Stochastic Continuum Mechanics- A Thermodynamic-Limit …

MATHEMATICAL

PERGAMON

COMPUTER MODELLING

Mathematical and Computer Modelling 36 (2002) 889-907 www.elsevier.com/locate/mcllr

Stochastic Continuum Mechanics- A Thermodynamic-Limit-Free Alternative

to Statistical Mechanics: Equilibrium of Isothermal Ideal Isotropic Uniform Fluid

E. MAMONTOV AND M. WILLANDER Laboratory of Physical Elect.ronics and Photonics. IvlC2

School of Physics and Engineering Physics Gothenburg University and Chalmers Ur1iversit.y of Technology

SE-412 96 Gothenburg, Sweden

J. WEILAND Transport Theory Group, Department of Electromagnetics

Chalmers University of Technology SE-412 96 Gotheburg, Sweden

(Received and accepted September 2001)

Abstract-“We do not know how to study finite systems in any clean way; that is, the thermodynamic limit is inevitable.” The lack of the clean way stressed in this sentence of Wsibois and de Leener means the following. To study finite systems, statistical mechanics and kinetic theory offer nothing but the formalism based on the thermodynamic limit (TDL) which in essence disagrees with notion of finite system. The present work proposes the model (almost-equilibrium in a certain sense) enabling one to construct continuous equilibrium descriptions of fluids, discrete multiparticle systems, with no application of TDL. For simplicity, the fluids are assumed to be isothermal, (rheologrcally) ideal, isotropic and uniform The core of the model is nonlinear Ito’s stochastic differential equation (ISDE) for the fluid-particle velocity The continuous equilibrium description is based on the stationary probability density corresponding to this equation. The construction is described as a simple analytical recipe formulated in terms of quadratures and includes the velocity (or momentum) relaxation times which can be determined theoretically, experimentally, or as the results of numerical simulations and depending on the specific nature of the fluid. The recipe can be applied to an extremely wide range of fluids of the above class. It is illustrated by the derivations of the Maxwell-Boltzmann and Fermi-Dirac descriptions for the classical and fermion fluids m arbitrary space domain, bounded or unbounded. In the particular, TDL case, the derived results are in a complete agreement with those of statistical mechanics. @ 2002 Elsevier Science Ltd. All rights reserved.

Keywords-Isothermal ideal isotropic uniform fluid, Thermodynamic-limit-free modelling, Non- linear Ito’s stochastic differential equation. Relaxation time, The Maxwell-Boltzmann and Fermi- Dirac descriptions.

1. INTRODUCTION

“We do not know how to study finite systems in any clean way; that is, the thermodynamic limit

is inevitable” [l, p. 3141. The lack of the clean way stressed in this citation means the following.

To study finite systems, statistical mechanics and kinetic theory offer nothing but the formalism

0895.7177/02/$ - see front matter @ 2002 Elsevier Science Ltd. All rights reserved. Typeset by &~S-TEX

PII: SO895-7177(02)00235-2

890 E. MAMONTOV et al

based on the thermodynamic limit (TDL) ( see Section 3.2) which in essence disagrees with notion

of finite system. There were many of attempts to eliminate this, generally speaking, inadmissible

contradiction (see Section 3.3). The present work is one more endeavor in this direction. In so

doing, it concentrates on the topic which was not in the research focus before. It proposes the

model (almost-equilibrium in a certain sense) enabling one to construct continuous equilibrium

descriptions of fluids, discrete multiparticle systems, with no application of TDL. For simplicity,

the fluids are assumed to be isothermal, (rheologically) ideal, isotropic and uniform. The core of

the model is nonlinear Ito’s stochastic differential equation (ISDE) for the fluid-particle velocity

studied by the authors before in connection with another problem. The continuous equilibrium

description is presented with the analogue of the one-particle distribution function formed as the

product of the stationary probability density corresponding to this ISDE and the fluid concentra-

tion also provided by this density. This construction is described with a simple analytical recipe

formulated in terms of quadratures and includes the velocity (or momentum) relaxation times

which can be determined theoretically, experimentally, or as the results of numerical simulations

and depending on the specific nature of the fluid. The recipe can be applied to an extremely wide

range of fluids of the above class. It is illustrated by the derivations of the Maxwell-Boltzmann

and Fermi-Dirac descriptions for the classical and fermion fluids, respectively, in the arbitrary

space domain. In the particular, TDL case, i.e., when the domain coincides with the whole phys-

ical space and the number of the particles in it is infinite, the derived results are in a complete

agreement with those of statistical mechanics.

Section 2 concerns some characteristics of fluids common in statistical mechanics (SM) and

kinetic theory (KT), the sciences based on TDL. The section presents such reading of these

characteristics which does not use TDL, thereby illustrating that their physical interpretation is

independent. Section 3 focuses on TDL, considers the limitations originated from it and lists some

possible ways to avoid it. The TDL-free almost-equilibrium model for the continuous equilibrium

distributions of the fluid-particle velocity is proposed in Section 4. This section also includes the

examples of application of the model and discusses its details. Section 5 summarizes the practical

procedure to use the proposed model, considers the general picture of the model, and presents

the concluding remarks.

2. INTERPRETATION OF COMMON FLUID CHARACTERISTICS WITHOUT THE THERMODYNAMIC LIMIT

This work considers an isotropic uniform fluid which occupies a domain (i.e., open connected

set) a(t) C R3 with piecewise smooth boundary do(t) where R = (-00,co) and t E R is the

time. The domain can be bounded or unbounded, in particular, can coincide with the whole

space R3. Points in this space are denoted with x E R3. The fluid is assumed to be ideal in the

rheological sense (e.g., [2, Sections 4.1.1 of Vol. I]). This means (e.g., [2, Sections 4.1.2 of Vol. I])

that the matrix of the components of the fluid stress tensor is -II1 where II is the fluid pressure

and I is the 3 x 3 identity matrix.

2.1. The Fluid Distribution Function and the Corresponding Probabilistic Picture

Fluids are multiparticle systems. They are usually modelled in probabilistic terms. In SM

and KT, the corresponding characteristic which, in the simplest approximation, describes the

fluid behavior is its so-called one-particle distribution function. The present section exemplifies

how the distribution function and the related quantities can be introduced in terms of probability

theory and continuum fluid mechanics (CFM), with no involvement of SM or KT.

Continuum fluid mechanics applies a general mathematical definition of the fluid concentration,

i.e., number of the fluid particles per unit volume (e.g., [2, Section 3.1 of Vol. I]). According to

it! the concentration is by definition the limit of the ratio of the number of the particles to the

volume occupied by the part of the fluid containing these particles as this volume tends to zero.

