A Specification of the Maxwell-Rayleigh-Heisenberg ...calvino.polito.it/~mcrtn/doc/lib/mamontov-mrh... · Modelling Fluids for Bioelectronic Applications E. MAMONTOV ... is illustrated

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Available online at www.sciencedirect.com MATHEMATICAL AND

8CIENCE~DIRECT • COMPUTER MODELLING

Mathematical and Computer Modelling 42 (2005) 441-470 www.elsevier.com/locate/mcm

A Specification of the Maxwell-Rayleigh-Heisenberg

Approach to Modelling Fluids for

Bioelectronic Applications

E . M A M O N T O V D e p a r t m e n t of Phys ics

Facu l ty of Science G o t h e n b u r g Unive r s i ty SE-412 96 G o t h e n b u r g , Sweden

yem©fy, chalmers, se

(Received and accepted September 2003)

A b s t r a c t - - T h e key question which any version of random fluid mechanics has to resolve is how to provide continuous probability distributions for the fluid particles. Each specific way is determined by one or another set of assumptions. Statistical mechanics proceeds on the thermodynamic-l imit assumption supposing tha t the domain occupied by the fluid is "macroscopically big" and the number of the particles in it is "statistically large". This picture cannot be the case in mesoscopic systems. The latter are common in many modern applications including bioelectronics. The present work develops a nonstatistical way to provide the above continuous distributions. It follows the vision formed by certain results of Heisenberg, Rayleigh, and Maxwell and specifies it by means of extending nonlinear nonequilibrium stochastic hydrodynamics (NNSHD) introduced by the authors earlier. The work concentrates on the following two generalizations: first, allowing for nonzero volumes of the particles, the feature typical in the biological parts of bioelectronic problems, and, second, accounting the general kinet ic-energy/momentum dependences, including the relativistic ones, which are usually necessary in the electronic parts of bioelectronic problems. The simplest case of the first generalization is exemplified with an evaluation of the electrochemical potentials and pressures of red blood cells in human blood in a recently published paper of the authors. The second generalization is illustrated in Section 10 of the present work with the relativistic distribution functions which take into account the general spin picture of composite particles by means of the model of composons, the flexible combination of bosons and fermions based on the generalized-kinetics (GK) methods. The above generalization is intended to be a framework rather than theory t ha t inherently includes the capabilities in coupling to other fluid-modelling t reatments like common hydrodynamics or stochastic kinetic equations. The issues on further extensions in line with GK and on the coupling to the latter are emphasized. A few directions for future research are discussed as well. @ 2005 Elsevier Ltd. All rights reserved.

K e y w o r d s - - M a x w e l l ' s single-particle probability density, Rayleigh's dissipation function, Heisen- berg's uncertainty principle, Distribution function for nonzero-volume disparate particles with the spin mixtures, It6's stochastic differential equation.

0895-7177/05/$ - see front mat ter @ 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2003.09.044

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442 E. MAMONTOV

1. I N T R O D U C T I O N

Recent developments in random fluid mechanics [1-10] focus on the treatments which are free from certain limitations of statistical mechanics. The present work is devoted to a generalization of NNSHD [8,9] (see also [10, Sections 6.2-8]), the nonlinear nonequilibrium stochastic hydrodynamics.

Random mechanics models the fluid-particle position-momentum (or velocity) vector in terms of continuous probability distributions. The following issue is important.

REMARK 1.1. The key question which any version of random fluid mechanics has to resolve is how to provide continuous probability distributions for the fluid particles.

Statistical mechanics including kinetic-equation theory is based on the so-called thermodynamic-limit (TDL) assumption (e.g., [11, pp. 39-41, Section 9.1]), the assumption that

quantity lim N(X(t)) exists and is finite, (1.1) x(t) 3 v(x(t))

where N(X(t)) is the number of the particles in bounded domain X(t) _C R ~, R = (-oo, oc), V(X(t)) is the volume of the domain, and the limit is uniform in time t. Since (1.1) implies N(X(t)) ~ oc, the random mechanics based oil (1.1) is called statistical mechanics. The limit relations V(X(t)) --* oo and N(X(t)) --~ oo in assumption (1.1) mean that [11, p. 39] statistical mechanics is the science of large systems. This science, however, does not include any criteria that would rigorously justify applications to the fluids where any of the limit relations V(X(t)) ~ co and N(X(t)) ~ co is not valid. In other words, the fluids with the particle number which is not statistically large or the fluids in the domains which are not macroscopicatly big are beyond statistical mechanics, classical or quantum. A deep discussion on the above lack can be found in various sections of [12] (see also the corresponding references in [8, Section 3.2]). Along with this, many problems in modern microtechnology, biology, electronics, and other fields deal with the fluids of low, down to a few units, numbers of particles located in fairly small, micrometer-size domains. In all such cases, assumption (1.1) is merely irrelevant.

The fact that this assumption is unnecessary to reveal a probabilistic nature of each single particle of a fluid stems from the Heisenberg uncertainty principle (HUP) (e.g., [13, Section 16; 14, Section 7]), the physical law well established in experiments. It is formulated as follows:

in the six-dimensional position-momentum space, any volume per particle that can be measured is not less than h 3 where h is Planck's constant. (1.2)

In the language of continuum mechanics, HUP (1.2) means that the position-momentum vector for a particle must be random. Law (1.2) is fairly universal: it is valid even for a single volumeless (or point) particle (like electron) and even in vacuum.

We cannot point out any theoretical derivation of this law. Statement (1.2) with h/2 where h = h/(2~r) in place of h is obtained (e.g., [13, (16.7), p. 48; 14, (7.14)]) within the Schr6dinger-wave version (cf. [15]) of quantum mechanics (QM). However, the corresponding value (h/2) 3 ~ 5 x 10-4h 3 underestimates Heisenberg's parameter h 3 and the experimentally extracted [16, the right column on p. 032109.3] value (0.89h) 3 ~ 0.Th 3 in about 2000(!) and 1400(!) times, respectively. Anyhow, the fact that the minimum volume per particle is, in line with (1.2), precisely h 3 is used as a quantitative result (e.g., [17, pp. 99,162]) in construction of modern statistical mechanics. This shows that it is reasonable to regard HUP (1.2) as merely a phenomenological law.

Law (1.2) does not point out in what specific way the randomness of the particle position- momentum vector is described. This is another question. There can be various answers to it. One of them is the Maxwell result below.

Maxwell was the first to anticipate that assumptions like the above limit relations V(X(t)) -~ oc or N(X(t)) ~ oo, not to mention TDL (1.1), are not inevitable to describe a continuous

A Specification of the Maxwell-Rayleigh-Heisenberg Approach 443

probability distribution for the particle momentum vector p E I~ 3. More than twelve decades ago, he derived the equilibrium probability density of momentum p of a single particle in the form

oo(p) = e x p - , = ( 1 . 3 )

that nowadays is known as the (nonrelativistic) Maxwell-Boltzmann (MB) density. It corresponds to a fluid where the particles are mutually noninteracting. Expression (1.3) applies Boltzmann's constant K, absolute temperature T, and dependence

pTp - ( 1 . 4 )

where u is the particle kinetic energy and m is the particle scalar rest mass. It must be stressed that Maxwell's derivation of (1.3) is available in modern textbooks in both physics (e.g., [18, pp. 36-37]) and mathematics (e.g., [19, Problem 8-5, p. 238]). There are other examples of the equilibrium fluid description not involving statistical mechanics. One of them is the Fermi-fluid theory [14, Section 36.1] based on the Dirac-Slater-determinant technique.

Thus, nonstatistical ways to the continuous probability distributions do exist. This in essence agrees with the above implication on the randomness stemming from HUP (1.2) in particular indicating a harmonious conjunction of Maxwell's and Heisenberg's particle paradigms. The picture is complemented by Rayleigh's notion of dissipation function appearing to be an in- dispensable component of the proposed treatment. The combined vision can be termed the Maxwell-Rayleigh-Heisenberg approach (MRHA) to modelling particle populations.

This approach can be implemented in different versions. We note that, as is well known (e.g., [11,18]), density (1.3) can be obtained as the stationary probability density corresponding to It6's stochastic differential equation (ISDE) for the particle-momentum vector. This fact is employed as the key idea underlying NNSHD, the MRHA version developed in [8] and [9] on the basis of the related results of [4]. Here the randomness in the particle position is described by means of the particle concentration (or volumetric density of the number of the particles). The latter is modelled thermodynamically at equilibrium and with respective ISDE at nonequilibrium. In other words, the NNSHD implementation of MRHA is built around ISDE.

Generally speaking, this line follows the well-known vision [11, p. 189]. "We thus change, from here on, rather drastically the philosophy o f . . . approach to statistical dynamics. We give u p . . . ambition of deriving macroscopic quantities rigorously for the Hamiltonian dynamics of the particle motion. We rather replace the description of the interactions by a reasonable stochastic model, which allows us to treat more complex problems. We thus accept "the physicist's compromise": either we attempt a mathematically rigorous treatment, but then we can only study oversimplified models of nature; or we want to attack complex realistic systems, but then we must admit some reasonable weaknesses in the physical description."

This vision may be the only choice in research on the biology-related fluids, the ones of un- precedented complexity (e.g., [20]) and of a great variety of physical interactions. These fluids include not only a wide range of particles, from zero-volume very light electrons and holes up to huge and heavy molecules and cells, but also a diverse family of chemical, mechanical, and electronic phenomena, many of which are inherently associated with geometric aspects and the structures of variable shapes.

The purpose of the present work is to generalize the NNSHD version [8,9] of MRHA in the following two directions: first, allowing for nonzero volumes of the particles, the feature typical in the biological parts of bioelectronic problems, and second, accounting the general kinetic- energy/momentum dependencies, including the relativistic ones, which are usually necessary in the electronic parts of bioelectronic problems. The most well-known relativistic dependence is

444 E, MAMONTOV

Einstein's relation (e.g., [21, (1.15b), p. 435, (1.3), p. 433])

u = m c 2 1 + L~CC j - 1 , (1.5)

where c is the velocity of light in a vacuum and the function 11 " II is the Euclidean vector norm. The above first and second parts are presented in Sections 2-6 and Sections 7-11, respectively.

This development allows for the mixed-spin picture of composite particles available in the model of composons [22], the flexible combination of bosons and fermions based on the generalized kinetics (GK) (e.g., [3, Section 10.3; 5,7,10]). Composons are summarized in Appendix A. An extended, nonmathematical-reader-oriented version of Section 6 is [23] which also includes the corresponding application to red blood cells in human blood. The model in Section 9 in the particular case (1.5) is described in Appendix B. One of the key notions in the second part, the particle kinetic-energy-determined Rayleigh dissipation function, is discussed in Section 8. The relativistic content of the equilibrium model of Sections 8 and 9 is illustrated in Section 10 with application to relativistic composons. Section 11 describes the incorporation of these results into the simplest relativistic generalization of NNSHD. This generalization is intended to be a framework rather than theory. One of the resulting benefits is the inherent capabilities in coupling to other fluid models like common hydrodynamics or stochastic kinetic equations. This is stressed in Section 12 that concludes the work. Section 12 also discusses the issues on further extensions in line with GK and suggests a few directions for future research.

