to assume that W takes values appreciably different from zero only over a restricted interval of values of jDv i j (i ¼ x; y; z). If we suppose that a convenient

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L. Ferrari / Chemical Physics 428 (2014) 144–155

dv > 0 exists, the situation may be summarized in the following way:

WðD~ v; ~ v þ D~ v ; tÞ  WðD~ v; ~ v ; tÞ for jD~ v j < dv ;

ð45Þ

WðD~ v; ~ v ; tÞ  0 for jD~ v j > dv :

ð46Þ

These conditions rather strictly parallel the assumptions made in Van Kampen’s book [2] to derive the Fokker–Planck equation from the master equation in the one-dimensional case, but here the subscript v has been introduced to signify that, at variance with Van Kampen [2], we intend that dv in general depends on v  j~ v j (cf. Eq. (28)). 2.3.3. Further considerations As we have seen in Section 2.2, for the evaluation of hDv i ið~v Þ ; hDv 2i ið~v Þ , hDv i Dv j ið~v Þ , and hDv i Dv j Dv k ið~v Þ (at ~ v ¼~ v 0 ) in the derivation of the Fokker–Planck equation, use has been made of the Langevin equation (4) as well as of the conditions (3) and (6), i.e. (39) and (40), supplementing the Langevin equation itself. Consequently, it is not particularly surprising to find that, as it really happens, the Fokker–Planck equation yields the same results (of Section 2.1) directly obtained from the Langevin equation itself under the conditions (15) and (16) (i.e. (43) and (44)) which are a direct consequence of conditions (39) and (40). We do not want to enter here into the problem of the equivalence of the two formulations [2]. Rather, we prefer to observe that, in spite of the fact that the assumed condition (40), and therefore (44) (i.e. (16)), imposes the relaxation of hv 2i i½~v 0  ði ¼ x; y; zÞ to its thermal value, it is not guaranteed in any way that the resulting temporal law of tendency to equilibrium is the correct one in all the physical situations one is allowed to devise. More in general, given the number of seemingly plausible, but unproved, assumptions used to arrive at the results of Section 2.1 (see also Section 1), we ask: are these results really correct (or at least reasonably accurate) in any situation in which the Langevin equation can actually be written? To answer this question, we pass to consider (as anticipated in Section 1) the cases of the Rayleigh gas and of the Maxwell interaction model in the framework of the Boltzmann equation.

As is well known, the behavior of test particles (of number density n  N) uniformly dispersed in a dilute gas of atoms in thermal equilibrium, can be well described by the (linear) Boltzmann equation

ð47Þ

V 0 are, respectively, the velocities of a test particle and where ~ v 0 and ~ of a gas particle after a collision initiated with the corresponding V. Moreover, v and g are the polar angles of vector velocities ~ v and ~ ~ V0  ~ V ~ VÞ is given by Eq. (8). g0  ~ v 0 with respect to ~g  ~ v , and F M ð~ Note that, since we suppose here that the collisions are elastic, it is g 0 ¼ g. Note also that from the momentum conservation in a collision one has

~ v0  ~ v ¼ l1 ð~g 0  ~g Þ with

ð48Þ

l1  M=ðm þ MÞ.

3.1. The Rayleigh gas In the particular case in which M=m  1 (and therefore

l1  1), one can replace in the Boltzmann equation (47) the collision integral Jðf Þ with its approximate differential form

ð49Þ

correct up to (and including) the first-order terms in l1 [24,19], or, alternatively, in M=m [25,26]. The resulting approximate kinetic equation is formally just the Fokker–Planck equation (37), but now k is that given by Eq. (7) (where l1 has to be replaced with M=m when M=m itself is chosen as small parameter in place of l1 ), and not a generic (assumed) friction coefficient as in the UO theory. Once the average value hað~ v Þit , at time t, of a quantity að~ v Þ is defined as

hað~ v Þit 

Z 1 að~ v Þf ð~ v ; tÞd~ v; n ~v

ð50Þ

if, according to the standard procedure, we multiply Eq. (37) by v i , v 2i , and , and integrate over the whole velocity space, we obtain the relaxation equations for the average quantities hv i it , hv 2i it and hit . Such equations are (i ¼ x; y; z)

d hv i it ¼ khv i it ; dt

ð51Þ

  d 2 kT ; hv i it ¼ 2k hv 2i it  dt m

ð52Þ

  d 3 hit ¼ 2k hit  kT : dt 2

ð53Þ

Their solutions, when all the test particles have initially velocity ~ v 0, i.e. when f ð~ v ; 0Þ is given by Eq. (38), are given by Eqs. (17), (21) and (22), as expected. But, as it follows from Eqs. (21) and (52), the relaxation of each hv 2i it ði ¼ x; y; zÞ proceeds independently from the other hv 2j it ’s (j – i), so that no transient influence on hv 2i it is due to a possible (initial) hv 2j i0 – 0 (with j – i). This is not particularly surprising if one considers that the fundamental solution Fð~ v0; ~ v ; tÞ of Eq. (37) [10] may be written as

Y Fð~ v 0; ~ v ; tÞ ¼ n F i ðv 0i ; v i ; tÞ

ð54Þ

i

with

3. Results following from the Boltzmann equation

Z Z 2p Z ph i @f V V 0 Þ  f ð~ VÞ ¼ Jðf Þ  d~ dg f ð~ v 0 ; tÞF M ð~ v ; tÞF M ð~ @t ~ 0 0 V  g rðg; vÞ sin v dv;

  kT J 1 ðf Þ  k div~v ð~ v f Þ þ r~2v f m

F i ðv 0i ; v i ; tÞ 



1=2 m 2pkTð1  e2kt Þ 2

mðv i  v 0i ekt Þ  exp  2kTð1  e2kt Þ

! ;