Stochastic Continuum Mechanics 891

In what follows, we consider the fluid characteristics at t 2 to where to E R is the initial time

point. The above definition means that the fluid concentration n = n(t,x) is nonnegative, i.e.,

n(t,x) 2 0, t 2 to, 5 E R(t), (2.1)

and number N(t) of the fluid particles in domain O(t) is expressed as

N(t) J n(t, x) dx > 0, t 2 to. n(t)

(2.2)

REMARK 2.1. We emphasize that both volume J&t) dx of domain n(t) and number N(t) in (2.2)

need not be small or large in any sense which is beyond the above-mentioned definition, in

particular, in any sense associated with statistical issues. If R(t) is bounded, then N(t) is finite.

Relation (2.2) shows that quantity n(t, x) can be presented as follows:

46 x) = N(t)eit, x:), t >to, x E R(t), (2.3)

where

e(4 xl 2 0, t 2 to, x 6 Q(t), (2.4)

J e(t,x) dx = 1, t > to. (2.5)

n(t)

This means that quantity e(t,x), as a function of x at every fixed t > to, is the probability

density of the particle position y (represented with variable x of function Q). This means that y

is a stochastic process, i.e., y = ~(5, t) where [ E Z is an elementary event and E is the space of

elementary events.

It follows from (2.3) that if 0 < N(t) < cm at t 2 to, then e(t,x) = n(t,x)/N(t), x E R(t).

Expression (2.3) points out the probabilistic meaning of the concentration: it is the product of

number N(t) of the fluid particles in the domain and the corresponding particle-position prob-

ability density e(t, x). This probabilistic picture is well known in statistical mechanics (e.g., [3>

(3.82), (3.73); 4, (6.4.13)]).

REMARK 2.2. Importantly, the notion or even availability of probability density ,Q of the particle

position in domain Cl(t) d oes not presume that the number of the particles in the domain at

time t is nonzero, i.e., N(t) > 0. A simple example illustrating this fact is the homogeneous

linear diffusion equation with constant coefficients. In general, the models for p when N(t) = 0 can be obtained as the limit cases of the models for N(t) > 0 as N(t) + 0 (uniformly in t > to).

However, this limit involves a nontrivial mathematical problem.

Similarly, to the particle position y, velocity u of the fluid particle is also assumed to be a

stochastic process, i.e., u = v(<, t), ,( E Z, t > to. The rest of this section focuses on joint

probability density of random variables y(., t) and v(., t) and the related quantities.

Function f of properties

f(GX>U) 2 0, t 2 to, (x,u) E R(t) x R3,

n(t,x) = J

f(t> 5, u) du, t 2 to, x E Wt), R3

(2.6)

(2.7)

where u E R3 represents random velocity w is called the fluid (one-particle) distribution function.

Relations (2.7) and (2.1) h s ow that j(t,x,u) can be presented as follows:

At, x> u) = 46 x:)dt, 5, u), t 2 to, (x,u) E Cl(t) x R3, (2.8)

892 E. MAMONTOV et al.

where function p is such that

This means that p(t, 5, u), as a function of u at every fixed t 2 to and z E 0(t), is a probability

density. Its meaning in terms of the particle velocity is revealed below (see the text below (2.14)).

It follows from (2.8) that if 0 < n(t, CC) < co at t 2 to, z E R(t), then p(t, z, U) = f(t, z, u)/n(t, z),

u E R3. Expression (2.8) points out the probabilistic meaning of the distribution function: it is

the product of the concentration n(t, X) of the fluid particles in the domain and the corresponding

probability density p(t, 2, u). This probabilistic picture corresponds to that in SM or KT (e.g.. [5,

Section 3.3.10]). It follows from (2.8) and (2.3) that

We denote the mass of the fluid particle and the fluid absolute temperature with m and T.

respectively. Mass m is assumed to be independent parameter.

REMARK 2.3. Equality (2.11) explicitly shows that the distribution function f is a scaled density

in the six-dimensional position-velocity space. According to the well-known Heisenberg uncer-

tainty principle (e.g., [6, Section ?I), the smallest volume in this space is (h/m)3 where h is the

Planck constant. Hence, f(t, 2, U) N (m/h)3. Moreover, N(t) - (2s + 1) where s is the spin of

the fluid particle (s = 0 for spineless particle). Thus, one obtains f(t,rc,~) - (2s + l)(m/h)3.

Equalities (2.4), (2.5), (2.9), and (2.10) show that quantity

rl(C 5, u) = e(t, z)p(t, 5, u), t >to, (x,u) E R(t) x R3, (2.12)

is of properties

and, as a function of vector (z,u) E R(t) x R3 at every fixed t > to, is the joint probability

density of random variables y(., t) and v(., t) or, that is the same, probability density of random

variable

(2.13)

Densities Q, p and random variable v(., t) are characterized in terms of (2.13) or (2.12) (cf. (2.15)

and (2.16) below) in the following way.

In view of (2.10), the particle-position probability density ~(t, .) is the corresponding marginal

probability density with respect to joint density q(t, ., .). Random variable ZJ(., t) is described

with the particle-velocity marginal probability density

q(t, x, u) dx = s e(t, X)P(L 2, u) dxc, t 2 to n(t)

(2.14)

Probability density p(t, 2, ,) at every fixed t 2 to and IC E R(1) is the conditional probability

density of the particle-velocity random variable v(t, .) under the condition that y(<, t) = IC.

Let v(E, t, ,) be the random variable corresponding to this density. Then, this variable is a

random field, i.e.,

v = v(<, t, X)> f L to, z E R(t). (2.15)


At every fixed t 2 to and x E O(t), it presents the velocity vector of the fluid particle under

the condition that its position y(<, t) at time t is equal to Z, i.e., y(<, t) = x. Subsequently,

we shall call velocity v = v(<, t, x) and conditional probability density p conditional velocity of

the fluid particle under the above condition and probability density of this conditional velocity,

respectively. Note, that the particle velocities u([: t) and v(<, t, x) are described with different

densities, marginal density s and conditional density p, and coupled according to equality

w(,c,t) = J 4E, 6 xc)dt, x) dx, t 2 to, (2.16) n(t)

which stems from (2.14).

2.2. Expectations and the Related Quantities

Analyzing one or another characteristic of the fluid, say. quantity C(t,x,u), one is usually

interested in its expectation. In connection with function f of property (2.11), the following two

expectations can be considered: expectation

J-W& ., .)I = L,,;,,, <(t> x: u)v(t, 2, u) dx du

=J ((6 x! u)e(t, x)p(t, x, ~1 dx du, t ‘1 to,

n(t) x R”

and conditional expectation

e[C(t, 5, ,) I XI = JR3 C(t, 5, u)p(t, 5, u) du, t 2 to, x E 0(t). (2.17)

In so doing, the former is expressed with the help of the latter similarly to (2.16), namely,

E[C(t, ., .)I = E[e(<(t, x! .) I x)1 = l,,, &(t, 5, .I I 4dt. 2) dx. t 2 to.