The work regards a one-component or multicomponent fluid as population of particles, identical for each fluid component, which is dispersed in a medium (that may be vacuum) and occupies domain (i.e., open connected set) X(~) C ]R 3, ~ = (-oo, oo), with a piecewise smooth boundary. This domain is not required to be "macroscopically big": it can be fairly small. However, it is assumed that the domain is sufficiently big to neglect the SchrSdinger-wave phenomena. More specifically, we assume that L(t) >> Lc(t) where L(t) is the characteristic minimum linear size of domain X(t) and Lc(t) is the critical length. As is well known [4, (6.3.32); 9, Section 1], if (1.4) holds, then Lc(t) can approximately be estimated as h/(2mvs) where vs is the velocity of sound waves in the fluid. (Note that the estimation applies "2" instead of "4" used in the mentioned works to allow for the SchrSdinger-wave-frequency doubling due to multiplication of the wavefunction by its complex conjugate.) In case of (1.3), vs = ~/m.

The work regards the positions of the fluid particles as stochastic processes. In so doing, vector x E X(t) is used to represent the values of these positions at fixed t. This in particular leads to Remark 7.2.

The work follows the conventions below. Equation (X.Y) is the yth equation in Section or Appendix X. The appendices are denoted

with letters. Remark X.Y is the yth remark in Section X. Similarly, Definition, Lemma, Theorem, or Corollary X.Y is the corresponding yth item in Section X. Sections 7-11 involve a series of assumptions. All these assumptions are denoted with the letter "/~" followed by the dot and number. Thus, Assumption ~.Y is the yth assumption in Sections 7-11. Assumptions/~..1-~.5 and ~.6, ~.7 are introduced in Sections 7 and 9, respectively.

For the sake of space, the conditions allowing to carry out most common operations in the course of consideration are assumed to hold and not formulated in the text. In particular, all the integrals dependent on parameters are regarded to be sufficiently smooth functions of respective parameters. It is also assumed that all the multifold integrals can be evaluated in any order of the integration, and all the corresponding intermediate integrals are sufficiently regular (e.g., smooth) functions of their variables.

For the reader convenience, all the function symbols are typed in bold. In so doing, the symbol is the same as the plain-typed symbol for the variable described by the bold-typed function. For example, if a variable is denoted with n (plain "n ') , then the function for it is denoted with n


(bold "n"), no matter whether the symbols are typed in italic or not. For brevity, only such variables as t, x, and p are explicitly listed as the variables of functions. This does not mean that the functions are independent of other quantities. For instance, ni(x) in Section 5 can depend on electrochemical potentials #i, particle volumes vi, or other variables.

2. T H E V O L U M E - S C A L I N G M E T H O D

F O R O N E - C O M P O N E N T F L U I D

The construction of probability theory does not enable one to allow for probability distributions of the nonzero-volume particles in a very simple way since these particles present domains rather than points. A possible, somewhat simplified solution of the probiem can be the volume-scaling method introduced in this section for a one-component fluid.

Since in the multicomponent-fluid case the nonzero-volume particles share the same domain, one can expect certain coupling of the component characteristics for the purely geometric reasons. What specific coupling is and how the approach of Section 2 can be generalized is shown in Section 3. Section 4 interprets the Heisenberg uncertainty principle in terms of the above coupling. It should be stressed that this interpretation is related to classical distributions and provides a criterion for their physical consistency. Section 5 specifies the volume-scaling method and the treatment in Section 4 in the equilibrium case. These results are applied to multicomponent fluids of mutually noninteracting particles in Section 6. The corresponding application to red blood cells is described in [231 .

Let ~(t , x,p) where x E X(t) and p E ]~3 be the nonequilibrium (reduced one-particle) distribution function (e.g., [11, p. 38]) for the fluid. Concentration n = n(t, x) at point (t, x) and the total number Ns(t) of the fluid particles in any (sufficiently regular) subdomain X~(t) C_ X(t) are determined as follows:

n ( t , x ) = dp, (2.1)

1 In so doing, it is assumed that 0 < n( t ,x) < oo but it is not required that Ns(t) < ec. In view of (2.1), expression (2.2) is equivalent to

N,( t ) = / n( t ,x) dx. (2.3) J x ~(t)

Equalities (2.1) and (2.3) point out that quantity

£3 (t,x,p)dp es(t,x) = fx~(t) [ f~ ~(t ,y ,p) dp] dy' at X~(t) such that N~(t) < oe, (2.4)

as a function of x C Xs(t), is the probability density of the position of a particle of the fluid at time t provided that the particle is in subdomain X~(t). All the above relations correspond to the fluid of point, or volumeless, particles.

If the particle volume v is nonzero, v > 0, then the picture is different. The key difference is that a certain fraction of the volume of domain X(t) is occupied by the bodies of the particles. Subsequently, the particles are distributed by means of random motion only in the part of the volume which is available for the motion of the particles, i.e., not occupied by their bodies. In terms of the volume dV of an infinitesimal domain which includes point x E X(t) and the particle number dN in this domain, the above fact can be expressed as follows. For the volumeless particles, quantity

dN d---V = n(t, x) (2.5)

446 E. MAMONTOV

determines the particle-position probability density (2.4) whereas this density for the nonzero- volume particles is determined by means of the quantity different from (2.5), namely,

dN n(t, ~) dV [1 - ( 1 / # ) ( V - ~ v ) ] ---- 1 - ( l / ~ ) v n ( t , x ) '

(z6)

In equality (2.6), the term in the brackets is the fraction of the infinitesimal volume dV which is free from the bodies of the particles and ¢ is the maximum possible value of vn(t, x) which is the fraction of dV which is occupied by these bodies, i.e.,

O <_vn(t ,x) <_ ¢ < l. (2.7)

REMARK 2.1. The volume scaling according to (2.6) does not prohibit the particles to move, loosely speaking, "through each other" or, more specifically, to participate in the coalescence and fragmentation (or recombination and generation) processes. However, the latter phenomena are to be taken into account in the corresponding chemical-reaction terms (for instance, the one denoted with "r," in (11.1) (or (11.11)) below).

Quantity ¢ presents the densest-package volume fraction. It generally depends not only on the shape of domain X(t) and the particle shape, but also on other properties of the particles. Indeed, if X(t) - R 3, then for hard spheres (e.g., [4, (C.1.17)-(C.1.19)])

~ /5456+ 2440V/5 0.7547,

whereas one can use value ¢ = 1 for a highly elastic particle (like a red blood ceil).

REMARK 2.2. Both quantities v and ( can be quite complex functions of other characteristics of the fluid. The present work does not specify these dependences.

According to the above passing from (2.5) to (2.6), we reinterpret the integral in (2.1) as follows:

n(t ,x) 3f R 1 - ¢- lvn(t ,x) ---- 3 ¢p(t,x,p)dp, (2.8)

keeping (2.3) and (2.4) unchanged. Since both (2.3) and (2.8) hold, relation (2.2) should be generalized to

N , ( t ) = Zs(t) /xs(t) [~a~a(t,x,p)dp] dx, (2.9)

where, obviously,

N~(t) < 1. Zs(t) = Jx~(t) [f~3 •(t,x,p) dp] dx - (2.10)

The inequality in (2.10) stems from (2.3) and (2.8).

Thus, we suggest to interpret integral f~3 q~(t,x,p)dp on the right-hand side of (2.8) as the scaled concentration n(t, x)/[1 - ( - lvn( t , x)] (see the left-hand side of (2.8)) rather than concentration n(t, x) (see the left-hand side of (2.1)). For this reason, we call the above integral the scaled concentration. The denominator 1 - ( - l v n ( t , x ) in (2.8) is due to the scaling of the aforementioned infinitesimal volume dV. The scaling effect on the volume of domain Xz(t) is presented with quantity (2.10) used in (2.9).


We point out the following equivalent form of (2.8):

n( t ,x) = f ~ o ( t , x , p ) d p 1 + (1/~)v fi~s ~(t, x ,p)dp' (2.11)

Importantly, this implies the second inequality in (2.7). Moreover, expression (2.11) shows that

n( t ,x) = j~3 ~o(t,x,p)dp,

n(t, ¢ 72

in the limit case as ( - i v jf~a qo(t, x, p)dp --* O, (2.12)

in the limit case as ( - i v j[~3 q~(t, x,p) dp ~ oo. (2.13)

In case (2.12), both the common and scaled concentrations are equal to each other like they are in case (2.1) of point particles. In case (2.13), the common concentration reaches its maximum value known from (2.7).

The above issues point out that interpretation (2.8) leads to exactly what one expects in the physical reality. The key formulas related to (2.8) are (2.3), (2.4), and (2.9) (or (2.10)).

3. T H E V O L U M E - S C A L I N G M E T H O D

F O R M U L T I C O M P O N E N T F L U I D

If one deals with a mixture of, say, u > 1 fluids which are described with respective sets of the corresponding functions, ni, ~i, and N~.i, then the multicomponent analogues of (2.6) and (2.8) a r e

dN~ n~ (t, x)

(1 1 dV - ~.j vj

n , ( t , x )

1 ~ -1 - jvj j(t,x) j = l

/2

1 - E (t, 5) j = l

= ]~3 ~'i (t, x, p) dp,

i = 1 , . . , , , (3.1)

/ = 1 , . . . , u . (3.2)

Note that the ~-values in (3.1) and (3.2) for the ith component depend not only on this component, but also on the entire content of the mixture. This is emphasized by the double index (u.i).

REMARK 3.1. In the fraction of the volume of domain X(t) available for motions of the fluid particles, the volumes per one particle of the components are gi[ni(t, x)]-l[1 - ~ j = l -1 " ~. jvjnj( t ,x)] , i = 1 , . . . , u . This follows from (3.1) and the fact that distribution function q¢~ in (3.2), i = 1 , . . . , u, is proportional (e.g., [8, Remark 2.3]) to g~ where (cf. (A.6)) g~ > 1, i = 1 , . . . , u, is the number of the spin orientations of the ith-component particles. More details on these numbers can be found in Appendix A.

One can show that (3.2) is equivalent to

ni(t, x) = f ~ ~oi(t,x,p)dp , i = 1, . . . ,u , (3.3)

j = l

that presents the multicomponent generalization of (2.11). The analysis similar to that in the text on (2.12) and (2.13) can be done in the multicomponent case (3.3) as well. The multicomponent

448 E. MAMONTOV

versions of (2.3), (2.4), (2.9), and (2.10) are

N~.~(t) = / x n~(t, x) dx, ~(t)

Q~.i(t,x) = f ~ ( t , x , p ) dp fX~(t) [fR~ ~(t,Y,p)dp] dy'

N~.~(t)=Z~(t) fx~(~) [£ ~(t,z,p)dp I d~, Ns.i(t) < 1,

Zs.i(t) = f x ~ ( t ) [ f a ~ i ( t , x , p ) d p l dx -

at Xs(t) such that Ns.~(t)< c~,

i = 1 , . . . ,~ , (3.4)

i = 1 , . . . , ~ , (3.5)

i ~" 1 , . . . ,/2)

i ---- 1 , . . . , u.