ð55Þ

i.e. as the product of three velocity distributions (in the one-dimensional case) which evolve independently one from the others. Con^ the evolution of sequently, if we take, for instance, ~ v 0 ¼ v 0z k, F x ð0; v x ; tÞ and F y ð0; v y ; tÞ towards their Maxwellian (equilibrium) form is always the same whatever the value of v 0z may be. Of course, this description of the relaxation process conflicts with the fact that the test (heavy) particles collide with the gas particles, and that, in such collisions, the heavy particles are (little) deflected. Therefore, in contrast with Eqs. (54) and (55), if it is v 0z – 0, some influence of the initial motion (along the z-axis) on the distributions of the v x - and v y -components has to be expected. Circumstances of this type were already put in evidence in Refs. [27,28] where the drawbacks of the replacement of the Boltzmann collision operator Jðf Þ with the Fokker–Planck operator J 1 ðf Þ were widely discussed. On the other hand, as already said, the Fokker–Planck equation (37) leads to the same results yielded by the Langevin equation (4) with the assumed statistical properties (43) and (44) (UO theory). So, the same circumstances indicate that the UO theory leads to some unsatisfactory results even in the Rayleigh-gas case, viz. in a case in which the Langevin equation (4) has been proved (in I) to

L. Ferrari / Chemical Physics 428 (2014) 144–155

be valid in the current approximation which neglects terms of order l1 (or M=m) with respect to unity. Further clarifications on this point follow from the discussion in the next subsection. 3.2. The Maxwell model When a Maxwellian interaction between test particles and gas particles is assumed, the Boltzmann collision operator Jðf Þ has the following property [26,29–31]:

JðfM ðv ÞLjlþ1=2 ðbv 2 Þv l Y lm0 ðh; uÞÞ ¼ klj fM ðv ÞLlþ1=2 ðbv 2 Þv l Y lm0 ðh; uÞ; j ð56Þ where

   m 3=2 mv 2 fM ðv Þ ¼ n exp  2pkT 2kT

ð57Þ

is the test-particle equilibrium distribution, Llþ1=2 are generalized j Laguerre polynomials (where b  m=2kT), h and u are the polar angles of vector ~ v , and Y lm0 ðh; uÞ are the spherical harmonics 0

0

im u Y lm0 ðh; uÞ ¼ Pm l ðcos hÞe

ðl ¼ 0; 1; 2; . . . ; m0 ¼ l; l þ 1; . . . ; lÞ; ð58Þ

0

Pm l being associated Legendre functions. Moreover, klj are the corresponding eigenvalues given by

Z pn 1  ½1  2l1 ð1  l1 Þð1  cos vÞl=2þj 0 !) 1  l1 ð1  cos vÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðg; vÞ sin v dv; 1  2l1 ð1  l1 Þð1  cos vÞ

klj ¼ 2pNg P l

ð59Þ

where P l are Legendre polynomials. This property of J allows one to easily obtain the evolution equations for the test-particle velocity averages (see Refs. [27,32]). The equations for hv i it ; hv 2i it and hit are found to be ði ¼ x; y; zÞ

d hv i it ¼ k10 hv i it ; dt

ð60Þ

d 2 1 kT hv i ¼ k20 hv 2i it þ ðk20  k01 Þhv 2 it þ k01 ; dt i t 3 m

ð61Þ

  d 3 hit ¼ k01 hit  kT ; dt 2

ð62Þ

k10 ¼ mm l1 ¼ kM ;

k20 ¼ k01 þ 3pNg l21

m k10 ; mþM

Z p

ð1  cos2 vÞrðg; vÞ sin v dv:

ð64Þ

ð65Þ

hv i it ¼ hv i i0 e

kT 1 ð1  ek01 t Þ þ hv 2 i0 ðek01 t  ek20 t Þ; m 3

3 hit ¼ hi0 ek01 t þ kTð1  ek01 t Þ: 2 As one can see,

so that the remarks made in points (a)–(c) must then be dropped and, consequently, the results of the UO description (Langevin equation and the related Fokker–Planck equation), as well as of the Rayleigh-gas theory based on the approximate Boltzmann equation (Fokker–Planck equation) are recovered. Note parenthetically that, when all the test particles have initially velocity ~ v 0 (as in Section 2.1), it is hv i i0  v 0i , hv 2i i0  v 20i ; hi0  0 . The circumstance stressed in point (d) is not surprising if one considers that (1) the Fokker–Planck operator J 1 ðf Þ, for any admissible test-particle–gas-particle interaction [34], is obtained from the Boltzmann collision operator Jðf Þ systematically neglecting terms of order l1 with respect to unity [19], (2) J 1 ðf Þ has the same eigenproperty (56) Jðf Þ has in the special case of the Maxwellian interaction, the only difference being that the eigenvalues of J 1 are, apart from the sign [26],

k lj ¼ ðl þ 2jÞkR ;

ð70Þ

in place of the klj ’s of Eq. (59), (3) in the Maxwell model, the first non-vanishing term of the expansion of klj in powers of l1 is just given by the k lj ’s of Eq. (70) for any l; j ¼ 0; 1; 2; . . . (see Ref. [34]), so that in such model one has (cf. Eqs. (63)–(65))

ð71Þ

and (cf. Eq. (69))

ð72Þ

As a consequence, in an approximate theory, the relaxation process described by all the three terms of Eq. (67) can effectively be observed for heavy test particles in a gas (in the Maxwell model) only if, in the theory itself, terms of order l1 are not neglected with respect to unity [27,19]. So, in such approximation, the shortcoming (pointed out in Section 3.1) in the description of the hv 2i it relaxation can be removed in the said model. 4. Some comments

ð66Þ

;

hv 2i it ¼ hv 2i i0 ek20 t þ

ð69Þ

0

From Eqs. (60)–(62) one has [27,32] k10 t

k20  k01  2k10 ;

k 20 ¼ k 01 ¼ 2k 10 : ð63Þ

k01 ¼ 2ð1  l1 Þk10 ¼ 2

(a) the evolution of h~ v it ; hv 2i it and hit involves altogether three 1 1 1 different time constants (k1 10 , k01 , and k20 ) and not two only [k and ð2kÞ1 ] as in the case of the UO theory; (b) the inverse time constant for the mean-energy relaxation (k01 ) can strongly differ from double the inverse time constant for the mean-velocity relaxation (k10 ) (cf. Eq. (64)), and this is particularly evident in the Lorentz limit m  M, the well-known case of electrons in gases [33]; (c) the third term to the right-hand side of Eq. (67) is a transient contribution which is zero both at t ¼ 0 and for t ! 1; but contributions of this type disappear when summing both Eq. (61) and Eq. (67) over the index i, since all the terms involving k20 mutually cancel (cf. Eqs. (62) and (68)); (d) only in the Rayleigh limit (l1  1) one has [cf. Eqs. (64) and (65)]

k 10 ¼ kR ¼ k10 ¼ kM

with (cf. Eqs. (7), (9) and (59))