One of the most important conditional expectations corresponds to quantity <(t, x;u) E u and is

the fluid-particle macroscopic velocity (cf. [5, (3.13a) on p. 1521)

(2.18)

which is the expectation of the conditional velocity of the particle at time t under the condition

that the particle-position y at t is equal to x, i.e., y(<, t) = x.

The two quantities related to conditional expectation (2.17) are

J&X 5, .)I = n(t, x)e[C(t, 5, .) I 4, t L to, x E n(t), (2.19)

and

e[c(t, x, .)] = ‘ff’if’ x;r” = e[C(t, x, .) I x] + n(t, x)ae’C(t’ x’ ‘) ’ x1 n ,X dn(t, x)1 (2.20)

t 2 to, x E R(t).

Characteristic k[c(t, 2, .)] is th e expected value of function <(t: x, .) per unit volume whereas

i?[c(t,x, .)] is the corresponding quantity per one particle. The derivatives in (2.20) are with

respect to explicit or implicit functional dependences on n. Relation (2.20) shows that

ifcK(t,x,~)I I’ f z 1s unctionally independent of n(t, x),

then i?[c(t, z, .)] K e[<(t, 2, .) I x]. (2.21)

894 E. MAMONTOV et al

An example of case (2.21) is velocity (2.18). Indeed, both n and v are usually the variables of

fluid-dynamic (or “hydrodynamic”) model where they are described as solutions of respective

partial differential equations (PDEs) and, thus, are functionally independent of each other.

An important example related to (2.19) and (2.20) is the case when <(t, 2, U) = U(t, 5, U) where

quantity

u(t x u) = mb - ~(WITb - 46x)1 > I

2

= mllu - v(t, x)l12 (2.22)

2 j t 2t0, (IC, u) E Q(t) x R3,

is the representation in terms of vector u (see the text below (2.7)) of internal energy U(t, r, V) of

the fluid particle and ]I // is Euclidean norm of vector. The internal energy is the kinetic energy

of the fluid-particle purely random (i.e., zero-expectation) motion! i.e., the motion with velocity

u - v(t, x). Then,

O(t, x) = E[<(t, x, .)] = n(t, x) J’ U(t, 5, u)p(t, x, ~1 du, t >to, z E R(t), (2.23) R”

qt, x) = $(t, 2, .)] = aOi(t, x)

ab(t, 41’ t 1 to, z E n(t). (2.24)

For the rheologically ideal fluid under consideration (see the beginning of Section 2), pressure

II = II(t, z) can be written as

rI(t, x) = ; ti(t,LI$ 0

t 1to, 5 E R(t), (2.25)

because of (2.22), (2.23), (2.8), and the well-known definition (e.g., [5, (3.15) on p. 1531). The

one-particle energy (2.24) can be represented similarly to (2.22), namely,

qt,x) = 0 2 m[dt, x)12, t >to, 5 E R(t),

where quantity (see also (2.25))

4w=/~=gpJ-; t2to, xER(t),

(2.26)

(2.27)

is the standard deviation of every entry of conditional velocity u. This deviation is one of the

most important deterministic characteristics of the fluid and is also known as the velocity of

sound waves in the fluid.

2.3. Stochastic Concentration

A substantially more general vision of distribution function (e.g., [7-10; 11, Sections 22 and 23;

121) allows it to be random, i.e., f(t, x, u) is extended to f(E, t, x, u) where, as before < E Z. In

this case, one can apply the following stochastic generalizations (marked with the subscript “s”):


of (2.6), (2.7), (2.1)-(2.3), and (2.11), respectively, where (cf. (2.4), (2.5), (2.9), (2.10)) ~~(<,t,r)

> 0, J&) es(E, t, x) dx = I, Ps(C,GX, u) > 0, JR3 ps(E,t, 5, u) du = 1, t 2 to, (2, u) E R(t) x R3,

provided that the two conditions below hold. First, particle position y(<, t) and conditional

velocity (2.15) are of the properties

y(<,t) = J xes(E, t, x) dx, t 2 to, n(t)

Second? for all (t,x,u) E [t O,CO) x0(t) xR3, randomvariables NS(.,t), eS(.!t,x), ps(.!trx,u) are

mutually stochastically independent (cf. [13! p. 161) and has finite expectations. This condition

enables one to evaluate N(t), e(t,x), and p(t,x,u) as expectations of N,(.,t). Q~(.,~,Lc): and

ps(., t, x, u), respectively.

We note that, in SM and KT, the random distribution function arises in connection with

the stochastic Boltzmann and Boltzmann-Enskog equations, both linear and nonlinear (see the

references at the beginning of this section). They were developed as the tools to model dense

fluids and those multiparticle effects which cannot be described with nonrandom one-particle

distribution function. These phenomena are usually beyond the modelling capabilities of common!

deterministic Boltzmann and Boltzmann-Enskog equations. Thus, their stochastic versions or

similar models, for instance, stochastic PDEs (SPDEs) should be regarded as the minimum

modelling at least in those problems which are targeted at the outcomes of a practical meaning.

3. A THERMODYNAMIC LIMIT AND POSSIBLE WAYS TO THE THERMODYNAMIC-LIMIT-FREE RANDOM MECHANICS

As a rule, distribution function f is determined by means of methods of SM or KT. Sec-

tion 2 shows that this function can alternatively be represented (see (2.8)) in terms of the fluid

concentration n and probability density p(t, 5, u) of the conditional fluid-particle velocity.

3.1. Some Examples Based on the Thermodynamic Limit

Let us consider the simplest examples of the above density for the case when the fluid is at

equilibrium. Regarding the state of equilibrium, the following two issues can be noted.

First, all the deterministic parameters of the fluid are independent of time t and

v(t,x) = 0, t 2 to, 5 E c&q. (3.1)

We denote the equilibrium versions with the subscript “eq” and without t in the corresponding

lists of variables. For instance, domain R,, (e.g., in (3.1)) is the equilibrium counterpart of R(t) in

Section 2 and density p(t, 2, u) at equilibrium is referred peq(x, u). In view of (3.1), energy (2.22)

is independent of (t, z), i.e., its equilibrium version is

Ueq(u) = $, u E R3. (3.2)

Second, the fluid at equilibrium is characterized with a special thermodynamic quantity /1 called

the chemical potential of the fluid (e.g., [14, Section 3.2.31). It is determined with expression (see

also [15])

II(x) = F - PCXi(5), 5 E G?q, (3.3)

where F is the absolute Fermi energy and peg(x) is the fluid potential energy at equilibrium

at point 5. (For two systems in contact such that the exchange of particles is allowed, the

corresponding values of energy F become equal at equilibrium. Until the equilibrium state is


reached, there is a flow of part)icles from the system with higher value of F to the system with

lower value of F.)