(3.6)

It follows from (3.2) and (3.4) that, if volumes vi = 0 for all i = 1 , . . . , ~, then each inequality in (3.6) becomes equality.

4. H E I S E N B E R G ' S U N C E R T A I N T Y P R I N C I P L E A N D T H E V O L U M E S C A L I N G

In view of Remark 3.1, HUP (1.2) can be presented as follows:

1 - ~ ~;~vjn j ( t ,x ) j=l Ti( t , n l ( t , x ) , . , n , ( t , x ) , x ) > h 3 (4.1)

g~ nii t ,x) ., _ , i = 1 , . . . , , ,

where T~(t, nl( t , x ) , . . . , n~(t, x) ,x) is the minimum momentum volume which can be detected experimentally. The latter means that T~(t, nl( t , x ) , . . . , n~(t, x), x) can be regarded as the expectation of the momentum volume, i.e.,

Ti( t , n l ( t , x ) , . . . , nu($,x),x) -- .]/~a iplp2palpi(t, n l ( t , x ) , . . . ,nu( t , x ) , x ,p )dp , (4.2)

i = 1 , . . . ,~ ,

where Pl,P2,Ps are the entries of vector p and quantity p~(t, nl(t , x ) , . . . , n~(t, x), x,p), as a function of p, is the conditional probability density of the momentum vector of a particle of the ira-component under the conditions that, at time t, the particle is at point x and the component concentrations n l , . . . , n . at point (t, x) are equal to nl( t , x ) , . . . , n~(t, x), respectively.

Note that the term ]P,P2P3] in the integrand in (4.2) presents the volume of the parallelepiped with the edge lengths IPll, IP21, and IP31. The parallelepiped shape is due to the fact that the three-dimensional formulation (1.2) (and hence, (4.1)) is obtained (see the references above (1.2)) from the one-dimensional formulation of HUP by means of Cartesian product of the corresponding intervals along the position and momentum axis.

Functions p~(t, n l ( t , x ) , . . . , n~(t, x) ,x , .) employed in (4.2) can be obtained in various ways, for instance, by means of relations (see also (3.2))

~(t,x,p) p~ (t, nl (t, x ) , . . . , n . (t, x), x, p) = fn3 ~ (t, x, p) dp

ni( t ,x) (4.3) = ~ai(t,x~p) : v , i = 1 , . . . ,u .

1- E C~.j~j(t,x) j = l

Indeed, these expressions show that

pi(t, n l ( t , x ) , . . . , n ~ ( t , x ) , x , p ) >_ O, i = 1 , . . . , v ,

f~3 pi(t, n l ( t , x ) , . . . , n , ( t , x ) , x , p ) d p = 1, i = 1,..

A Specification of the Maxwel l -Rayle igh-Heisenberg Approach 449

i.e., each function pi(t, nl(t , x ) , . . . , n , ( t , x), x, .) is a probability density. A specific example of volumes (4.2)is (6.6) below.

One can show that (4.1) is equivalent to

Vi ~ Vmaq.i Uj -1 n i ~__ Umaq. i + ~,.-----~ ÷ , i = 1 , . . . , U, (4.4)

j=l;jT~i Umaq'J ~ ' j

where quantities

h 3 Vmaq.i = [giTi(t, n l ( t , x ) , " . . , n v ( t , x ) , : : g ) ] , i = 1 , . . . , ' , (4.5)

are the minimum average quantum volume per particle in the position-space domain X(t). The quantum nature of (4.5) is due to Planck's constant h. Quantum volumes (4.5) are not taken into account in classical mechanics since the latter, by definition, does not "recognize" Planck's constant. (The case when Ti(t, nl( t , x ) , . . . , nv(t, x), x) - 0 for some i is not interesting since it, in view of (4.2), corresponds to pi which is the Dirac delta-function in momentum p.)

Inequalities (4.4) present the formulation of HUP for the components of the multicomponent fluid where the particles are generally of nonzero volumes v l , . . . , v u . Violation of any of these inequalities means that the corresponding component enters unphysical behavior.

5. E Q U I L I B R I U M O F M U L T I C O M P O N E N T

F L U I D A N D T H E V O L U M E S C A L I N G

At equilibrium, domain X(t) and ai1 distribution functions qO~ are independent of t, i.e., X ( t ) - X and ~i(t,x,p) --~i(x,p), i = 1,. . . ,L,. As is well known (e.g., see [8, Section 4.5] and the references therein), equilibrium concentrations ni under the assumption (ef. (1.4)) that

the particle kinetic energy for the i t h component is --,P-cP i = 1,. . . , u, (5.1) 2mi

are determined as the solutions of ordinary-differential-equation (ODE) system

n ~ ( x ) - dIIi(x) dp~ ' i = 1 , . . . , ~ , (5.2)

with initial condition

lim ni(x) = 1, i = 1 , . . . , L,, (5.3) , , / (KT)~-~ Ni exp[#i/(KT)]

where the limits are uniform in x E X and

Ni = ~-g ai , (ri = , i = 1 , . . . , v . (5.4)

Pressures Hi (x) in (5.2) are evaluated by means of the common expressions

2 I~i(x) = 5U~(x), i = 1 , . . . , ~, (5.5)

U~(x) = £ [ P % ] (5.6) 3 [2-~miJ ~i(x,p)dp, i= 1 , . . . , , ,

where Ui(x) is the volumetric density of the kinetic energy of the i t h component of the fluid.

450 E. MAMONTOV

Application of the volume-scaling method in the previous sections to the multicomponent fluid at equilibrium means the following. In line with the passing to (3.2) or (4.3), the equilibrium version of ~oi(x,p) in (5.6) is

~o~(x,p) = ni(x) p i (n l (x ) , . . . ,n,,(x),x,p), i = 1, . . . ,u , (5.7)

1 - k #~.~vjnj(x) 5=1

whereas ODEs (5.2) are generalized to

n~(x) dII~(x)

j = l

- - , i = 1 , . . . , u. (5.8)

In view of (5.7), expressions (5.5),(5.6) for IIi(x) in (5.8) are equivalent to

2 £ hi(X) pmppi(nl(x), n,,(x),x,p)dp, i = 1, u. (5.9) = 5 1 - c.l.j j(x) ' " " '

j = l

Thus, in the general case the equilibrium concentrations are determined from initial-value problems (5.8),(5.3) where the pressures are calculated according to (5.9).

6. E X A M P L E OF A P P L I C A T I O N OF THE V O L U M E - S C A L I N G METHOD: EQUILIBRIUM OF THE M U L T I C O M P O N E N T FLUID OF M U T U A L L Y N O N I N T E R A C T I N G PARTICLES

To illustrate the volume-scaling method, we consider the equilibrium state of the multicomponent fluid of mutually noninteracting particles. The latter property means that the particle- momentum probability densities pi(nl(x) . . . . , n . (x) , x,p)in (5.7) and (5.9) are of MB type (1.3), i.e,,

pi(nl(x) , . . . , n u ( x ) , x , p ) : (V/-~cri)-3 exp [--(2~r2)-lpTp] , i = 1 , . . . , u . (6.1)

In this case,

I I i ( x ) = KTni(x)

j = l

/ = 1 , . . . , u , (6.2)

because of (5.9). Subsequently, ODE system (5.8) becomes of the following form:

re(z) =KT~--~[ ni(x) 1-- k -1 ~v'.3 vjnj (x) 1 -- k ~..lvjnj (x)

j=l j=l

, i = 1 , . . . , . . ( 6 . 3 )

For the sake of simplicity, we assume that each quantity (~TJvj in (6.2) is independent of #1 , . . . , # . . Then one can show that the solution of initial value problem (6.3),(5.3) is n~(x)/[1 -

v ~ y = l C . ) v j n s ( z ) ] = N~ exp[p~/Kr], i = t , . . . , u, or

ni(x) = Ni exp[pi/KT] i = 1 , . . . , u. (6.4) 1 + ~ --1 (~,.j ~3jN i exp[#j / KT]

j=l


The corresponding expressions for pressures (6.2) in terms of electrochemical potentials ~1,. . - , ]-tu are

r I i ( x ) = K T N i e x p [ - ~ T ] , i = 1 , . . . , ~ . (6.5)

Note that vi, ~,.~, and #~ in (6.4) and (6.5) as well as g~ and m~ in (5.4) may depend on x thereby determining the x-dependence of the concentrations and pressures.

Substituting (6.1) into (4.2), one obtains

Ti ( 2 K T r n i ) a/2 = - , i = 1 , . . . , , , (6.6)

7F

for the particle momentum volumes. The corresponding particle minimum average quantum volumes (4.5) for the fluid components become Vmaq.i = gi-l(rc/ki) 3, i = 1 , . . . , L,, where ,ki = h/x/2rcKTm~ are (e.g., [21, (3.6), p. 3511) the so-called thermal de Broglie wavelengths. Allowing for this, one can show by means of (6.4) that the HUP inequalities (4.4) are equivalent to

exp ~ _< ~ , i = 1 , . . . , (6 .7 )

Thus, the MB probability densities (6.1) are physically meaningful if and only if (6.7) holds, no matter if the particle volumes are zero or nonzero. To our knowledge, result (6.7) is the first exact estimation for the physical validity of the MB distribution. This result, i.e., p i / K T <_ - ln(rr a) -3.43, i = 1 , . . . , u, presents the rigorous form of the well-known heuristic criterion [24, the right column on p. 154]. Relations (6.4) and (6.2) show how geometric quantities vi and (..~ affect the concentration and pressure, respectively. Interestingly, the HUP condition (6.7) does not explicitly include any geometric characteristics.

Paper [23] exemplifies application of the above representations to the evaluation of the electrochemical potentials and pressures of red blood cells in human blood. The results of Sections 5 and 6 are also used in the analysis in Sections 7-11 below.

7. THE RELATIVISTIC ISDE MODEL: P R E L I M I N A R I E S A N D A S S U M P T I O N S

The equilibrium distribution function can be constructed by means of the component concentrations and momentum probability densities (cf. (5.7)). The latter densities can in turn be described in various ways. One of them is the description by means of the stationary solutions of certain ISDEs for momenta of the particles of the fluid components. This is precisely the idea underlying NNSHD treatment discussed in Section 1. Until now, it is developed under assumption (5.1). However, the biology-related systems include a great variety of physical structures and phenomena (e.g., [20]). They in particular comprise dielectric, conducting, and semiconducting systems. The latter are often associated (e.g., [25]) with the dependences of the particle kinetic energy u on the particle momentum vector p which are neither parabolic nor isotropie, like that employed in (5.1) (or (1.4)). Einstein's dependence (1.5) points out that the nonparabolicity is a manifestation of the relativistic nature. The present section generalizes the equilibrium part [8] of the NNSHD model to the u-p dependences of this kind. In so doing, only the one-component fluids are considered and the following assumptions are involved.