149

ð67Þ

ð68Þ

At this point, some comments on the results reported in the preceding Sections have to be made. First of all we observe that the procedure by which the Fokker– Planck equation (37) is derived from the Boltzmann equation (47) in the Rayleigh-gas limit l1  1 [26,17] confirms, in the domain of the dilute gases, the fact that the UO procedure of obtaining the same Eq. (37) must really refer only to situations in which the test-particle velocity change D~ v in collisions is very small. In the

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L. Ferrari / Chemical Physics 428 (2014) 144–155

Wang Chang and Uhlenbeck derivation of J 1 ðf Þ from Jðf Þ [26], in fact, the smallness of the parameter M=m (i.e. of l1 ), and therefore of D~ v ¼~ v0  ~ v (cf. Eq. (48)), is essential to truncate after few terms the expansion of

(with fM given by Eq. (57)) in Taylor series about ~ v0 ¼ ~ v . Note moreover that the near-equilibrium condition

To definitively clarify this issue, a more general formulation of the theory is necessary which, starting from the Newton’s law, can lead to the Langevin equation (when it can really be written), or, more in general, to a Langevin-like equation, and, at the same time, can provide relaxation equations (for the average quantities) agreeing with those following from the Boltzmann equation. Such a theory has been implemented by us some years ago for particles in a gas in external fields [13] and will be briefly recalled and discussed in the present simpler case of a field-free gas in the next Section.

mv 2 MV 2

5. A more general approach

hð~ v 0 ; tÞ  f ð~ v 0 ; tÞfM1 ð~ v 0Þ

ð73Þ

ð74Þ

is also assumed that allows one to expand g rðg; vÞ in Taylor series about v =V ¼ 0, i.e. again in powers of M=m (or of l1 ) (see Refs. [26,17]). The lowest-order non-vanishing contribution [to Jðf Þ] resulting from these expansions is then J 1 ðf Þ. Obviously, if only l21 , but not l1 , is negligible with respect to unity (quasi-Rayleigh gas), more terms in the expansions have to be retained so that the obtained appropriate, approximate collision operator J 1;2 ðf Þ has a fourth-order differential form correct up to (and including) the terms of order l21 (see Ref. [19]). Of course, for increasingly larger values of l1 such a procedure becomes rapidly impracticable or even impossible, as in the case of the Lorentz gas (m=M  1, viz. l1  1). In this case the vector ~ v rotates in a collision without suffering appreciable variations in magnitude. So, D~ v is very often large, but dv  v 0  v is very small. Only in the speed (or energy) space, therefore, a Fokker–Planck equation can then be built up [35–37] which corresponds to the evolution equation (following from the Boltzmann equation) for the isotropic part of the test-particle velocity distribution (see Refs. [38–40]). Returning now to the main subject of this paper, we observe that the Maxwell interaction model gives us the occasion for an interesting reflection. We recall, in fact, that in such model, for test particles in a dilute gas (see I) (1) the Langevin equation (4) can be exactly derived from the Newton’s law for any value of the test-particle–gas-particle mass ratio, (2) the (~ v -independent) friction coefficient k (Eq. (9)) is explicitly obtained, (3) the fluctuating force (due to collisions) is ~ v -independent since, as is well known, the test-particle–gas-particle collision frequency is constant in the model; moreover, the same force is position-independent, since we are tacitly considering in this paper test particles in a uniform gas, (4) the condition (39) (and therefore (43)) is automatically (and exactly) satisfied owing to the above point (1), (5) the supplementary condition (44) (coming from (40)) can be imposed to get the correct value of hv 2i ðtÞi½~v 0  for t ! 1. As one sees, all the assumptions underlying the UO theory are satisfied. In spite of this, as it follows from point (b) of Section 3.2, the temporal relaxation of hðtÞi½~v 0  yielded by the UO theory is generally incorrect, unless the Rayleigh-gas limit is considered. In addition, even in this limit, the UO relaxation of hv 2i ðtÞi½~v 0  is in some respect unsatisfactory, as already said in Section 3.1. This follows from the fact that from Eq. (13) only integer multiples of the inverse time-constant k can rule the relaxation of hv pi ðtÞi½~v 0  (with integer p), and this can really happen only in the Rayleigh-gas limit (cf. Eq. (70)). One can reasonably suspect that the failure of the UO theory in describing the above relaxation processes is due to some inadequacy of the currently assumed statistical properties of the fluctuating force in taking into account the test-particle collisions, a drawback certainly not suffered by the theory based on the Boltzmann equation (47) when the test particles are dispersed in a dilute gas.

5.1. The general evolution equation As observed previously [13], if Að~ v Þ  Aðv x ; v y ; v z Þ is a scalar ~ ~v A, we obtain the quantity, taking the dot product of Eq. (1) by r equation

dA ¼ dt

  dA ; dt coll

ð75Þ

which simply affirms that the rate of change of A is (obviously) that produced by the collisions of the test particles with the gas particles. Of course, also this equation (as well as the Newton’s law (1)) refers to test particles having velocity ~ v at time t. But, the test particles have in general velocities which, at the generic time t, are arbitrarily distributed in velocity space. As a consequence, if we want to get an evolution equation for hAð~ v Þit , the first step consists in writing the Að~ v Þ-evolution equation for the average test particle of velocity ~ v . The latter equation can be obtained replacing in Eq. (75) the term ðdA=dtÞcoll with its appropriate average calculated according to the following procedure. First of all, we consider a particular collision dynamics for V of the colliding particles are changed which the velocities ~ v and ~ to ~ v 0 and ~ V 0 , respectively. Then, we should average ðdA=dtÞcoll on the (large) number of test particles undergoing the chosen collision dynamics. However, it must be observed that (i) ðdA=dtÞcoll is different from zero only when a collision really occurs, (ii) for test particles dispersed in a dilute gas the collisions are distributed in time, and (iii) in dilute gases the collision time sc is much lesser than the mean time of flight s, so that one is generally allowed to assume that sc ! 0, i.e. that the collisions are instantaneous. In these conditions, it is practically impossible to obtain a reliable ensemble average (of ðdA=dtÞcoll ) different from zero at the considered time t. This difficulty can be overcome observing that every test particle (of velocity ~ v ) suffers, on the average, one collision in a mean time of flight. So, it is possible to replace the searched ensemble average with the temporal average (indicated with an overbar) of ðdA=dtÞcoll over a mean time of flight of a test particle in the gas. In other words, we take [8,11,13]