The first example of density peq based on TDL is the Maxwell-Boltzmann (MB) probability

density

Peq MB(Z, u) = w+Ueq(u)I(KT)I

(27r)347: ’ (5, u) E f&s, x R3, Cl,, = R”. (3.4)

with concentration (at s = 0; see Remark 2.3)

P(X) Gq. MB(X) = N* exp ~

[ 1 (KT) ’ x E Qsq, R,, = R3.

where K is the Boltzmann constant,

and

N, = F 3 (2T)3/2ag ( >

(3.5)

(3.6)

(3.7)

Density (3.4) d escribes the classical fluid which is also called the MB fluid.

REMARK 3.1. The MB fluid is special in the following respect. Any nonrandom characteristic

for any fluid (of the type considered in this work) in the classical (or nondegenerate) limit case

as p(x)/(KT) + -cc at z E a,,! R,, = R3, tends to the product of the same characteristic for

the MB fluid and a certain constant (which depends on the characteristic). For example.

lim neq (xl iL(.c)l(KT)--co (2s + l)neq, MB(x) = ”

uniformly in IC E R,,, R,, = R3, (3.8)

where neq MB is described with (3.5). If the above-mentioned characteristic depends on u E R3.

then the classical limit is also uniform in U. An example is limit relation (3.12) below.

Another example of the equilibrium probability density based on TDL is the one for the fermion

fluid. Particles with spin s such that 2s is an odd number are called fermions. (Particles with

spin s such that 2s is an even number are called bosons.) There are only the following two

differences of the fermion fluid from the classical fluid. The first one is that s # 0 for fermions,

whereas the second one is that fermions obey the so-called Pauli exclusion principle. It states that,

in aggregate of fermions, no two particles can be in the same quantum state. Electrons, protons,

and neutrons are examples of fermions (protons and neutrons are also known as nucleons). The

equilibrium probability density for the fermion fluid is the Fermi-Dirac (FD) density

P-4. FD(xl u, = 1 (2T)3,2aB~

l/2 [dx)IKTl (1 + exp [ueq(u) - PL(x)/KTI) ’ (3.9) (x-u) E R,, x R3. R,, = R3,

with concentration (at s = l/2; see also (3.6))

%q.FD(x) = (2s + l)N*@l/Z I(: E Gq, C12,, = R3, (3.10)

where Q1,2 is the FD function of index l/2 (e.g., see [16] for the review on the FD functions).

Density (3.9) is spherical symmetric in vector u and has exactly one local maximum. This

maximum is achieved at u = 0 and is also the only global maximum.

At fixed x such that P(Z) > 0, the sphere Ueq(u) = p( 2 in the velocity space is called the )

Fermi sphere and its radius UF(Z) = Jm z m is called the Fermi velocity. In so doing, value


Ueq(UF(X)) = PC .r is called the (relative) Fermi energy (i.e.! counted from the potential energy ) in (3.3)).

The equilibrium version of energy (2.26) is

(3.11)

In the case of the fermion fluid, the equilibrium standard deviation geq FD(IC) is described wit.h

the well-known equality (e.g., see (3.7) above and (C.l.12) and (C.1.23) in [17])

where @_I,2 is the Fermi-Dirac function of index -l/Z. The fermion fluid at equilibrium is

called the Fermi fluid if and only if &-,(x) = p(x), x E R,,. This equality holds only in the

quantum (or degenerate) limit case as p(x)/(KT) + co, i.e., as T 1 0 at 0 < p(x) < W. If the

fluid is the Fermi one, then the velocity standard deviation geq.Fb(x) is coupled with the Fermi

velocity UF(X) by means of equality oeq FD(Z) = p(r)/&, and is known (e.g., [5, p. 3771) as the

so-called zero-sound velocity.

In the classical limit case as p(s)/(l<T) + -w, the FD density becomes the MB one, i.e.,

lim IL(Z)I(JW’--m

Peq.~~(x,U) =Peq.~~(x,u): (x,~) E %, x R3, CL?,, = R3. (3.12)

This follows from (3.4), (3.9), and the well-known fact (e.g., [16]) that

One can readily check that the limit in (3.12) is uniform with respect to u E R3 (cf. Remark 3.1).

In the quantum limit case as T 1 0 at 0 < P(Z) < 00, probability density pes FD(x, u) is zero

for all u which are outside the Fermi sphere and is equal to unit divided by the volume of this

sphere for all u which are inside the sphere.

Notion of the Fermi fluid and all the related characteristics are also developed with no con-

nection to SM or KT by means of the Dirac-Slater method in quantum mechanics (e.g.. [6.

Section 36.11).

3.2. The Thermodynamic Limit

Importantly, according to the SM or KT vision, the condition &., = R3 in any of (3.4), (3.5),

(3.9), (3.10) cannot generally be omitted. The point is that these formulae are the results based

on TDL.

REMARK 3.2. In SM or KT, all the continuous distributions of the fluid-particle velocity and the

related quantities (like functions f, Q, and p in Section 2) can be interpreted only in connection

with statistically large values of N(t) (e.g., [18, Remark on p. 21 or 15, Section 1.4.11). More

precisely, the above continuous characteristics are based on the so-called thermodynamic limit

(TDL) (e.g., [l, (VII.68), pp. 1866187, 313-314; 19, pp. 7-8; 20, Section 7 of Chapter 2 of Part I

and Sections 4.e, 8.g of Chapter 3 of Part I]). In terms of Section 2, this limit can be expressed

as the condition that, for every t 2 to, quantity

N(t) lim ~ n(t)+n3 Jflctj dx

(3.13)

exists and is finite. The thermodynamic limit is merely a tool which enables one to eliminate

certain nonphysical assumptions of the SM formalism preceding the introduction of this limit


(e.g., [l, p. 204; 20, pp. 48-491). Relation R(t) + R3 in (3.13) and the implied relation N(t) + co

are the SM/KT limitations which, as is stressed in Remark 2.1, are not required in CFM.

The case R(t) = R3 corresponds to the universe. Clearly, the overwhelming majority of the

applied problems is far beyond the particular vision prescribed by (3.13). What allows us to

apply SM or KT built on TDL to the problems where TDL merely cannot hold?

Most of works in SM or KT ignore this question (for the reasons which are not considered in

the present work). So, the corresponding discussions are extremely rare in the literature. There

are, however, a few helpful texts on the topic (e.g., [20, Sections 5 and 6.e of Chapter 3 of Part I]).

They provide some elucidating ideas but do not overcome the frames of semiheuristic reasonings.

The rigorous analysis of the general conditions of applicability of TDL to the fluids of finite

numbers of particles in bounded domains is not available.