ASSUMPTION A1. The fluid is one component and is at the equilibrium state. Scalar T is independent of (t ,x,p). Domain X(t) is independent o f t and denoted with X, i.e., X(t) - X. T~.e equilibrium one-component probability density Pl (e.g., see (4.3) at ~, = 1) is denoted more briefly, i.e., p(x,p) = pl(nl(x) ,x ,p) . Expressions (5.7)-(5.9) are also rewritten in the more

452 E. MAMONTOV

compact forms, namely,

n(x) ~(x ,p) ---- 1 -- •- lvn(x) p(x ,p) , (7.1)

n(~) d n ( x ) - - - (7.2)

1 - ( - l v n ( x ) d# '

2 ~ n(x) II(x) = 5 a 1 - ( - l v n ( x ) u (x ,p )p ( x , p ) dp, (7.3)

where function u generalizing dependence (1.4) is described in Assumption ~ 2 below (e.g., see (7.4)), and both quantities v and ( are independent of p.

NOTES. As follows from the text below (4.2), function p(z , .) is the conditional probability density of the particle momentum vector p under the condition that the particle is at point x E X which is independent of time.

ASSUMPTION A2. Scalar u is described as follows:

= u(x , p),

where (e.g., [25, (4.4), p. 95, Section 3.4])

(7.4)

Moreover,

u(x, -p) = u(x, p),

u(x,p) > 0, if and only if p 7£ O,

U(X, ') C C 3 (m3).

(7.5) (7.6)

(7.7)

symmetric matrix ( V p V f ) u ( x , p ) is positive definite, (7.8)

where Vp is the Hamilton differential expression with respect to the entries of vector p, i.e., Vp = (bo m 0 a T , o ;~ , ~ ) "

NOTES. Relations (7.5), (7.6), and (7.8) are common features of the p-dependence of the particle kinetic energy (7.4) (e.g., [25, Section 3.4; (8.4), p. 111, (8.5), p. 112, (8.6), p. 113, Section 3.8; (9.2), p. 118, Section 3.9]). These relations mean that point p = 0 is a point of a local minimum of function u(x, .) and this point is also a point of the global minimum of the function.

Also note that for photon (e.g., [12, (3.8), p. 351]) u(x ,p) -- c H • [[ (that can be obtained from (1.5) at IIp[]/me --+ ec). In this case, the function u(x, .) is not differentiable with respect to p at p = 0. Subsequently, any treatment involving the p-differentiability of u(x,p) , in particular, the approach of the present work (cf. (7.7)), is inapplicable to photons.

ASSUMPTION A.3. The particle-velocity vector v is determined as follows (e.g., [12, p. 357; 25, (1.3), p. 121, Section 4.i]):

v = VpU(X,p). (7.9)

NOTES. In general, image Y(x) C_ ]~3 of function Vpu(x, . ) need not coincide with the entire space N3. For instance, in the relativistic case (1.5), it is bounded, more specifically, J(x) = {v E R 3: It~[I -< c}.

ASSUMPTION A.4. There exists function p of x and v which is defined for all x C X , v E J(x) and is such that dependence

p = p(x, v) (7.10)

is a solution of equation (7.9).

NOTES. In view of (7.8) and the well-known theorem (e.g., [26, Section 5.4.4, p. 143]), solution (7.10) of equation (7.9) is the unique solution of this equation. Relations (7.9) and (7.10) are equivalent to each other. In the case of (1.5), they are (B.1) and (B.2), respectively.


ASSUMPTION A.5. The particIe-mass matrix is determined (e.g., [12, p. 357; 25, (1.11), p. 123, Section 4.1]) as M(x,p) where

M(x,p) = [(VpV ) -1 (7.11)

In view of (7.8), matrix (7.11) is symmetric and positive det~nite. The (i,j)-entry, i , j = 1, 2, 3, of matrix (7.11) is denoted with mij(x,p) .

NOTES. As follows from (7.9)-(7.11),

---r-~ = M (x, p). (7.12) Ov

In view of (7.7) and (7.11), M(x, .) E C 1 (Ra). (7.13)

The derivatives of function M(x,-) can be evaluated without the corresponding differentiation of the inverse matrix in (7.11) by means of the well-known formulas

OM(x,p) _ M(x,p)0 [(VpV ) u(x,p)] M(x,p), = 1,2,3. Opi

In the case of photons, i.e., when (see Notes in Assumption/~.2) u(z, .) -- cll. ]], det[(VpV~) × u(x,p)] = 0 for all p. Subsequently, matrix (7.11) does not exist, and hence, the mass of photon is not defined at any p.

Assumptions /~.1-~.5 are common in any one-component-fluid treatment of the particle u-p dependence (7.4). For example, one can easily check that Einstein's version (1.5) of (7.4) satisfies all these assumptions.

We also note that (1.5) is commonly used not only when the particle is in a vacuum but also when it is in other media. For example, expression (1.5) is employed in semiconductor theory in the following equivalent form u(1 + c~u) = [[pN2/2m where a = 1/2Eg, Eg is the energy-band gap of the semiconductor material, m is the effective mass of the mobile charge carrier, i.e., the electron in the conduction band or the hole in the valence band of the material, c = V ~ 9 / m is the velocity of light in the conduction (or valence) band.

REMARK 7.1. As follows from Notes in Assumption A.4, the particles under Assumptions ~ .1- /~.5 can be modelled in terms of velocity vector v or momentum vector p. These descriptions are equivalent. However, the p-treatment may be more convenient than the v-treatment since the set of values of p is always the whole space ]R 3 whereas set J(x) of values of v can be bounded and, in principle, of a complicated shape.

REMARK 7.2. In connection with Notes in Assumption ~.1 (see also Section 1), we emphasize the following. The NNSHD model (proposed in [4, Appendix C, Section 4.10], further developed in [8,9], and generalized in the present work) describes fluids by means of Eulerian variables. The distinguishing features of this approach are well known [27, p. 25].

"Concentrate attention on a given point of space at which arrive at different times different particles. In essence, this is the Eulerian approach to the study of the motion of continua. For instance, the motion of water in a river can be studied either by following the motion of each particle from the upper reaches to the mouth of the river (the Lagrangian approach), or by observation of the changes of the flow of water at definite locations in the river without tracing the motion of individual water particles along the entire river (the Eulerian approach).

From a Lagrangian point of view, the laws of change of the velocity, acceleration, temperature, etc., of a given particle of a continuum are of interest. In the Eulerian

454 E. MAMONTOV

approach, one studies the velocity, acceleration, temperature, etc. at a given location, i.e., one focuses attention on some region of space and wishes to know all the information on the particles which pass through it."

In line with the Eulerian approach, NNSHD models the dynamics of fluids at given locations. In so doing, no equation for stochastic trajectories of the fluid particles are included. These trajectories can be obtained by means of the particle-position probability density (eft (2.4) or (3.5)). Note that the absence of the stochastic-trajectory equation and the emphasis on the densities are also the basic features of the SchrSdinger-wave version of QM (e.g., [14]).

The above assumptions and considerations are used in the next sections.

8. THE PARTICLE K I N E T I C - E N E R G Y - D E T E R M I N E D RAYLEIGH DISSIPATION F U N C T I O N A N D T H E ODES FOR

THE PARTICLE VELOCITY A N D M O M E N T U M V E C T O R S

Before formulating the relativistic model for the particle-momentum vector p, we need to introduce the notion of a special type of the particle Rayleigh dissipation function (see Definition 8.1 below). The present section discusses this notion. Section 9 describes the relativistic model and points out its equivalent form.

DEFINITION 8.1. Let Assumptions A.1-~.5 hold. Also, let scalar function ¢ of the particle position x 6 X and the particle kinetic energy u >_ 0 be of the following properties:

• the physical dimension of ~(x , u) is work, • is continuous for ali (x, u), and

• (x, 0) - 0; (8.1)

• • is continuously differentiable in u for all x and almost all u with the positive u-derivative, i.e.,

O~(z, u) > o; (8.2)

Ou

• time parameter v(x , u) is determined as follows:

[~(x,u)]_~ = O~(x,u) . (8.3) Ou '

• if the only force acting on the particle in the asymptotic equilibrium case is the dissipative force, then ODE

Ov -- [~'(x, u(x ,p))]- lv , at fixed x and (7.9) or (7.10), (8.4)

Ot

is the corresponding asymptotic representation for velocity vector v = v( t, x) of a particle passing fixed, time-independent point x at time t;

• if ODE (8.4) holds, then any solution of it with the initial time point to C R is defined for

all t 2 to.

Function ,~ of the above properties is called the particle kinetic-energy-determined Rayleigh dissipation function or, brieIty, E-RDF. Time 1-(x, u) is cMled the particle characteristic dissipation time (CDT).

Equation (8.4) is the ODE for the particle-velocity vector v. Note that the meaning of this velocity in the text below (8.4) is in a complete agreement with Eulerian picture in Remark 7.2. The term "Rayleigh dissipation function" (RDF) in Definition 8.1 is due to the function introduced by Lord Rayleigh (J.W. Strutt, Third Baron Rayleigh, 1842-1919) in 1873 [28, Section II] (see also [29, Section 81 and the footnote on p. 1031). We explain below why Definition 8.1 applies this notion.


REMARK 8.1. In view of (8.2), (8.3), (7.4), and (7.9), equation (8.4) is equivalent to

C~V 0---~ = -VpO(x,u(x,p)), at fixed x and at (7.9) or (7.10), (8.5)

that is another form of the velocity ODE (8.4). Multiplying (8.5) from the left by the particle- mass matrix (7.12) and accounting the property of this matrix to be symmetric (see (7.11),(7.8)), one obtains the following ODE for the particle-momentum vector p:

Op - V ~ ( x , u ( x , p ) ) , at fixed x and at (7.9) or (7.10), (8.6) Ot

where Vv is the Hamilton differential expression with respect to the entries of vector v, i.e., Vv -- (01 o o )T. Subsequently, vector - V v ~ ( x , u ( x , p ) ) at fixed x and at (7.9) or (7.10) ' Dv2 ' Or3

is the dissipative force acting on the particle. This expression for the force is the well-known distinguishing feature of common RDF (e.g., [30, pp. 21-22]). This fact as well as features (8.1) and (see (8.2)) ~(x , u) > 0 at u > 0 point out that function ~(x , u(x, .)) (under condition (7.9) or (7.10)) in Definition 8.1 is precisely a RDF. Since it is related to a particle and governed by its kinetic energy, it is called the particle kinetic-energy-determined RDF (or E-RDF). Importantly, the RDFs of the forms similar to that of the kinetic-energy (and hence, the ones which may be expressed in terms of this energy) were outlined as early as in 1877 by Rayleigh [29, p. 130] (e.g., see [31, (2.3)] for an example of the particle RDF which is in fact the simplest E-RDF). In this respect, Definition 8.1 can be regarded as a specification of Rayleigh's ideas. The systematic involvement of RDFs in the present approach is exemplified with the relativistic results (10.11) and (10.12) below.