! dA dt

 coll

1

s

Z s  dA 1 dt ¼ ½Að~ v 0 Þ  Að~ v Þ; dt coll s 0

ð76Þ

where

s ¼ sðgÞ  ½NQ ðgÞg1

ð77Þ

is the mean time of flight relevant to test particles colliding with the chosen dynamics, and

QðgÞ 

Z 2p 0

dg

Z p

rðg; vÞ sin v dv

ð78Þ

0

is the related total scattering cross section. Subsequently, we average ðdA=dtÞcoll on all the possible collision dynamics undergone by test particles of velocity ~ v . Since we are tacitly considering only elastic collisions, so that g 0 ¼ g, from Eq.

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L. Ferrari / Chemical Physics 428 (2014) 144–155

(48) we have that, at fixed ~ v; ~ v 0 depends on g; ~ and V). Therefore, our (ensemble) average is

*

! dA dt

+

 v;g;~ V

coll

1 N

Z

F ð~ V; tÞd~ V

~ V

Z 2p 0

v and g (i.e. on v; g

! Z p dA dg dt 0

Pðv; gÞdv; coll

ð79Þ where

Pðv; gÞdv dg 

rðg; vÞ sin v dv dg

ð80Þ

Q ðgÞ

is the probability of scattering with polar angles between v and v þ dv, and between g and g þ dg, while F ð~ V; tÞ is the (normalized to N) gas-particle velocity distribution which, in principle, could v (see Ref. [13]). also depend on ~ In conclusion, the final, appropriate average value of ðdA=dtÞcoll , for test particles of velocity ~ v , is (cf. Eqs. (76)–(80))

*

! dA dt

+

¼

coll

Z

F ð~ V; tÞd~ V

~ V

v;g;~ V

Z 2p 0

dg

Z p

½Að~ v 0 Þ  Að~ v Þg rðg; vÞ

0

 sin v dv ð81Þ and the equation

dA ¼ dt

*

! dA dt

+

coll

ð82Þ v;g;~ V

is the searched Að~ v Þ-evolution equation for the average test particle of velocity ~ v. Averaging now, at time t, over the whole swarm of test particles (of any velocity), we get (see Ref. [13])

dhAit ¼ dt

**

! dA dt

+

+ ;

coll

v;g;~ V

d~ v ¼ dt

*

! d~ v dt

which is the desired evolution equation for hAð~ v Þit . Note that this equation is valid for any value of the mass ratio m=M and for a large class of test-particle–gas-particle interactions. Note also that, with the exception of the hard-sphere model, the total cross section Q ðgÞ, as defined in Eq. (78), is in general infinite in classical mechanics (see Ref. [41]). This fact, however, does not constitute a difficulty since the Q ðgÞ’s appearing in the time average (cf. Eqs. (76)–(78)) and in the average over the polar angles (cf. Eqs. (78) and (80)) mutually cancel. In this way, in analogy with the Boltzmann equation, only the differential scattering cross section appears in the final result (81) and (82). In any case we point out that, as customary in kinetic theory, also in our equations rðg; vÞ may in general be the differential cross section following from quantum–mechanical calculations or from experiment. We finally observe that the possible divergences in the values of Q ðgÞ are due to the fact that in Eq. (78) angles v ! 0 are also considered. Since angles v lesser than a certain minimum value vmin are not experimentally measurable, it is reasonable to disregard these angles in the evaluation of the integral over v in Eq. (78). This not only leaves the final result (81) and (82) unchanged, but also ensures that the values of the mean time of flight sðgÞ do not tend to zero. 5.2. Connection with Langevin, Fokker–Planck and Boltzmann theories At this point it is convenient to explore the relations between this approach and Langevin, Fokker–Planck, and Boltzmann theories. First of all we observe that if we take Að~ v Þ  v i ; ði ¼ x; y; zÞ, Eq. (82) yields the equations of motion for the average test particle of velocity ~ v , i.e., in vector form,

ð84Þ

: v;g;~ V

coll

When the background gas is assumed to be (and remain) in thermal equilibrium, and (1) the Maxwell interaction model is assumed, or (2) the Rayleigh-gas case is considered, Eq. (84) becomes, after evaluation of the averages involved (see Ref. [8])

d~ v ¼ k~ v: dt

ð85Þ

with k ¼ kM in the Maxwell model, and k ¼ kR in the Rayleigh-gas case. As one sees, Eq. (85) agrees with the Langevin equation (4), once the appropriate average of the same equation (4) over a large number of test particles of velocity ~ v is performed. This implies, obviously, that Eq. (39) (i.e. (3)) really holds, and that

h. . . ið~v Þ  h

coll iv;g;~ V:

ð86Þ

But some difference exists between the above two cases. While in the Maxwell model the Langevin equation (4) is exact, since in such model the result (85) is obtained exactly, in the Rayleigh-gas case the right-hand side of Eq. (85) is only approximate since it is got in the limit l1  1 (see Ref. [8]). Consequently, in such case, the drag force mkR~ v and, therefore, the Langevin equation (4) are only approximate. Moreover, while in the Maxwell interaction model the test-particle–gas-particle collision frequency mðgÞ  s1 ðgÞ is g-independent (and therefore v-independent), so that it is really L as reasonable to regard, as in Eq. (4), the fluctuating force ~ ~ v -independent, in the Rayleigh-gas case, in which more general test-particle–gas-particle interactions can be considered, the ~ v -independence of ~L is in general only due to the approximations one is allowed to make. In such case, in fact, the meaningful v , is the average quantity to evaluate, for test particles of velocity ~ collision frequency3