We also note that condition Q(t) + R3 in (3.13) to a great extent explains popularity of

the wavevector-based formalisms in statistical mechanics and classical kinetic theory. However:

the wavevector related to the points in domain R(t) can be introduced in a common way only

if 0(t) = R3. If the latter does not hold, then introduction of the wavevector is inherently

associated with the boundary layers in a neighborhood of boundary Xl(t), thereby requiring

thorough and complicated physical and mathematical analyses. Special care has to be taken

if the domain is bounded. Regrettably, all these precautions are usually disregarded in both

fundamental and applied fields of the above sciences that makes the corresponding results open

to question.

Remark 3.2 points out that the statistical-mechanics or kinetic-theory approaches cannot serve

as the comprehensive bases to describe f(t,x,u) (see (2.11)). The question is which theory can

be used. Possible answers are discussed in Section 3.3 and described in Section 4.

3.3. In Search for the Clean Way:

Possible Approaches to the Thermodynamic-Limit-Free Random Mechanics

The search for the clean, i.e., TDL-free way mentioned in Section 1 stimulates SM to revise

its traditional role of a dogma and to turn to the problems of its internal development. For

instance, works [21-241 report some preliminary results of the TDL-free application of SM. A

remarkable and instructive peculiarity of these results show that one can in principle consider the

task to formulate the basis of the special version of SM devoted to finite systems. Development

of this version would open a way for application of the future capabilities of the above version to

practically important problems.

REMARK 3.3. It should be emphasized that chemical potential (3.3) is not associated with TDL.

It is an intensive thermodynamic quantity (i.e., it does not scale with the volume of the domain or

the number of the particles in the domain) and is introduced in thermodynamics before passing

to TDL (e.g., [14, pp. 64-65; 20, Sections 6.b-6.d of Chapter 31). Thus, the chemical potential

can be used in the treatments which do not apply TDL.

Another possible implementation of the clean way is exemplified with the vast treatment of

random mechanics based on stochastic differential equations (SDEs) (e.g., (25-271). The SDE

alternative is used in many subject fields, for instance, gas theory (e.g., [8]), synergetics [28],

physics of complex systems (e.g., [29]), and heavy-ion physics (e.g., [30, Section 3.11). The

distinguishing feature of the SDE approach is that incorporation of the stochastic sources into

the deterministic equations (which are often regarded as the so-called phenomenological from

the SM viewpoint) endows the resulting models with a probabilistic meaning in a compact and

efficient way. In so doing, there is naturally no need to pass to TDL since continuous probabilistic

distributions originate from the above sources. This direction is developed theoretically much

deeper than the SM-based research on finite systems.

However, not all the related physical aspects are developed equally well. One of them is

the continuous equilibrium distributions. In other words, there is still no systematic way to


construct the continuous equilibrium characteristics such as pes and neq for fluids in bounded

domains containing finite numbers of particles. A possible solution of this problem is proposed

in the next section.

4. THE THERMODYNAMIC-LIMIT-FREE MODEL FOR THE

FLUID-VELOCITY PROBABILITY DENSITY AT EQUILIBRIUM

Let us consider the following physical picture. Let the fluid be at equilibrium in domain R,,.

The equilibrium state is characterized with the equilibrium probability density peq(x, u) for the

conditional velocity Y (see (2.15)), x E R,,. Assume now that, due to one or another excitation,

the fluid finds itself at some time point, say, t = to in the perturbed state such that all the fluid

parameters except the particle-velocity distribution are the same as in the equilibrium state,

but the mentioned distribution is different from the equilibrium one. We call states of this kind

almost-equilibrium. They are similar to the states in the evolutions which are quasi-static in the

sense of [20, Section 4.a of Chapter 3 of Part I]). In terms of probability densities, the almost-

equilibrium state at t = to can be described with density po(s, u) which is generally not equal to

peq(x, IL). Assume also that the above excitation is no longer active for all t > to, so in the course

of increase in t from value to, the fluid returns to equilibrium.

4.1. The Model and Theorem

The main assumption of the proposed approach in this section is that the evolution is governed

by nonlinear Ito’s stochastic differential equation (ISDE) of the following form (see [17, (C.l.l),

(C.1.26), and (C.1.27)]):

udt

dv = - 7JZ, (w(v))“] +

2

7&J2, (zu(v))2] O* dW(tlt), x E %q, t > to, (4.1)

where u* is determined with (3.7), quantity

w(u) = Ues(u) (m4)/2’

u E R3, (4.2)

is the square root of the normalized value of the particle internal energy (3.2),

Teq(X, z2) > 0, x E G!q, z E R, (4.3)

and W(<, t) is the three-dimensional Wiener stochastic process. Frequency T~~[x, (w(~))~]-’ is,

in the SM or KT terms, analogous to the velocity-dependent frequency in the Bhatnagar-Gross-

Krook-Welander (BGKW) equation discovered in [31,32] ( see also [5, Section 4.2.1 and (3.35)

on p. 3591). Note, that the BGKW equation is also regarded as the so-called relaxation-time

approximation for the Boltzmann equation. The stochastic version of the BGKW equation for

the fermion fluids is discussed in [30]. We consider ISDE (4.1) in connection with the stochastic

CFM (SCFM).

The physical meaning of the Wiener-process-related term in (4.1) is revealed with the follow-

ing fact. If one multiplies both sides of (4.1) by mass m of the fluid particle, then the term

J2/{7,J5, (w(v))21)mo* dW(c, t) on the right-hand side of the resulting equation represents the

random force acting on a particle by its surroundings. Another term in ISDE (4.1) which also

couples the particle and its surroundings is the first term on the right-hand side. It includes

relaxation time req[x, (u~(v))~] of the particle velocity v (or momentum mv). The subscript

“eq” stresses that this time parameter can depend only on the equilibrium characteristics of the

fluid, for instance (see Remark 3.3), on chemical potential (3.3) (see more on time parameter

T~~[z, (,w(~))~] in Remark 4.2 below). This completely agrees with the physical picture described

900 E. MAL~ONTOV et al

above in connection with the almost-equilibrium states. In this sense, ISDE model (4.1) is

almost-equilibrium.

Since, in the above perturbed state at t = to and in the state at every t > to, all the fluid

parameters except the velocity distribution are equal to their equilibrium values, function 7ees is

the same as that at equilibrium and the particle position x is independent oft. This independence

in particular allows us to consider z merely as a parameter in ISDE (4.1).

The corresponding initial condition is associated with the above density po(x, U) and can be

written as follows:

&” = YO(<> t), x E Cl,,. (4.4)

where vO(<, .) is the random variable with probability density po(x, ,).