The above E-RDF is, however, less and more general at the same time than the common RDF. The key features in the notion of E-RDF are the fact that ~ does not explicitly depend on momentum p, and hence, the velocity v and the fact that, if the dissipative force is the only force acting on the particle, then the velocity and acceleration vectors are collinear (ef. (8.4)). These features enable one to introduce scalar CDT T(x, u) (see (8.3)). In contrast to this, the common RDF allows a more general dependence on the velocity (e.g., [29, Section 1; 30, p. 22; 32, (28)]) resulting in the matrix dissipation time. Along with this, common RDF is usually assumed to be parabolic in the velocity. Definition 8.1 of E-RDF does not involve this restriction and even any restriction on the specific form of the velocity dependence. In particular, it allows for the relativistic dependence (B.3) equivalent to (1.5). The fermion E-RDF (10.11) below and the text on it exemplify that. Much earlier examples of nonparabolic RDFs can be found in [33].

Note that force - V ~ ( x , u(x,p)) determines the power associated with dissipation. Indeed, differentiating (7.4) with respect to t and allowing for ODE (8.6), one obtains that, at fixed x and under condition (7.9) or (7.10), ou(x,p) =-p(x , u(x,p)) where scalar P(x,p)----vTVv~(x, u(x,p)) Ot

is the above power.

Definition 8.1 does not specify the physical nature of E-RDF. For instance, it can be due to the scattering of the particie with the surrounding particles if they are present in the same medium, the friction of the particle with the medium, or the combination of these phenomena. From this point of view, the above physical picture is fairly universal (see also [29, Section 45] for a general discussion on the dissipative force).

Quantity ~-(x, u) is in Definition 8.1 called the particle characteristic dissipation time to avoid confusion with the notion of the particle relaxation times. Indeed, the velocity relaxation times are the entries of the matrix inverse to the matrix of the partial derivatives of the right-hand side of ODE (8.4) (or (8.5)) with respect to the entries of vector v. Similarly, one can determine the momentum relaxation times by means of ODE (8.6). However, in spite of the fact that E-RDF is associated with scalar CDT, the E-RDF notion does not imply isotropicity of the dissipation. Indeed, the isotropicity in velocity (momentum) usually means that the matrix of the velocity

456 E. MAMONTOV

(momentum) relaxation times is scalar. No consequence of this kind stems from equations (8.4) (or (8.5)) and (8.6) since the kinetic-energy function u(x, .) does depend on momentum and hence (cf. (7.9)) on velocity. In connection with the Definition 8.1, we also note the following.

REMARK 8.2. Not every positive function r of x ~ X and u _> 0 meets the physical requirements formulated in Definition 8.1. For instance, if (1.4) is the case and [4, (C.4.1)]

r ( x , = (8.7)

then the last condition in Definition 8.1 is equivalent (e.g., [4, Remark 4.2]) to inequality s k 0. The latter also assures that, in case (8.7), ' ~ -1 hm~10 f0 [r(x,w)] dw = 0 holds, i.e. (see also (8.1)), aS(x, .) is continuous at p = 0. The power dependence (8.7) is, however, rather idealized. Indeed, there are problems where parameter z may depend on kinetic energy u. For instance (e.g., [34]),

= 1 at low u and e = 1/2 at high u. A discussion on more general forms of the u-dependence of function r can be found in [4, pp. 214-216, Appendix C.4; 8, Section 4.4], and the references therein.

The next section applies E-RDF to determine terms in ISDEs. Interestingly, conjunction of Rayleigh's dissipation functions with stochastic differential equations is not a very common topic in the literature. A few examples are [32,35].

9. T H E R E L A T I V I S T I C I S D E S F O R T H E E Q U I L I B R I U M

P A R T I C L E V E L O C I T Y A N D M O M E N T U M V E C T O R S

A N D T H E C O R R E S P O N D I N G P R O B A B I L I T Y D E N S I T Y

This section proposes the relativistic ISDE for the equilibrium particle-velocity vector that generalizes the nonrelativistic velocity ISDE [4, (C.I.I); 8, (4.1)], on the one hand, and the deterministic ODE (8.5) (or (8.4)), on the other hand. The section also derives the ISDE for the momentum vector which is equivalent to the relativistic velocity ISDE. The velocity ISDE is

introduced in the assumption below.

ASSUMPTION A.6. All the assumptions in Definition 8.1 hold and there exists E-RDF • such that the equilibrium velocity v of a single particle is determined as the unique stationary solution of the following ISDE:

dv = - V p ~ (x, u(x, p))]p=p(~,~) dt

i (9.1) + 2KT { [ O ~ ( x ~ x ' P ) ) 1 [ ( V p ~ ) u ( x , p ) l } dW( , , , ) , at f ixedx,

p=p(x,v)

where ~ E E is an elementary event, S is the space of elementary events, and W(~, t) is the three- dimensional Wiener stochastic process, i.e., the vector of the mutually stochastically independent scalar Wiener processes. In so doing, the equilibrium velocity probability density is the stationary probability density of the above solution.

Assumption ]k.6 is one of the main assumptions in the present approach (it is included in Assumption/1~.7 below). It postulates the It6 stochastic generalization of ODE (8.5). Why the second term on the right-hand side of equation (9.1) is of the very form shown in the equation becomes clear in Section 10 (see Remark 10.3 for a concise summary). Equation (9.1) is the main equation in the present approach. Notion of E-RDF is the key notion.

The present approach does not postulate the microcanonical, canonical, or grand canonical ensembles. It applies neither TDL (1.1) nor a "statistically large" number of particles or a "macroscopically big" domain. It does not presume to construct the kinetic-equation collision integral, equate its integrand to zero, and resolve the resulting equation. It instead involves the assumption on the stochastic nature of the equilibrium particle-velocity probability distribution. We stress this more specifically below.


REMARK 9.1. The NNSHD approach models the equilibrium probability distribution of velocity vector v of a single particle as is described in Assumption/1..6 in terms of the following input characteristics: absolute temperature T and two functions, u and O, where the latter represents the equilibrium interaction of the particle with its surrounding.

The stochasticity of v in (9.1) makes the corresponding v-dependent dissipative force (see (8.5)) random, highly irregular. This conceptually agrees with many experimental results (e.g., see [36,37]). We also emphasize that the nonlinearity of the above v-dependence is an inherent feature. It is present even in extremely small, nanometer-scale systems (e.g., [38]).

The principle in Remark 9.1 is in line with the following issues [11, pp. 321-322].

"Probabilistic arguments are not to be excluded from statistical dynamics, but they should not be declared as 'fundamental': they can only be considered, more modestly, as helping to model the behavior of complex dynamical systems. Knowing that stochastic laws can be derived for simple systems, a probabilistic law of evolution can be conjectured, without proof. It then acquires the logical status of a postulate, whose con- sequences must be tested directly against experiment. Once a reasonable set of working assumptions are accepted, the way is open for nontrivial physics. These assumptions have to do with the 'degree of randomness' admitted a priori for the description of the system's law of evolution . . . . Thus, one could adopt a ~semi-dynamicaF point of view like in the Langevin equation: a Newton equation with a random force. Or else, one could replace the dynamics by purely probabilistic rules, like in the continuous time random walks . . . . In all these cases the working assumptions are sufficiently clear and their adoption (or experimental verification) poses no serious problems of conscience. .. . We are only beginning to uncover an extremely rich domain of problems."

The assumptions and terms in Definition 8.1 enable one to rewrite ISDE (9.1) as follows:

d v z V

u(x, p))Ip=,(x,v) dt + ~/2KT { {'r (x, u(x, p))[M(x, P)]}[p=p(~,v)}-1 dW(~, t),

at fixed x. (9.2)

This form of ISDE (9.1) explicitly shows that it is a stochastic generalization of ODE (8.4).

REMARK 9.2. Eulerian paradigm inherent in (8.4) (see the text below Definition 8.1), and hence, in the velocity ISDE (9.1) or (9.2) makes them different from all the Lagrangian-variables-based velocity ISDEs which are related to the Kramers model (e.g., [39, Section 5.3.6.a)]) and usually used (e.g., [40]) in connection with statistical physics.

The nonrelativistic version of ISDE (9.2) is this equation under assumption (1.4). Equa- tion (9.2) in this case was introduced in [4, Appendix C.1] and studied in [4, Sections 4.10,4.11; 8, Section 4]. It is also used in [9] as the equilibrium part of the NNSHD model. The results of the equation testing can briefly be summarized as follows.

The nonrelativistic version of ISDE (9.2) enables one to reveal the "long", nonexponential "tail" of the particle-velocity-vector covariance for the hard-sphere fluid by means of approxi- mate analytical technique [4, Appendix C.4, Section 4.10]. Moreover, as is shown in [8, Sec- tions 4.2,4.3,4.5,3.1], the above version provides not only the MB distribution but also includes a complete formulation of the Fermi-Dirac (FD) velocity probability density including the degenerate, Fermi-fluid limit case.

According to Assumption 1.6, ISDE (9.2) describes the equilibrium velocity v. The equilibrium momentum p and the momentum probability density p(x, .) pointed out in Assumption/~.1 can also be described with (9.2) but only after application of change of variable (7.10) to it. Indeed, the following lemma is valid.

458 E. MAMONTOV

LEMMA 9.1. Let Assumption ~.6 hold and tr(.) be the trace of the matrix. Then, ISDE (9.2) for the velocity vector v of a single particle is equivalent to ISDE

M(x, p)Vpu(x, p) - KTq(x, p) / 2 K T M ( x , p) at fixed x, (9.3)

for the particle-momentum vector p where the i th entry q~ (x, p) of vector q(x, p) is described as qi(x,p) = tr{[(V~V~)p~][M(x,p)]-l}, i = 1, 2, 3, or

3

q,(x ,u) = E L OPt J ' j=l i = 1, 2, 3. (9.4)

The equilibrium particle momentum is determined as the unique stationary solution of ISDE (9.3). The equilibrium probability density p(x, .) of the particle momentum is the stationary probability density corresponding to this solution.

PROOF. The proof is based on the well-known (e.g., [41, (5.311),(5.3.10)]) It6 theorem applied to stochastic differential dp of vector (7.10) (see also (7.12)). The derivation of expression (9.4) involves (7.12) and (7.13).

REMARK 9.3. The velocity ISDE (9.2) does not include the convective acceleration {°--~v Sub- \ O x / "

sequently, the momentum ISDE (9.3) does not include the corresponding term either. The reason is that the related convective effects are accounted with the convective force f , (see (11.13)) in the nonequilibrium momentum equation (11.12).

Probability density p(x, .) in Assumption .~.1 and Lemma 9.1 is evaluated with the help of Theorem 9.1 below. This theorem applies the following assumption.

