stochastic continuum mechanics- a thermodynamic-limit …
TRANSCRIPT
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 thermody- namic 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 prob- ability 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)
Stochastic Continuum Mechanics 893
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”):
Stochastic Continuum Mechanics 895
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
896 E. MAMONTOV et al.
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
Stochastic Continuum Mechanics 897
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
898 E. MAMONTOV et al.
(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
Stochastic Continuum Mechanics 899
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.
902 E. MAMONTOV et al.
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)
Stochastic Continuum Mechanics 903
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.
904 E. MAMONTOV et al.
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
Stochastic Continuum Mechanics 905
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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