In terms of equation (4.1), the equilibrium state of the fluid is underst.ood as the unique

stationary solution of this equation. The theorem below formulates the conditions which enables

initial-value problem (4.1),(4.4) t o model the physical picture described at the beginning of the

present section.

THEOREM 1. Let the assumptions below hold.

(1) Inequality (4.3) is valid.

(2) If, for any initial time point to and any initial probability density po, initial-value prob-

lem (4.1),(4.4) has the unique solution, then this solution is global, i.e., is defined for all

t > to. (3) Quantity ~~~(x, z2), as a function of z, is Lebesgue integrable over (0, co) uniformly in x

and

r 2

z2T,,(x,z2)exp 5 ( ) dz cc%, 5 E G., 0

Then, the following assertions are valid.

(1)

(2)

(3)

(4)

Equation (4.1) is equivalent to linear 1SDE

dv = --v dB + v%, dW(<, Q). x E Qq, L9 > 0,

where Bit+, = 0,

dt

(4.5)

(4.6)

(4-7)

For any initial time point to and any initial probability density po, the unique solution of

initial-value problem (4.1),(4.4) defined for all t > to is diffusion stochastic process (DSP)

with drift vector -v/{.T,~(w(v))~]} and diffusion matrix 2a,21/{~,,[~:, (W(V))“]}.

Equation (4.1) has the unique stationary solution which is specified by stationary proba-

bility density

p (x u) zz Teq[x, (w(“))21, ,MB(x u) -2 7

Teq *(x) eq ’ ’ (x,u) E f&, x R3,

where density peq. MB is described with (3.4) without limitation R,, = R3 and

Density peq(x, u) is an even function of every entry of vector u.

If

f 0 M z4~eq (x, z”) exp (<) dz < co! x E &,

(4.8)

(4.9)

(4.10)

Stochastic Continuum Mechanics

then the standard deviation (see also (3.2))

aes(r) = &$GZ= /w.

of the stationary solution is finite and is expressed as

901

x E Qq, (4.11)

(5) Relation

IL,(~) = mG&)[G&)12, x E Gqr

holds where III,,(x) is the Auid pressure at point x at equilibrium

5 E cl,,. (4.12)

(4.13)

PROOF. Assertion (1) follows from Assumption (1) and the well-known consideration (e.g., [33])

on random time change in ISDEs. Assertion (2) stems from Assertion (l), Assumption (l), and

the related results of theory of ISDE (4.6) (e.g., [13, Section 8.31). Assertions (3) and (4) follow

from Assumption (3), notations (3.4) and (4.2), and [17, Corollary 1.2 and Section C.31). (The

particular versions of (C.3.3), (C.3.1), (C.3.2), (C.3.6), and (C.3.5) in (171 are (4.5), (4.8), (4.9),

(4.10) and (4.12), respectively.) Accountin g result (4.11) and the equilibrium versions (2.23))

(2.25) shows that Assertion (5) is valid. I

Assumptions (1) and (3) of Theorem 1 are fairly mild conditions from the physical viewpoint,.

Regarding Assumption (2) therein, we note the following.

It is not difficult to involve the conditions in terms of function req which are sufficient, for the

globality in Assumption (2) (e.g., [34, pp. 58-591). H owever, these sufficient conditions may be

too restrictive to treat a practically important range of the fluids by means of Theorem 1. It does

not include any more specific conditions for the globality because at present, we are not aware

of any condition equivalent to the globality rather than sufficient for it. The following necessary

condition can be a helpful test since it enables to us reveal inappropriate descriptions for res.

REMARK 4.1. In view of Assumption (2), the unique solution of initial-value problem (4.1),(4.4)

in the noiseless case, i.e., when the last term on the right-hand side of (4.1) is replaced with

zero, has to be defined for all t > to. The noiseless version of (4.1),(4.4) is equivalent to (cf. [17,

(4.10.2), (4.10.3)] nonlinear ordinary differential equation

dl- 2T

dt=- res(“, T) ’ .1: E Gq, (4.14)

with initial condition

Tlt=t” = To(z). 5 E Gq, (4.15)

where variables Y and TO represent the nonrandom counterparts of random normalized energies

(w(v))~ and (w(vo))~: respectively. (see (2.15) and ~a(5.r) in (4.4) for v and IQ). Thus, if the

solution of (4.14),(4.15) is not defined for all t 2 to, i.e., is not global, then Assumption (2)

cannot hold, and hence, Theorem 1 cannot be applied. For example, if rees(x; T) N YP’. then

the above solution is global if and only if E 2 0 (see [17, Remark 4.21). Moreover, the globality

is the property which cannot be removed from the physical picture (at least until it is physical).

So, the lack of the globality points out that the corresponding model for the normalized-energy

dependence of re4 is unphysical. The discussion on this topic as well as on the examples of the

models which were published long ago and are turned out to be unphysical can be found in [17,

Remark 4.21.

The key point of Theorem 1 is that it established a one-to-one correspondence (4.8) (see

also (3.4)) and (4.9) between relaxation-time function rees and equilibrium probability density pes.

In so doing, as follows from (4.8), multiplication of req[5, (w(u))~] by any u-independent quantity

does not affect the u-dependence of the equilibrium density at all. In this respect, the (w(u))~-

dependence of function rees is crucial. This issue is exemplified in Sections 4.2 and 4.3.


4.2. Illustration:

The Classical Fluid in the Domain

Which Need Not Coincide with the Whole Space

This section considers the following application of Theorem 1.

COROLLARY 1. If the hypothesis of Theorem 1 holds, then probability density peq(x, .) is equal

to the MB one (3.4) if and only if function req is independent of the normalized energy (w(v))‘:

say,

7-eq [? (W(u))“] = Teq.MB(J:), x E %.qr u E R3. (4.16)

Relation (4.16) is equivalent to the linearity of ISDE (4.1) in V.

PROOF. The proof follows from (3.4), (4.8), and (4.9). I

The result of Corollary 1 is in a complete agreement with the well-known linear-ISDE approach

to the MB probability density for the classical fluid (e.g., [l, Section 2 of Chapter 21). This

approach also applies ISDE (4.1), h owever, only under the independence condition mentioned in

Corollary 1 (cf. [l, (11.3)]. F rom this viewpoint, our nonlinear ISDE (4.1) can be regarded as the

corresponding generalization.

The importance of both the linear-ISDE treatment and Corollary 1 is that any of them is not

associated with TDL (3.13) and thereby extends notion of the classical fluid and the MB distri-

bution to the spatial domains which need not coincide with the whole space R3. In particular,

they can be bounded.

4.3. Illustration:

The Fermion Fluid in the Domain

Which Need Not Coincide with the Whole Space

A less simple and more important example of application of Theorem 1 concerns the FD

probability density (3.9) without limitation R,, = R3.