ASSUMPTION A7. The hypothesis of Lemma 9.1 is valid. Solutions of lSDE (9.3) are diffusion stochastic processes which

• are defined for all t > to where, as before, to the initial time point; • have drift vector -[~-(x, u(x,p))]-l[M(x,p)VpU(X,p) - K T q ( x , p ) ] and diffusion matrix

2KT[7"(x, u(x,p) )l- l M (x, p); • and are described with the corresponding Kolmogorov forward equation (KFE) (or Fokker-

Planck equation) at every t at which they are defined.

Moreover, relations

A~.MB(X)=f~aexp [ u~TP)] dp < oo,

p~.MB(X,p)=[A~.MB(X)]-lexp[ u~TP) ] ,

T~.MB(X) = J~R~ T(X, U(x,p))p~.MB(X,p) dp< oo,

/ ~ u(x,p)~'(x, U(x,p))pr.MB(Z,p) dp < oc,

(9.5)

(9.6)

(9.7)

(9.8)

are valid.

The conditions formulated in Assumption ~.7 above (9.5) are not very restrictive. Indeed, as is well known in ISDE theory (e.g., [41, (9.3.1)]; see also [4, Theorem 1.3] and the references therein), similar requirements are met under rather mild conditions. Assumption /~.7 enables one to formulate the theorem on density p(x, .).


THEOREM 9.1. Let Assumption ~.7 hold. Then, probability density p(x, .) verifies the detailed- balance equation system corresponding to the stationary version of the KFE in Assumption ~. 7, i.e., equation system,

3 rou( ,p l E n j(x,p) t opJ J -KTq (x,p) j=l

u(x, ;)) £ ° [2gTm,j(x,p),(x,p)] = o, j=l ~pj [ T(X,U(X,p)) J

i = 1,2,3.

Moreover, density p(x, .) exists for all p C ~3 and is evaluated as follows:

p(x ,p) -- T(X, U(X,p))pr.MB(X,p), (9.9) Tr.MB(X)

where Pr.MB(X,p) and Tr.MB(X ) are described with (9.6) and (9.7), respectively. Density p(x, .) is an even function of p, i.e.,

p (x , -p ) = p(x,p). (9.10)

The pressure corresponding to this density, i.e., quantity (7.3), exists.

PROOF. The proof is an application of the well-known theorem (e.g., see [4, Corollary 1.2, Re- mark 1.8] and the references therein) to ISDE (9.3). Property (9.10) follows from (9.9), (9.6), and (7.5). Pressure (7.3) exists because of (9.8).

Theorem 9.1 is the generalization of Theorem 1 in [8] for the cases when the u-p dependence (see (7.4)) need not be of property (1.4). More specifically, (9.6), (9.7), and (9.9) are the generalized versions of [8, (3.4),(4.9),(4.8)], respectively. Subsequently, all the analysis in [8] that follows Theorem 1 therein can easily be reformulated for the present case. In particular, we complement Remark 9.1 with the following one.

REMARK 9.4. According to Theorem 9.1, the equilibrium probability density p(x, .) of the momentum of a single particle is determined not only by absolute temperature T and function u(x, .) but also by the CST function T(x, .) (see Definition 8.1) which enables one to take into account the particle interaction with its surrounding. Result (9.9) establishes the one-to-one correspon- dence between functions p(x, .) and r (x , .) (cf. [8, p. 901]). Indeed, p(x, .) can be evaluated if r (x , .) is known (see [8, Section 4.4] for a discussion on the latter function). Inversely, if p(x, .) is known, then one can evaluate v(x, .). The latter gives E-RDF (see Definition 8.1) by means of (8.3) and (8.1), namely,

/o u dw (I)(x, u) = v(x, w) ' u > 0. (9.11)

Thus, expression (9.9) is able to take into account virtually any interactions of the particle with the surrounding.

What is not considered in [8] but is available now are the relativistic representations. This topic is discussed in the section below.

10. T H E RELATIVISTIC C O N T E N T OF THE P R O P O S E D MODEL: A P P L I C A T I O N TO RELATIVISTIC C O M P O S O N S

The equilibrium probability density (9.6) of the momentum of a single particle is precisely the relativistic MB density. Density (9.6) (see also (9.5)) in the particular case of Einstein's relation (1.5) coincides with the well-known density [21, (3.10), p. 450] described in detail in [21, Section 6.3] (see also [42, Chapter VIII, Sections 3,4; 39, Section 14]). In so doing, the expression for (9.5) in terms of (1.5) is also available in [21, (3.10), p. 4501.

However, the present result is more general since it does not confine the u-p dependence to (1.5). Note that, as follows from (9.9), p(x,p) =_ p,-.MB(X,p) if and only if function ~- (introduced in

460 E. MAIvlONTOV

Definition 8.1) is independent of u, i.e., the fluid particles are mutually noninteracting. In this case, the role of Ni (see (5.4)) is played by

N(~I = ( ~ ) h ~ ( ~ ) . (10.~)

Subsequently, the corresponding version of condition (5.3) is

lim n(x) = 1. (10.2) ,/(KT)--+-~ N(x) exp[tt/(KT)]

The relativistic versions of the momentum-volume expectation (4.2) and the minimum average quantum volume per particle are

Yr(z) = £3 [plp2pnlp(x,p) dp, (10.3)

h a Vmaq- [g.~r(X)] (10.4)

REMARK 10.1. The procedure to determine the distribution function and concentration from the momentum probability density described at the end of Section 5 is in the general, relativistic case extended as follows. First, the fluid pressure is evaluated by means of functions u and p according to (7.3) (cf. (5.9)). Next, the fluid concentration is determined as the solution of ODE (7.2) with initial condition (10.2) (cf. (5.8),(5.9)). The concentration makes it possible to verify the one-component version of HUP (4.4), i.e.,

[ ( ~ ) l -1 n(x) _~ Vma q -Jr , (]-0.5)

where Vm~q is described with (10.4) (see also (10.3)). Density p can be used if and only if (10.5) holds. Finally, the fluid distribution function is constructed as shown in (7.1).

Relativistic distribution functions are also well known (e.g., [43]) for the quantum-mechanical particles mutually noninteracting in the QM sense. The latter means that they are noninteracting up to the interactions inherent to their QM nature. These particles are bosons and fermions (see Appendix A). From the classical point of view, they are mutually interacting: fermions obey the Pauli-exclusion interactions whereas bosons interact with each other according to their bosonic nature. Strictly speaking, the particles are mutually noninteracting if and only if they are the MB ones. Any deviation from the MB distribution points out that the interaction is the case. There are many examples of the non-MB fluids where, however, neither boson-specific nor fermion-specific features are pronounced. For instance, a series of the examples (obtained by the molecular-dynamics simulation) are pointed out in [44, p. 227] (see also the references therein).

The boson and fermion relativistic distribution functions in [43] are almost the same as the nonrelativistic ones, i.e., (A.1) and (A.2), respectively. The only but very important difference is that the form of dependence (7.4) in the relativistic case is not tied down to (1.4) and, thus, is fairly general. The most well-known relativistic form is (1.5). Since the method of [22] is independent of the form of dependence (7.4), it can straightforwardly be applied to the relativistic QM distributions. This results in the composon relation (A.5) under condition (7.4), i.e., (see also (9.6) and (10.1)),

g exp[(u(x,p) - #)/KT] + z ~(x,p)[v=o = ~-g {exp[(u(x,p) - #)/KT] - 1}{exp[(u(x,p) - , ) / I (T] + 1}

{exp[(u(x, p) - #)/KT] + z} exp[(u(x, p) - #)/KT] (10.6) {exp[(u(x,p) - p ) / K T ] - 1}{exp[(u(x, p) - p ) / K T ] + 1}

x N(x)exp[#/KT]pr.MB(X,p), # _< O, at z > -1 .


Applying the recipe in Remark 9.4, one can easily reveal what density (9.9) involved in (9.9) correspond to (10.6).

Indeed, one can determine the density as follows:

and function ~"

p(x, p) = [ ~ ( x , p) l . =- 0] [ - ( ~ ) l v - - 0 ] ' (10 .7 )

where the numerator is expressed with (10.6) and the denominator is the equilibrium version of (2.1), i.e., n(x)l,_=0 = f e~(x ,p ) l ,=_odp . Then it follows from (10.7) and (10 .6) tha t density (9.9) is

p(x, p) = {exp[(u(x, p) - # ) /KT] + z} exp[(u(x, p) - ~ ) /KT]

{exp[(u(z, p) - # ) /KT] - 1}{exp[(u(x, p) - # ) / K T ] + 1} (10.8)

x[n (x ) i , -o l - lN(x )exp[p lKT]pr .MB(X ,p ) , # < 0, at z > -1 .

The corresponding expression for CDT r (x , u(x,p)) comes from comparison of (10.8) and (9.9), namely,

T(X, U(X, p)) = rr.MB (x)In(x)Iv----o]- 1N( x ) exp[tt /KT]

x {exp[(u(x,p) - #) IKT] + z} exp[(u(x, p) - # ) I K T ] (10.9) { e x p [ ( u ( z , p ) - v)/KT] - 1 ) { e x p [ ( u ( x , p) - v)/KT} + 1}'

#_< 0, at z > -1 ,

where r~.MB (x) is described with (9.7). In the limit case pointed out in (10.2), composon representations (10.8) and (10.9) become the MB ones. This is the only case when r (x , u) is independent of u. The versions for bosons and fermions are (10.8) and (10.9) at z = 1 and z = -1 , respectively.

For instance,

_ , , N ( x ) e x p [ p l K T ] e x p [ ( u ( x , p ) - p ) l K T ] , ( x , u(x,p)) = ,r.MBLX) n--~-v~-_-o exp[(u(x,p) -- # ) I K T ] + 1' (10.10)

at z = - 1 .

The Pauli-exclusion component T~.pe (x, u(x, p)) of CDT ~-(x, u(x, p)) for relativistic fermions can be determined from the well-known equality (e.g., [4, (18)]) [~-(z, u(z,p))] -1 = [r~.MB(X)]-I + [~'~.pe(x, u(x,p))] -1 that, in view of (10.10), leads to

exp[(u(x, p) - ~ ) / K T ] q ' r . P e ( X , U ( X , p ) ) : Tr.MB(X )

( 1 - ~)exp[ (u (x ,p ) - # ) / K T ] + 1'

= N(x)exp[# /KT] at z = -1 . n ( x ) l . ~ o '

In the particular nonrelativistic case when dependence (7.4) is presented with (1.4), expression (10.10) is reduced to [8, (4.17)]. In the FD case (10.10), E-RDF (9.11) is of the following form:

U -(x)l,,_:o ~+ KTn(:~)lv-o L ',/JP1-exp(-~--~51 • (x,u) = ~'~.MB (x)N(x) exp[#lKT] ~'r.MS (x)N(x) (10.11)

at u > 0 and z = -1 .

This is the E-RDF for fermions. It is valid in both the nonrelativistic and relativistic cases since they differ only in the specific forms of dependence (7.4) (cf. (1.4) and (1.5)). We are not aware of any other expression in the literature for Rayleigh's dissipation function for fermions. Note that

¢~(X,U) -~ ['Fr.MB(Z)]-lit, in the MB limit case (10.2), (10.12)

i.e., E-RDF (10.11) in the MB case becomes linear in u as it should be.