ð83Þ

t

+

hmðgÞi~V 

Z 2p 0



dw

Z 2p 0

Z p 0

dg

sin /d/

Z p

Z

1

F M ðVÞV 2 dV

0

g rðg; vÞ sin v dv;

ð87Þ

0

V with respect to ~ where w and / are the polar angles of vector ~ v, and

  v v 2 1=2 V ~ g ¼ j~ v j ¼ V 1  2 cos / þ 2 : V V

ð88Þ

As shown in I, hmðgÞi~V does not depend on v in the usual approximation of the Rayleigh-gas limit, so that it is still correct to ignore, in such limit, any possible ~ v -dependence of the fluctuating L. force ~ The above circumstances are confirmed by the fact that, when only l21 (and not l1 ) is neglected with respect to unity (quasi-Rayleigh gas), Eq. (85) can still formally be written, but k must then be replaced with the v-dependent friction coefficient [7].

KQR ðv Þ  kR 

M mv 2 ðkR  nÞ m 2kT

ð89Þ

with

n

 2 Z 1 Z p 8 2 M p l1 F M ðVÞV 7 dV rðV; vÞð1  cos vÞ sin v dv: 15 kT 0 0 ð90Þ

3 As one can see, Eq. (87) involves the total cross section Q ðgÞ (Eq. (78)) which, as already said, diverges classically except that in the hard-sphere model. The reasoning made below on the v-independence of hmðgÞi~V in the Rayleigh-gas limit is anyhow correct. If we want to avoid divergences, we can make a cut-off on the value of the smallest angles v as already said in Section 5.1.

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L. Ferrari / Chemical Physics 428 (2014) 144–155

In such an approximation, moreover, hmðgÞi~V depends in general on so that the fluctuating force is also v-dependent. As a consequence, in place of Eq. (4), we can write the (approximate) Langevin-like equation [7]

v,

m

d~ v ¼ mKQR ðv Þ~ v þ ~Lðv ; tÞ: dt

ð91Þ

Note that in the Maxwell model it is n ¼ kR ¼ kM , KQR ¼ kM , and ~ L is v-independent. So, the usual Langevin equation is recovered. This is in accord with the fact that, in such model, the friction coefficient kM is exact and, therefore, no corrective term is needed. If now, as in Eq. (83), we average Eq. (85) (i.e. (84)) over the whole swarm of test particles, we get Eq. (51) for the Rayleigh gas, and Eq. (60) (cf. Eq. (63)) for the Maxwell model. In these cases, therefore, as regards h~ v it , all the results of the UO theory (Langevin equation and Fokker–Planck equation) and of the approximate (Fokker–Planck) or exact Boltzmann equation are reobtained. We recall finally that in Ref. [13] it is proved that, under the said assumption of the thermal equilibrium for the background gas, the above general Eq. (83) coincides with the hAð~ v Þit -evolution equation following from the Boltzmann equation. Of course, we do not report here the general proof which is given in the said paper. Instead, for a useful illustration of the calculation procedure, and for further considerations, we shall give, in the next Subsection, the explicit derivation of the evolution equation of hv 2i it on the basis of our Eq. (83). 5.3. The evolution equation for hv 2i it

dhv 2i it ¼ dt

! 2

dv i dt

+

+ ð92Þ

; v;g;~ V

coll

t

and to calculate then (cf. Eq. (81))

*

2

dv i dt

!

+ ¼ coll

v;g;~ V

Z

1

F M ðVÞV 2 dV

0



Z 2p

dw

0

Z 2p 0

dg

Z p 0

Z p

Z 2p Z 2p 2 sin /d/ dw ðv 02 i  v i Þdg 0  0  0   3 ¼ 8p2 l1 v 2i 2 þ l1 1  ð1  cos vÞ 2

1 2 2 2 2 2 þ l1 v ð1 þ cos vÞ þ l1 V ð1  cos vÞ: 2 3

Z p

sin / d/

0

2 ðv 02 i  v i Þg rðg; vÞ sin v dv:

ð93Þ

Note that in writing this equation we have supposed that the background gas is and remains in thermal equilibrium. Now, observing that from Eq. (48) one can easily deduce that

v 02i  v 2i ¼ 2l1 v i ðg0i  g i Þ þ l21 ðg0i  g i Þ2 ;

2

dv i dt

!

+ v;g;~ V

coll

1 kT ¼ k20 v 2i þ ðk20  k01 Þv 2 þ k01 : 3 m

ð98Þ

Therefore, the explicit form of Eq. (92) is just the evolution Eq. (61) following from the Boltzmann equation. Another simple case is that of the Rayleigh gas for which, expanding g rðg; vÞ in Taylor series around v =V ¼ 0, one can write, as customary,



v

rðV; vÞ  cos / V

@ ðV rðV; vÞÞ : @V

ð99Þ

Then, once the Eqs. (96) and (99) are introduced into Eq. (93), and the integration over / is carried out, one easily arrives, in the limit l1  1 (i.e. M=m  1) at the result

*

2

dv i dt

!

+

 ¼ 2kR

coll

v;g;~ V

v 2i 

 kT : m

ð100Þ

So, the explicit form of Eq. (92) coincides in this case with Eq. (52) which follows, in turn, from the Fokker–Planck equation. We want to stress that the above results constitute only a particularly interesting example. More in general, in fact, the type of procedure we have followed here (as well as the observations we have made) can be applied to find the explicit form of the evolution equation of any hAð~ v Þit . For instance, all the remaining results of Section 3.1 (in the Rayleigh-gas limit) and of Section 3.2 (in the Maxwell model) can easily be obtained.