COROLLARY 2. If the hypothesis of Theorem 1 holds, then there exists the unique dependence of

T~,,[x, (w(u))~] on the normalized energy (w(u))~ (see (4.2)) which makes probability density (3.9)

without limitation R,, = R3 and probability density (4.8) equal to each other. Under the

condition that (cf. Remark 3.1 without limitation R,, = R3) limpcz),(KT)__m req[x, (w(u))~] =

req. Mn(.r), (IC, U) E s1,q x R3, where the limit is uniform in u E R3, the above unique dependence

is of the following form:

Teq [T (W(~,,“] = Teq. FDb(W(u))21 w{FLq(~) - ~CL(~)lI(~T)~

= Teq.MB(5)1 + exp{[U,q(u) - p(S)]/(KT)}’ (x,u) E R,, x R3.

(4.17)

PROOF. The proof follows from (3.9), (4.8), and (4.9). I

One can also check that relaxation time (4.17) passes the test described in Remark 4.1.

Since the only difference of the fermion fluid described with the FD density (3.9) from the

classical one is that fermions obey the Pauli exclusion principle (see the text between (3.8)

and (3.9)), quantity (4.17) can be represented as

Teq FD,k’&~~2, =

1

Teq. MB(Z) + (x,u) E Re, x R3, (4.18)

where the subscript “Pe” stands for the Pauli exclusion and the last term on the right-hand side

of (4.18) is due to this phenomenon alone. It follows from (4.17) and (4.18) that

r eq. pe [cc, (W(U))“] = 7eq. MB(x) exp { ‘ueq(;&;(z)‘} , (? u) E %, x R3. (4.19)


This expression for the Pauli-exclusion relaxation time is exact in the present approach and, to our

knowledge, is new. Inequality Tag. p,[~(w(u))~] 2 ~eq.MB(Z)exP[-~(z)/(KT)], (z,u) E f&XR3, shows that contribution of relaxation time (4.19) into (4.18) is zero in the classical limit case (i.e..

as ,u(z)/(KT) + -oo). The fact that the Pauli exclusion is of no importance for the classical

fluid is well known in SM or KT. Relations (4.18) and (4.19) also show that relaxation time (4.19)

dominates the relaxation time r eq.~s(~) for vectors u inside the Fermi sphere when P(Z) > 0)

i.e., under the conditions close to the quantum limit (see Section 3.1).

In general, the idea to allow for the Pauli exclusion principle in the ISDE-modelling is not

new. It goes back to at least [35]. However, the purpose, spirit, and techniques of the present

work are altogether different from those of [35]. The Pauli exclusion principle in connection with

stochastic phenomena is also investigated experimentally (e.g., [36]).

4.4. Accounting a Few Relaxation Mechanisms

The right-hand side of expression (4.16) includes only one relaxation mechanism, namely, that

in the MB fluid. The right-hand side of expression (4.18) takes into account two relaxation

mechanisms, one of them is the same as that in (4.16), i.e., associated with the relaxation in the

MB fluid, whereas the other one is due to the Pauli exclusion. In the real fluids, relaxation time

reeq[~, (w(v))~] in ISDE (4.1) ’ g is enerally contributed by a few, say, T 2 1 mechanisms. They can

be described with respective relaxation times

Teq, kb-, bW21 > 0, (T u) E Gq x R3, k=l,...,r-. (4.20)

Thus, the generalization of expressions (4.16) or (4.18) is (cf., [37! (6f.l)], [17, (C.1.8)])

1

G&G (w(u))21 = (2,~) E 02,, x R3. (4.21)

Each of the relaxation terms summed on the right-hand side of (4.21) can in turn be represented

with a similar sum depending on the specific nature of the fluid and the specific modelling purpose.

So, in general, one has a hierarchy of the relaxation mechanisms. This hierarchy always includes

at least one type of the relaxations, the interparticle one. (If, in so doing, no other types of the

relaxation are present, the fluid is called simple.) The corresponding examples are discussed in

Sections 4.2 and 4.3. More complex representations can be obtained by means of generalization

of more simple ones. Thus, the resulting hierarchy can be fairly extensive. For example, the

classical-fluid and Pauli-exclusion terms in (4.18) for the electron fluid can be accompanied by

the electron-electron scattering due to the Coulomb interaction. In this case, the corresponding

term should be added to the right-hand side of (4.21). Accounting the extra phenomena, for

instance, the electron-photon interaction leads to the additional terms.

Fortunately, in many problems, not all relaxation mechanisms are equally important. So, it is

in many cases reasonable to keep in the sum in (4.21) only the dominating terms.

The following crucial role of function req in our almost-equilibrium model (4.1), (4.4) should

be noted.

REMARK 4.2. The equilibrium properties of the fluid are completely described with equilib-

rium probability density (4.8) ( see also (4.9)) which in turn depends on quantity (3.7) and

relaxation-time function req on the left-hand side of (4.21). Thus, the overall adequacy of ISDE

model (4.1),(4.4) is determined by the physical relevance and numerical accuracy of the summed

terms on the right-hand side of (4.21). The relaxation times in these terms are in our model

assigned to depend not only on the specific properties of the fluid but also on the shape and size

of domain R,, and the conditions at its boundary a&,. There is, in contrast to the SM or KT

formalisms, no need to involve limitation R,, = R3 associated with TDL.


The summed terms in (4.21) can be obtained in different ways. One of them is theoretical

derivations of the terms. Many examples of the derivations can be found in [37] or [38]. Another

option is to extract the expressions for the above terms from the numerical simulations, for

instance, those based on the molecular-dynamics or Monte-Carlo approaches. The extraction

can be applied to the data on the covariance of the stationary solution of nonlinear ISDE (4.1)

mentioned in Theorem 1. In so doing, one can apply the approximate analytical-numerical

technique to evaluate the covariance in case of ISDE (4.1) developed in [17] (see the summary in

Section 4.10.4 therein) on the basis of the so-called deterministic-transition approximation. The

analytical part of this technique is formulated in quadratures. (In so doing, the modification

concerning the passing discussed in the text around (4.24) below should be incorporated.) The

covariance can also be obtained experimentally, for example, by means of the well known diffusing-

wave spectroscopy (e.g., [39]).

4.5. Equilibrium Concentration

It follows from definition of the fluid pressure (e.g.. (2.25)), the physical meaning of chemical

potential (e.g., [20, p. 100)) and the fact noted in Remark 3.3 that relation

(4.22)

is valid. Under the TDL condition, this equality is well-known in SM (e.g., [20, p. 1011). The

pressure in (4.22) is expressed by means of the concentration as is shown (4.13). Thus, rela-

tion (4.22) presents the ordinary differential equation (ODE) for equilibrium concentration n,,-,.