462 E. MAMONTOV

The analysis similar to that in (10.10)-(10.12) can also be carried out for bosons (see (10.9) at z = 1) or composons (see (10.9) under condition (A.7)). The relativistic-composon pressure YI(x), concentration n(x), and distribution function ~(x,p) at zero v (used in (10.6)-(10.11)) or nonzero v are calculated as described in Remark 10.1. In so doing, existence of pressure (7.3) is assured by Theorem 9.1.

The above considerations show that the model in Section 9 agrees with the well-known results (e.g., in [21,43,44]) on the probability distributions for relativistic fluids. They also explain how composons (see Appendix A) can be treated by means of the above model as relativistic particles.

REMARK 10.2. The notions of potential energy, kinetic energy, and RDF are common in both the Lagrange equation [29, Section 80-82] (see also [30, pp. 19-22; 46]) and the related models in mechanics. They are also employed in hydrodynamics [47, Section 345] (see also [48]) and mechanics of ensembles of particles (e.g., [32,35]). The potential and kinetic energies of a single particle have enormous application in both classical mechanics and QM. In a sharp contrast with this, RDFs for a single particle are underrepresented in the particle/statistical-physics literature (in fact, the only example we are aware of is the RDF [31, (2.3)] which can, in terms of Defini- tion 8.1, be recognized as the simplest E-RDF). This is the gap filled by the above ISDE (9.1) or (9.3) and the analysis in the present section. The systematic involvement of the E-RDF notion including evaluation of E-RDFs (e.g., (10.11),(10.12)) from Theorem 9.1 on ISDE (9.3) empha- sizes the role of the single-particle RDFs. This new feature is one of the advantages of the present approach.

Coming back to MRHA discussed in Section 1 and to the topic of Remark 1.1, we note the following.

REMARK 10.3. As is discussed in Section 1, the three main components of MRHA, the Maxwell- Rayleigh-Heisenberg approach, are certain results of these authors. They determine the key features of the models proposed in Sections 8-10.

First, Heisenberg's principle (1.2) points out that the characteristics of a single particle are random. This fact is in the present treatment met by ISDE (9.3) for the random momentum of a single particle and other quantities related to it and described in Remark 10.1. In so doing, the above principle is also used quantitatively (see (10.5)) as the criterion for the physical validity of the distributions.

Next, Theorem 9.1 on the stationary solution of ISDE (9.3) extends the idea of the derivation of probability density (1.3) developed by Maxwell for a single particle. Section 10 shows that the resulting probability density can reproduce not only many distributions common in statistical mechanics including those for relativistic fluids, but also more general distributions which combine the above ones. This explains why the ISDE postulated in Section 9 is of the form (9.1). In fact, the family of the densities granted by (9.9) comprises virtually any densities of the momentum of a single particle.

Finally, the ISDE (9.3) is formulated in terms of two functions, u and 7- (see (7.4) and (8.3)). Function u is common in nonrelativistic and relativistic mechanics (e.g., see (1.4) and (1.5)), semiconductor theory (see the paragraph above Remark 7.1), and other fields. Function T is less common but, being associated with E-RDF ~ (see Definition 8.1), is of a sharp physical meaning (see also Remark 10.2). As is noted in Remark 8.1, ~ is a certain generalization of the dissipation function introduced by Rayleigh.

Importantly, availability of functions u and r in conjunction with ISDE (9.3) enables one to determine the equilibrium probability density (9.9) of the momentum of a single particle without any statistical ensemble and TDL (see the text on (1.1)) or without equating the collision-integral integrand to zero and resolving the resulting equation. The statistical-ensemble/TDL techniques are questionable if the particle number is low (e.g., see the discussion in [8, Section 3.3; 21-24]) or if the particles are mutually interacting. The latter feature considerably complicates the formulation of the above integrand. In other words, ISDE (9.3) is the very tool that grants


the continuous equilibrium probability distribution of the particle momentum (cf. Remark 1.1) without the above "statistization".

The next section shows how the equilibrium model (9.3) is used in the corresponding nonequilibrium model.

11. T H E ROLE OF ISDE (9.3) IN T H E C O N S E R V A T I O N E Q U A T I O N S OF T H E R E L A T I V I S T I C N N S H D M O D E L

The relativistic NNSHD model is obtained from the nonrelativistic NNSHD model [9] by means of the passing from the velocity to the momentum in line with the settings in Section 7. The present section shows how ISDE (9.3) is incorporated into the relativistic NNSHD model. For the sake of simplicity, the momentum-conservation equations are confined to the case when the only force acting on the fluid particle is the dissipative force (like it is in Assumption ]k.6). We in what follows distinguish the nonequilibrium quantities from their equilibrium counterparts with the subscript "*" and regard p and v as the stationary solution in Lemma 9.1 and quantity (7.9), respectively.

Then the resulting versions of the fluid-particle-number and fluid-momentum conservation equations [9, (17),(18)] are

On. + V : ( n . v . ) - V : (nv ) = - r . , (11.1) Ot

O(p. ~n,) + v : (;, ~n.v,) at - V : (p~nv) at

[M.(t ,x,p.)v.-- KT.q.(t,x,p.)] Ot = - r . ; . ~ 0~ - n , ~ . ( t , x , ~ . ) j~ (11.2)

[ /2KT, M.(t,x,p,) 1 +n. L V ~,(t ,~u-~ dW(~,t) , i---1,2,3, i

where Vx is the Hamilton differential expression with respect to the entries of vector x, i.e., V~ = (__oo o o ) T r . is the chemical-reaction rate, Pi, p..i, and the quantities in the brackets

Ox~ , Ox2 ' Ox3

are the i th entries of the corresponding vectors, and the equilibrium velocity v and concentration n are determined as before, i.e., with (7.9) and Remark 10.1, in particular,

On Ot - O. (11.3)

Functions u . (t, x, p), T, (t, X, p), and O, (t, x, u) are of the properties analogous to those of functions u, T, and ,I, described in Sections 7 and 8 with the exception that u . , r . , and (see (9.11)) ~.(t,x,u) = fo[~'.(t,x,w)]-I dw, u > 0, are nonequilibrium dependences which can be influ- enced by various nonequilibrium variables (e.g., the nonequilibrium concentration n.) . Other characteristics in (11.1) and (11.2) are described as follows:

v . = V p u . ( t , x , p , ) , (11.4)

N-~3 [Om..~ (t,x,p.) ] M.(t,x,p.) = [(VpVVp )u.(t,x,p.)] -1, and q..~(t,x,p.) = z..,j:lL av..j j, i -- 1,2,3.

We note that the last terms on the left-hand sides of (11.1) and (11.2) are according to the equilibrium-flux principle [9, Remark 2]. They present the equilibrium random fluxes of the particles and the momenta of the particles. These fluxes always exist but cannot be accounted for within any deterministic fluid model.

The equation for nonequilibrium absolute temperature T. used in (11.2) (and (11.1)) is in fact the kinetic-energy-conservation equation [9, (28),(29)]

I I . (t, x, n . , T.) = YI(x, n . , T.) (11.5)

464 E. IViAMONTOV

in the new, relativistic reading, i.e., when

2 L n. u . ( t , x , p . ) p , ( t , x , p , n . , p . ) d p . (11.6) I I . ( t , x , n . , T . ) = -~ 3 1 - ~ , l v . n ,

and II(x, n,T) = II(x) is evaluated with (7.3). In so doing, probability density p.(t , x ,p, . , .), the conditional density of random variable (n., p.) conditioned with value p of the equilibrium momentum, is granted by stochastic system (11.1), (11.2). Expression (11.6) is of a nonlocal, "mean-field" nature (because of its integral form determined by p . ) and, due to the involvement in (11.5), creates an additional relaxation, or "memory", effect in system (11.1),(11.2). (Note that the term "n" in [9, (29)] should be in the integrand and typed in bold italic.) Further details on (11.5) and (11.7) can be found in [9, Sections 2,3].

The common feature of all the nonequilibrium variables is that each of them at equilibrium coincides with its equilibrium version. More specifically, r , , u , , "r,, and T, are such that they at equilibrium coincide with their equilibrium versions, i.e.,

at equilibrium the following relations hold: p, -= p, n, = n, r , = 0,

u . ( t , x , p , ) - u(x,p) , r . ( t , x , u . ) - r (x ,u) , "~. ( t ,x ,u . ) = ~2(x,u), T, -- T. (11.7)

It follows from (11.1) that

O (p,.in,) = n,Op,.i - r,p,.iOt - V : (n,v,)p, . iOt + V~ (nv)p,.iOt, i = 1,2, 3. (11.8)

Also note that

v : (p. ~n.v.) = v : (n.v.)p. ~ + (vxp. d T n.v.,

v : [;,n%~(=, p)] = v J [~%~(x, ;)] ;~ + (v=;,) v n%~(x, p),

i = 1,2, 3, ( 11 .9 )

i = 1 , 2 , 3 . (11.10)

In view of (7.9) and (11.4), equation (11.1) is equivalent to

On.o__.~_ + v=T [n.Vpu. (t ,x,p.)] -- V~ [nVpu(x,p)] = --r, , (11.11)

whereas substitution of (11.8)-(11.10) into equation (11.2) transforms it into

. . . . . . 2 K T . M . ( t ,x ,p . ) OP* + f * O t = - M * ( t ' x ' p * ) v * - KT*q*(t 'x 'P*) ot + ~-~.-(L,'x-~.) dW(~, t ) , (11.12)

where vector ( f . = n2 -~ ~V~ [n.Vpu. ( t ,x,p.)] (p. - p )

k (11.13)

+ L k o ~ /

is a force. It is of a convective origin that is pointed out by the velocity vectors (see (7.9)

and (11.4)) ~pU(X,p) and Vv, u,(t,x,p,) on the right-hand side of (Ii.13). Expression (11.13)

shows that force f, is purely nonequilibrium since

f . = 0, at equilibrium. (11.14)

Thus, the main relativistic NNSHD equations are Itb's stochastic partial differential equations (SPDEs) (11.11) and (11.12) for nonequilibrium concentration n. and nonequilibrium momentum p. of a single particle where nonequilibrium temperature T. is described with (11.5) and equilibrium momentum p is determined according to Theorem 9.1, i.e., as the unique stationary solution of ISDE (9.3). Generally, r . , u . , and T. depend on T,. In view of the nonlocal, "mean-field"


coupling of (11.5),(11.6) with random distribution function (cf. (5.7)) n . p . ( t , x , p , n . , p . ) / ( 1 -

~ , lv .n . ) , the above SPDE system (9.3),(11.11),(11.12),(11.5) presents a system of ItS's equations of the McKean type [49,50]. The above nonlocality creates an additional memory effect in the McKean-It5 SPDE (ISPDE) system. The role of ISPDE (9.3) is that it determines the equilibrium-momentum random filed p which along with the Wiener process W forms the stochastic driving excitation in the model. At the equilibrium state (see (11.3), (11.7), and (11.14)), the concentration SPDE (11.11) and temperature equation (11.5) become identities whereas the particle-momentum SPDE (11.12) becomes ISPDE (9.3).