ð94Þ

from the rules on the integrations over the azimuthal angle [19,26] we have

6. Discussion

Z 2p

6.1. Our kinetic approach and the Uhlenbeck–Ornstein theory

 2 ðv 02 i  v i Þdg ¼ 2p 2l1 v i ðV i  v i Þð1  cos vÞ 0  

1 1 2 2 þ l21  sin v þ ð1  cos vÞ2 ðV i  v i Þ2 þ g 2 sin v 2 2

ð97Þ

The subsequent integration over V yields finally (cf. Eqs. (63)–(65))

*

g rðg; vÞ  V

To obtain the explicit form of the evolution equation for hv 2i it (i ¼ x; y; z), it is sufficient to put A ¼ v 2i in Eq. (83), which becomes

**

Of course, insertion of Eq. (96) into Eq. (93) can constitute a first step to get an explicit form of Eq. (92). In effect, further progresses in this direction can be achieved if a particular form of rðg; vÞ is chosen (i.e. a particular test-particle–gas-particle interaction is assumed), or if some approximation is made allowing to replace g with V in the functional dependence of r. The simplest choice consists in supposing that g rðg; vÞ is gindependent, i.e. in assuming the Maxwell interaction model. In this case we easily get

ð95Þ

and, using again the said rules, Z 2p Z 2p 2 dw ðv 02 i  v i Þdg 0 0    V ¼ 4p2 2l1 v 2i ð1  cos vÞ cos /  1 v !  "  2 1 V 1 V2 2 2 2 2 cos /  1  sin / þl1  sin v þ ð1  cos vÞ v 2 2 v2

1 1 v 2i þ V 2 sin2 / þ l21 g 2 sin2 v : ð96Þ 2 2

We are now in the position to clarify the issue put in evidence at the end of Section 4. The considerations made in Sections 5.2 and 5.3, in fact, indicate that, at least in the case of test particles homogeneously dispersed in dilute, uniform gases, our general approach (of Section 5) can constitute a solution of the age-old problem [15] of formulating a self-consistent theory which adequately relates Brownian motion theory and kinetic theory of gases (i.e. Langevin, Fokker–Planck, and Boltzmann equations). In particular, as regards the Langevin equation, our approach allows to unequivocally define the correct averaging procedure (cf. Eq. (86)), and so, to explicitly calculate the friction coefficient (see Eqs. (84) and (85)). But, while the statistical property (39) necessarily follows from our approach, the same does not happen for the other property (40) which appears to be substantially extraneous to our formulation, the correct test-particle mean-energy

L. Ferrari / Chemical Physics 428 (2014) 144–155

relaxation process towards the thermal equilibrium being automatically predicted by our theory (cf. Section 5.3). Such a circumstance is due to the fact that, in our theory, the determination of the friction coefficient (and, therefore, of the Langevin equation) and the derivation of the evolution equation for any hAð~ v Þit are unified, since they follow from the sole Eq. (82) which, by an appropriate statistical procedure, takes adequately into account all the possible collision dynamics experienced by test particles of velocity ~ v (see Section 5). On the contrary, in UO theory the Langevin equation is simply postulated, no statistical procedure to arrive at it being established. LðtÞ are disconIn these conditions, the statistical properties of ~ nected from the (statistical) determination of k, and, consequently, they may be (and have been) established only on the basis of plausibility reasons and of agreement with well-known results pertaining to equilibrium conditions. In this way conditions (43) and (44) are chosen which consider test particles having velocity ~ v 0 at time t ¼ 0. This choice allows a rapid evaluation of the relaxation processes (Section 2.1), but leads to some logical inconsistencies in the derivation of the Fokker–Planck equation so that the above conditions have to be replaced with those of Eqs. (39) and (40) (Sections 2.2 and 2.3). In addition, as shown in Section 3.2, condition (44) (i.e. (16)) gives rise to a relaxation law for hit (as well as for hv 2i it ) which is unacceptable unless a Rayleigh gas is considered. It must also be observed that from Eqs. (44) and (20) it follows that

k¼

1 6mkT

Z

þ1

1

0 h~ LðtÞ  ~ Lðt0 Þi½~v 0  dt ;

ð101Þ

i.e. a formula giving the friction coefficient in terms of the autocorrelation function of the fluctuating force (cf. Ref. [42]). It must be noted that, to arrive at Eq. (101), Eq. (20), i.e. the energy equipartition theorem (19), has been employed. This entails that in Eq. (101) we can consistently take t ! 1, viz., we can regard the ensemble average h. . . i½~v 0  as an equilibrium average. Then, obviously, the tdependence of k in Eq. (101) is only apparent (cf. Refs. [43,44]). We want to stress that, in the framework of the UO theory, relation (101) (not explicitly given in the UO paper) solves the problem of the mathematical compatibility of the assumed Eqs. (4), (43) and (44) with the energy equipartition theorem (cf. Section 2.1). However, from the physical point of view, Eq. (101) is subject to caution. In fact, this result can have some physical meaning only when there are well-founded (theoretical and/or experimental) reasons to deem that the Langevin Eq. (4) can really be written, and, in addition, that condition (44) actually guarantees an acceptable mean-energy relaxation process of the test particles. In consequence of our discussion of Sections 1–4, and taking also into account that Eq. (101) can only produce a v-independent k, we must conclude that Eq. (101) has no precise meaning (and should not be used) unless very small values of M=m, i.e. of l1 , are considered. All these observations lead us to the conclusion that, except in the case of a Rayleigh gas, the UO theory based on Eqs. (4), (43) and (44), and leading to Eq. (101), is surely questionable when referred to test particles in dilute gases. On the other hand, the Rayleigh gas just constitutes the physical situation in which the kinetic theory of gases (as well as our theory of Section 5) better approaches the case of Brownian particles in dense fluids,4 i.e. the case in which the UO theory has been proved to be extraordinarily successful [1]. So, it is perhaps not too hazardous to think that just the successes achieved by the Langevin and Fokker–Planck equations 4

It must be noted that, for a Brownian particle in a dense fluid, the Fokker–Planck equation (37) with a friction coefficient of type (101) was obtained, in the same limit M=m  1, from the Liouville equation [45,46].