The concentration is deterrnined from this equation under initial condition (3.8). In particu-

lar, one can check with the help of the properties of FD functions (e.g., [16]) that initial value

problem (4.22),(3.8) in the FD case leads to (3.10).

Concentration neq(z) and probability density pes(z, u) provide the equilibrium version

fecI(x, u) = %q(Xhq(X, u), (x, ~1 E fL, x R3, (4.23)

of distribution function (2.8). Concentration neq(z) also enables one to pass from the equilib-

rium versions of the e-expectations (see (2.17)) to the equilibrium versions of the &expectations

(see (2.20)) by means of (2.19). F or example, standard deviation oeq(z) in (3.11) is obtained

from a,,(x) in (4.12) ( see also (4.11)) with the help of equality

(4.24)

which stems from the equilibrium version of (2.27).

5. SUMMARY OF THE PRACTICAL PROCEDURE, DISCUSSION ON THE PROPOSED MODEL AND CONCLUDING REMARKS

This section considers some general aspects of the SCFM model proposed in Section 4.

5.1. Summary of the Model

The almost-equilibrium fluid model in Section 4 can serve as the basis of the practical procedure

to evaluate the equilibrium probability density and the corresponding standard deviation of the

fluid-particle velocity as well as the equilibrium fluid concentration. This procedure includes the

following six steps.

(1) Determine number T of the relaxation mechanisms which should be taken into account

and the corresponding relaxation times (4.20) to be used in (4.21) (see the discussion in


Section 4.4). In so doing, the results of Sections 4.2 and 4.3 can be applied. For example,

the relaxation time for the MB (or classical) fluid is independent of the particle velocity

(see (4.16)). Th e relaxation time for the FD (or fermion) fluid is described with (4.17).

The relaxation time corresponding to the Pauli exclusion principle alone is (4.19).

(2) Determine quantity rees[zrr (W(V))“] by means of (4.21).

(3) Make sure that the hypothesis of Theorem 1 holds for function rees resulting from Step (2).

Regarding the globality pointed out in the hypothesis. check at least that this function

passes the test described in Remark 4.1.

(4) Determine the particle-velocity equilibrium probability density pes with (4.9) and (4.8).

(5) Evaluate ces according to (4.12) and then determine the particle-velocity standard devi-

ation geq by means of (4.24).

(6) Evaluate concentration neq from initial value problem (4.22),(3.8) (see also (4.13)) and

distribution function fes from (4.23).

The above procedure is not very complex and does not involve TDL (3.13). The domain R,,

occupied by the fluid may be bounded or unbounded. In the latter case, it may or may not

coincide with the whole physical space R3.

5.2. Discussion on the Proposed Model

The core equation in the almost-equilibrium model is ISDE (4.1) coupled with ISDE (4.6) by

means of (4.7). The dimensionless time scale 0 in (4.6) is abstract. The world in this time scale

is not very interesting. For example, it includes only one fluid, namely, the one with the almost-

equilibrium behavior described with ISDE (4.6). Th e individual features of a specific fluid are

associated only with the actual time t and manifest themselves only in the passing to it by means

of (4.7). In so doing, all these features are represented with a single scaling parameter, relaxation-

time function reTes which, however, can have a fairly complex structure discussed in Section 4.4.

The corresponding expression (4.21) enables one to take into account various physical relaxation

phenomena. Depending on a specific content of the right-hand side of (4.21), one obtains different

equilibrium probability densities as is described in the procedure in Section 5.1.

The core equation (4.1) is, strictly speaking, an assumption. However, it is not of uncertain

features. Indeed, it admits a clear physical reading (see the part of Section 4 above Theorem 1)

and enables one to derive (see Sections 4.2 and 4.3) the characteristics well known in SM or

KT but in the TDL-free way. Moreover, ISDE (4.1) turned out to be helpful even in the cases

which are not very well suited for application of SM or KT. For example (see [17, Sections 4.9

and 4.10]), it underlies the approximate analytical derivation on the long, nonexponential asymp-

totic representation for the particle-velocity covariance in the hard-sphere fluid. This derivation

is fully-time domain (i.e., does not involve any time-space Fourier-like frequency techniques ap-

plicable to only certain, very special responses and under very special conditions), and is valid

regardless of whether the domain is bounded or unbounded. At present, we cannot point out any

result of this kind in SM or KT.

5.3. Concluding Remarks

Various models available in SM or KT are also derived independently, for instance: within

the CFM (or “phenomenological”) approach (e.g., [2]) or the extended thermodynamic theory

(e.g., [40]). In so doing, there is usually no need to involve TDL. An interesting example of

the phenomenological derivation of the kinetic equations is presented in the survey [41j devoted

to tumor dynamics in competition with immune system (see also, recent developments in the

framework of mean field theory in (421). Th us, the resulting models are not a distinguishing

feature of the SM/KT paradigm.

Its cornerstone is the imperative requirement to describe the phenomena from first principles.

This feature, in conjunction with the severely limited set of the employed mathematical models,

906 E. MALIONTO\~ et al.

makes it necessary at certain stages of constructing the comprehensive theory to apply such

assumptions as TDL.

The SCFM model developed in Section 4: summarized in Section 5.1 and discussed in Sec-

tion 5.2, is not concentrated on first principles. The notion of first principles is related to the

feature “to be absolute”. However, allowing for the present state of physics where many major

theoretical fields include a series of still unsolved fundamental difficulties, one has no sufficient

ground to apply the term “first principles” (or the equivalent ones).

The Landau-Lifshitz (LL) stochastic fluid model was proposed in [43] and described in detail

in [44, Section 88 of Part 21. Its key component is a system of stochastic partial differential

equations (SPDEs). The model was developed under the TDL assumption. Hence, it can only be

used within the TDL-based treatments and applied to the problems relevant to TDL. The linear

form of equation [44, (88.11) in Part 21 underlying the derivation (see also, [44, Section 122 or

Part I]) is in fact an assumption. Moreover, this equation is involved heuristically.

Our model is based on nonlinear ISDE (4.1) which is beyond the frames of SM or KT. The

model does not lead to TDL. If necessary, it enables one to derive the TDL-based descriptions (see

Corollaries 1 and 2). In so doing, it provides the corresponding exact expressions for the relaxation

times for the fermion fluid (see (4.17)) and due to the Pauli exclusion alone (see (4.19)). The

model is applicable to the fluids in both unbounded and bounded domains containing arbitrary

large or small (up to the quantum corrections) numbers of the fluid particles. The corresponding

boundary conditions can be accounted as is noted in Remark 4.2. The model provides considerable

flexibility (see Section 4.4) to different user communities (from the basic-aspects researchers to

engineers in industry), to construct the relaxation-time function 7eq depending on the specific

problems, specific purposes of the analysis, admissible complexity, desirable accuracy! and other

issues. The future application of the model will give the material to improve it.

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