153

(UO theory) in the study of Brownian motion (in some way also confirmed by the Rayleigh-gas theory of Wang Chang and Uhlenbeck [26]5) can have been the reason discouraging in the past any attempt to investigate the limits of validity of the UO theory itself. 6.2. Role of the Maxwell interaction model In our analysis the Maxwell interaction model between testand gas-particles plays a very important role (see Sections 1, 3.2, 4, 5.2 and 5.3). Such model (also called ‘‘constant mean free-time’’ model [49]), which corresponds to an interaction potential varying as the inverse-fourth power of the distance between a test particle and a gas particle, has often been considered in theoretical studies. The simple properties of the collision operator in this model allow, in fact, to make detailed investigations on the solution of the Boltzmann equation in ionized gases (see, for instance, Refs. [24,49– 53]), as well as on the related ion transport properties (see, for instance, Refs. [32,54–57]). It is known, however, that, for ion-atom interactions, the said model works well only at very low temperatures (and at very low values of E=N, when an electric field ~ E acts on the gas [56]). In different conditions, an inverse-fourth power potential can only constitute a part of the more realistic ion-atom interaction potential [56,58] and the results obtained for the Maxwell model can then be exploited to develop approximate procedures in ion transport theory [56]. On the other hand, the same model cannot in general be employed to describe the interactions between test- and gas-particles in a neutral gas. Nevertheless, also for neutral particles in neutral gases, the Maxwell model has been used to study the particle behavior (see, for instance, Refs. [59,60]) with the aim to obtain simple analytical results giving a first description of the phenomena. At first sight our present analysis could seem to belong, in some respects, to this category of studies. But, in our analysis, the Maxwell model plays a more particular role since (i) only in such model the Langevin equation can exactly be written for particles (of any mass) dispersed in a dilute gas (see I), (ii) only in the same model exact analytical results can be extracted from the linear Boltzmann equation, and (iii) just these results often contrast with those following from the UO theory, putting so in evidence the drawbacks of such theory, which, as already said in Section 4, is unsatisfactory in the Rayleigh-gas approximation ðM=m  1Þ and inapplicable for larger values of M=m. For different interaction models (except in the Rayleigh-gas limit) the Langevin Eq. (4) should not even be written (see I and Section 5.2). Consequently, in the most general case (i.e. for a large class of interaction potentials, and for arbitrary mass ratios) the UO theory fails, and the linear Boltzmann equation (or, equivalently [13], our general kinetic approach) appears to be the only known, reliable theoretical tool able to investigate the behavior of test particles moving in a dilute gas in thermal equilibrium. On the other hand, just the linear Boltzmann equation has been extensively used to successfully study the transport properties of charged particles in gases (see Refs. [56,57,61]). 6.3. Possible applications of our approach In Section 5.1 we have written the evolution Eq. (83) for the average value hAð~ v Þit (at time t) of the generic quantity Að~ v Þ. Such V; tÞ an equation has been obtained for an arbitrary distribution F ð~ of the gas-particle velocities and has been studied, in particular 5 We observe in this regard that the UO theory has recently been extended to charged Brownian particles in a fluid in external fields [47,3] in agreement with the results obtained, for heavy ions in a dilute light gas, from the Rayleigh-gas approximation of the Boltzmann equation [21,48].

154

L. Ferrari / Chemical Physics 428 (2014) 144–155

cases, in Sections 5.2 and 5.3 when F is assumed to be the MaxVÞ. wellian equilibrium distribution F M ð~ The possibility of taking an arbitrary F ð~ V; tÞ allows one, in principle, to apply our kinetic approach to any physical situation in which F is a non-equilibrium stationary distribution F ð~ VÞ, or even V; tÞ. The former a non-equilibrium time-dependent distribution F ð~ case occurs, in practice, when the test particles move, for instance, in a gas (at a constant temperature T) flowing with constant velocity ~ u, so that one can write that

VÞ ¼ N F ð~



M 2pkT

3=2

2! V ~ uÞ Mð~ exp  : 2kT

ð102Þ

The latter case, on the other hand, occurs, for instance, when the test particles move in a background gas having a time-dependent temperature TðtÞ and/or a time-dependent flow velocity ~ uðtÞ. Clearly all these possibilities cover a wide spectrum of experimental situations, so that our approach can be used to interpret the results of experiments provided a pertinent form of F ð~ V; tÞ can be either reasonably devised, or calculated by solving, for instance, the Boltzmann equation for the F ð~ V; tÞ itself (viz., for the pure background gas uninfluenced by the test-particle motion). Of course, all these cases can also be treated in the framework of the current Boltzmann theory6 But our general approach, as stressed in Ref. [13], holds also when there are correlations in velocity space between test- and gas-particles, so that the Boltzmann equation cannot be employed, the hypothesis of molecular chaos being then violated. In other words, we can in general take F ¼ F ð~ v ; tÞ in Eqs. (79) and (81). V; ~ In particular, if one takes

V  n~ F ¼ F M ð~ vÞ ¼ N



M 2pkT

3=2

exp 

2! V  n~ Mð~ vÞ ; 2kT

ð103Þ

with 0 < n < 1, from Eqs. (81)–(83) it is possible, in principle, to get the evolution equation for hAð~ v Þit for large test particles in a gas in any regime, the molecular regime (considered throughout this paper) being attained for n ! 0 and the continuous or hydrodynamic regime being attained for n ! 1. In the latter regime, in fact, it is expected that the Stokes’ law holds, so that the gas particles surrounding the test particle (of velocity ~ v ) move also, on the average, with velocity ~ v (no slip). Of course, the intermediate values of n correspond to transition regimes. Moreover, if the test particles are heavy (with respect to the gas particles), and the near-equilibrium condition (74) is satisfied, it is possible to expand (to the first order in v =V) the product V  n~ F M ð~ v Þg rðg; vÞ in Taylor series about v =V ¼ 0 (cf. Refs. [8,11]). In this way, from Eqs. (81) and (82) one obtains, in general, for Að~ v Þ  v i ði ¼ x; y; zÞ the equation of motion

d~ v ¼ kR~ v ð1  nÞ dt

ð104Þ

for the average (large, heavy) test particle of velocity ~ v (cf. Eq. (85)). On the other hand, by elementary arguments one can conclude [8,11] that it is necessary to take

n¼

kR ; kR þ kS

ð105Þ

in order that the Stokes’ law be satisfied in the continuous regime. Consequently, Eq. (104) may be rewritten as (cf. Eq. (12))

d~ v ¼ kF ~ v; dt

ð106Þ

6 Note that also a Fokker–Planck equation, and the related Langevin equation, were obtained [62] from the Boltzmann equation in the case of Eq. (102).

which formally coincides with Eq. (85). Note that in the Maxwell model this result is obtained, without approximations, for any value of M=m. In complete analogy with the remarks of Section 5.2, we can conclude that, in the case of Eq. (103), our general kinetic approach of Section 5.1 includes the Langevin equation (4) with k ¼ kF . The Fokker–Planck equation (37), and therefore the approximate relaxation Eqs. (51)–(53), can then be obtained (cf. Ref. [12]). Of course, also in this case, Eq. (53) is unsatisfactory for the same reasons given in Section 3.1. Nevertheless, it must be observed that, when taking k ¼ kF , the Langevin equation (4) and the Fokker–Planck equation (37) are at present, as far as we know, the best, simple theoretical tools able to describe non-equilibrium processes undergone by large, heavy particles in a gas in any regime. We recall in this regard that the action of external forces on the test particles can be included in our theory without particular difficulties (see Refs. [8,11,13]) and that our theory then leads to a number of useful results [8,11,12], among which we want to mention the correct generalization of the Cunningham mobility formula for large, heavy particles in gases of any density when a constant force acts on the test particles themselves (see Ref. [8]). In general, therefore, Langevin and Fokker–Planck equations can be used to interpret a variety of experiments, and in particular to examine, for instance, the temporal behavior of aerosol particles liberated in a gas of any density [63] or the behavior of macroions (charged aerosol particles) in a drift tube [56] when the electric field is suddenly removed. Finally we observe that, if the test particles are not sufficiently large (so that it is not certain that the Stokes’ law holds in the continuous regime) and/or are not sufficiently heavy (so that the ordinary approximations of the Rayleigh gas break down), one can try to directly use our Eq. (83) taking F M ð~ v Þ in place of V  n~ V; tÞ in Eq. (81). Obviously, the results will involve the parameter F ð~ n which could then be inferred from experimental data.

7. Conclusions In this paper we have critically analyzed the UO theory for test particles in a fluid. In order to test the correctness of such theory, we have supposed that the fluid is a dilute gas, so that many results and methods of the kinetic theory of gases have been exploited to this end. In this way, not only virtues and defects of the UO theory have been put in evidence, but also the conceptual and practical advantages of the more general approach discussed in Section 5 have been pointed out. Such an approach, in fact, includes Langevin, Fokker–Planck and Boltzmann theories in all cases in which each of these theories is acceptable. The same approach constitutes, therefore, a unique simple tool to interpret experimental results. But, what is much more important, our general kinetic approach can be applied to situations in which the Boltzmann equation cannot be used. In such cases, our approach not only appears to be the framework in which the approximate Langevin and Fokker–Planck theories can be developed for large, heavy particles in a gas of any density [12], but also can constitute the general tool to study the behavior of many types of test particles in a gas in any regime. Our analysis has concerned only homogeneous systems, and therefore has referred to that part of the UO paper which deals with phenomena in velocity space only. It is possible that we treat the complete problem of the test-particle swarm motion in the whole phase space in the near future. Anyhow, we believe that already the present analysis not only can constitute a first answer (in a particular case) to the general problem originally posed by Uhlenbeck and Ornstein [6] (and recalled in Section 1), but also

L. Ferrari / Chemical Physics 428 (2014) 144–155

can give a useful contribution to the formulation of a satisfactory (unified) kinetic theory of particles dispersed in dilute gases. Finally we want to stress that in this paper we have followed the current belief that all the possible quantum effects are incorporated in the collision cross section rðg; vÞ, so that the pure and simple use of quantum-mechanically correct cross sections certainly gives rise to a quantum-mechanically correct kinetic theory [56]. It must be observed, however, that in recent years a quantum version of the linear Boltzmann equation has been derived to describe the quantum motion of a test particle in a background gas. This subject, however, falls beyond the scope of the present paper. For a comprehensive discussion of such subject we defer the readers to the very recent papers by Vacchini and Hornberger [64] and Clark [65]. References [1] W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation, World Scientific, Singapore, 2004. [2] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, NorthHolland, Amsterdam, 1994. [3] J.I. Jiménez-Aquino, M. Romero-Bastida, Phys. Rev. E 76 (2007) 021106. [4] J.I. Jiménez-Aquino, R.M. Velasco, F.J. Uribe, Phys. Rev. E 77 (2008) 051105. [5] A. Baura, M.K. Sen, B.C. Bag, Eur. Phys. J. B 75 (2010) 267. [6] G.E. Uhlenbeck, L.S. Ornstein, Phys. Rev. 36 (1930) 823. [7] L. Ferrari, Chem. Phys. 336 (2007) 27 (ibid. 337 (2007) 177 (Erratum)); Paper I. [8] L. Ferrari, Chem. Phys. 257 (2000) 63. [9] P. Langevin, C.R. Acad. Sci. 146 (1908) 530. [10] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [11] L. Ferrari, Chem. Phys. 274 (2001) 255. [12] L. Ferrari, Chem. Phys. 327 (2006) 506. [13] L. Ferrari, Phys. Rev. E 68 (2003) 021103. [14] M.C. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323. [15] J.G. Kirkwood, J. Chem. Phys. 14 (1946) 180 (ibid. 14 (1946) 347 (Erratum)). [16] P. Mazur, I. Oppenheim, Physica 50 (1970) 241. [17] L. Ferrari, Physica A 115 (1982) 232. [18] L. Ferrari, Physica A 127 (1984) 194. [19] L. Ferrari, Physica A 142 (1987) 441. [20] L. Ferrari, Chem. Phys. 115 (1987) 187. [21] L. Ferrari, Physica A 163 (1990) 596. [22] J. Keilson, J.E. Storer, Q. Appl. Math. 10 (1952) 243. [23] P.R. Berman, Phys. Rev. A 9 (1974) 2170. [24] T. Kihara, Rev. Mod. Phys. 25 (1953) 844. [25] M.S. Green, J. Chem. Phys. 19 (1951) 1036.

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