The above McKean-It5 SPDE system is the simplest form of the relativistic generalization of NNSHD. Many phenomena can be taken into account in this system by means of the corresponding extra terms or extra equations. For instance, the influence of general, nonstationary electromagnetic field can be incorporated in the way of [9] (with the interpretation of velocities v and v. as shown in (7.9) and (11.4)). This leads to both the extra equations and extra terms. To allow for various forces acting on a particle, it is sufficient to add the corresponding force terms (e.g., [51, Chapters 4-7; 52, Chapters 5,6,11,18]) to the right-hand side of SPDE (11.12). In many other respects, the above relativistic extension of NNSHD is also similar to its nonrelativistic counterpart in [9].

12. C O N C L U D I N G R E M A R K S

The present work is devoted to the specification of MRHA discussed in Section 1 in the following two directions. The first one is allowing for nonzero volumes of the particles, the feature typical in the biological parts of bioelectronic problems. The second direction is accounting for the general kinetic-energy/momentum dependencies, including the relativistic ones, which are usually necessary in the electronic parts of bioelectronic problems.

The first part of the development is presented as the volume-scaling method proposed in Sections 2-6. The main results are

• the specific representations (see Sections 2 and 3) which explicitly show how the nonzero volumes of the fluid particles and the densest-package volume fractions affect all the basic characteristics of one-component and multicomponent fluids;

• the sharp connection of the above geometric parameters to Heisenberg's uncertainty principle and the related criterion for the physical validity of the probabilistic distributions for the particles (see Section 4);

• the corresponding specification for the equilibrium case (see Section 5) illustrated with an example of the multicomponent fluid of mutually noninteracting particles (Section 6 and [23]).

tn the multicomponent-fluid case, we emphasize the coupling (3.2), (3.3), (4.4) of the component characteristics for the purely geometric reasons (originated from the fact that the nonzero- volume particles share the same domain). We also note an inherent interplay of the classical and quantum parameters (see (4.5) for the latter) in the validity condition (4.4) resulting from Heisenberg's principle.

The second part of the development is presented as the relativistic generalization of NNSHD. (NNSHD was introduced in [9].) This analysis is proposed in Sections 7-11. The concise summary on the equilibrium treatment in Sections 7-10 can be found in Remarks 10.1-10.3. Section 10 presents the application to composons [22].

Regarding the generic nonequilibrium version in Section 11, we note that the relativistic effects are accounted by means of function u. in McKean-It5 SPDE system (9.3), (11.11), (11.12), (11.5). Many terms in it can depend on the nonzero volume v of the fuid particles and the densest- package volume fraction ~. They are explicitly included, for instance, in (11.6). The general spin picture provided by the equilibrium treatment for composons [22] can be incorporated into the above nonequilibrium model in the way analogous to that in the equilibrium case (see the

466 E. MAMONTOV

text on (10.6)-(10.9)) in terms of nonequilibrium CDT function v, (see also Definition 8.1). The latter is coupled (see the text between (11.3) and (11.4)) with the nonequilibrium version ~ . of E-RDF ¢, i.e., the nonequilibrium kinetic-energy-determined Rayleigh's dissipation function for a single particle. Importantly, the availability of random distribution function (see the paragraph below (11.14)) makes the above McKean-It5 SPDE system comparable with the stochastic Boltzmann (or Boltzmann-Langevin) equation discovered in [53]. However, the presence of the equilibrium random fluxes in (11.11) and (11.12) as well as the nonlocal, McKean-type nature of our system points out that it is, in a certain respect, even more general than the aforementioned equation.

The above specification of MRHA (in particular involving the McKean-It5 SPDE system) is not intended to be a theory. It is thought to be a framework, i.e., [54, the middle column on p. 902] "a basic ideational and narrative structure; a systematic set of relationships; a conceptual scheme, structure or system; the limits or outlines especially of a particular set of circumstances". Unlike what is usually understood as a theory, the present framework is intended to be flexible, free of dogmas, and open to incorporation of new ideas, visions, and models which, in particular, may replace the previous ones (including all those developed in the present treatment) thereby facilitating future development in various directions by independent researchers. For instance, the framework approach does not specify (at least, at present) the (x, u)-dependence of the E-RDF function ¢ or CDT function ~-: they can be derived theoretically, evaluated by computer simulation, or determined experimentally. None of these options is rejected. The specific ways can be very different and need not comply with one or another theory prescribed in advance. Here we arrive to other distinguishing features of the framework. We emphasize two of them.

First, the framework approach is not hunting the so-called first principles (loosely speaking, the formal rules that the nature must follow). It instead searches for the forms, methods, and tools which can improve understanding of the actual features of the fluids for the purpose to predict them. In fact, the first-principle strategies are often not very fruitful.

Indeed, statistical mechanics, a science based on the stiff first principles, still fails to introduce and evaluate Rayleigh's dissipation function for a single particle (the published example [31, (2.3)] of such function is not related to statistical mechanics). Along with this, the single-particle function of the above kind, E-RDF (see Definition 8.1), is in the heart of the present development. Another field where the first principles are nothing but quite relative entities is QM. The reading of physical characteristics based on the wave function is unquestionably one of the first principles associated with the QM SchrSdinger equation. However, this equation is applicable only until one deals with the so-called pure QM states (e.g., [15]). If the analysis involves the mixed QM states, the above first principle has nothing to do with them since in the mixed-state case the entirely different, density-matrix formulation must be used (e.g., [15]). The first principles related to the latter in turn fail when a QM problem includes dissipative phenomena. Here another equation, the GSran Lindblad equation [55, the equation on p. 129, Section 6, (4.3)], comes into the focus considerably altering and enriching the corresponding first-principle interpretations. The next stage in the QM first-principle revision is a nonlinear QM noted (with the related references) by Wigner more than sixty years ago [56, p. 149]. The development of this nonlinear science would formulate the first principles unseen today and also noticeably change the whole "landscape" of QM and all the activities around it.

Second, the present framework approach inherently includes the capabilities in coupling the developed models to other fluid-modelling treatments like common hydrodynamics or kinetic equations, both deterministic and stochastic. This feature is assured by that the above relativistic NNSHD includes all the quantities necessary for such coupling, in particular, the basic fluid variables and the distribution function which all are random. In this respect, of a special interest is a combination of the present specification of MRHA with the GK theory (e.g., [3, Section 10.3; 5,7,10]) mentioned in Section 1 and developed especially for complex fluid problems such as those arising in biology [20].


This combination may include various aspects. One of them is a description of the fluids with high numbers of the components. The corresponding research can be implemented in line with the advanced multicomponent treatment presented in [10]. Another aspect is bioeleetronie problems with the chemical-reaction rate r . in (11.11) (or (11.1)) of an increased complexity. Here, the results of [3,5,7] would be of a prime importance. An example of a specific bioeleetronie problem in this field is the competition of the neoplastic immunogens and the antibodies in the course of a growth of a tumor, benign or malignant. The possible outcomes of the above combination can extend the results of the previous studies (e.g., [57,58]) and contribute to development of the anticancer immunotherapies and related drugs.

A P P E N D I X A

C O M P O S O N S All the representations below are for the case when particle volume v is identically zero. If a particle is elementary, it is either boson or fermion (e.g., [17, Section 56]). Bosons are

described with the Bose-Einstein distribution function, ( )1 g u - # ~o(x,p) = ~-5 exp KT 1 , # _< 0, (A.1)

where g is positive odd, u is the kinetic energy of the particle, and # is the electrochemical potential. Fermions are described with the FD (Fermi-Dirac) distribution function,

( )--1 g u - # qO(x,p) = ~-- 5 e x p ~ + 1 , (A.2)

where g is positive even. Note that in the FD case, # need not be nonnegative. The QM intrinsic angular momentum number s, also known as the spin of the particle, is coupled with g as follows:

g = 2s + 1. (A.3)

Quantity g in (A.1) and (A.2) is the number of the spin orientations of the particle. Each of the distribution functions (A.1) and (A.2) can be reduced to

g # - u # - u ~,(x,p) = ~ e x p K T ' exp--ff~- << 1, (A.4)

that is called the Maxwell-Boltzmann distribution function. If the particle is composite, i.e., either elementary or composed of elementary particles, then

it generally need not be either fermion or boson. In connection with certain experimental data, work [22] suggests to treat a composite particle as composon, the particle described with the following distribution function [22, (3.3),(3.4)1:

~(x,p) = ~-5 - - exp K T

p<_O,

where

- - - 1 + - - - ~ e x p - ~ - - + 1 ,

at z > -1 ,

(A.5)

is real 'rather than an integer as it is in (A.1) and (A.2)), quantity

z e [-1, 1] (A.7)

determines the distribution type of the particle, and s is described as before, i.e., with (A.3). Clearly, (A.5) is reduced to the particular cases (A.1) and (A.2) at z = 1 and z = -1 , respectively. According to [22], even at these values of z, quantity g need not be integer since the physical picture underlying the composon model is more complex than that of elementary-particle models (A.1) and (A.2).

Representation (A.5)-(A.7) is derived in [22] by means of the GK methods as a probabilistic interpretation of the related experimental data. More details on composons can be found in [22]. We also note that scalars g and z in (A.3) can generally be quite complex functions of various characteristics of the particle.

g > 1 (A.6)

468 E. MAMONTOV

A P P E N D I X B

T H E M O M E N T U M I S D E (9 .3) I N C A S E O F E I N S T E I N ' S R E L A T I O N (1 .5)

This appendix presents the example of ISDE (9.3) corresponding to relation (1.5) derived by Einstein in his special relativity theory. Accordingly, dependence (7.4) is understood as (1.5) down to the end of the appendix.

Relation (7.9) becomes

v = VpU(Z,p) = P my~1 + [[IPll/(mc)] 2 . (B.1)

Then, function (7.10) is of the following form,

m y

p = p(x, v) = X/1 _ (]lvll/c)2, (B.2)

that, in particular, transforms (1.5) into

1 _ _1 . U = 7 r ~ e 2

Entries of mass matrix (7.11) are written as

mij (x, p) = m 1 + k (me) J ~ij + mc m c / , i, j = 1, 2, 3. (B.4)

Matrix (B.4) is obviously positive definite for all p E R 3 (cf. the text below (7.11)) and is of property (7.13). Expressions (B.4) imply

Ornij(x 'p) - l ~ { pjSij ÷ pi[pj/(mc)]2 ÷ x / l + [''p]f/(mc)]2pi rnc2 V~ 1 ÷ ["P'[/(mc)] 2 , i , j = 1 , 2 , 3 . (B.5)

Substitution of (B.5) into (9.4) transforms the latter into

4v/l + [ll;lt/(-~c)?.p. (B.6) q = rnc2

Relation

M v = 1 + p (B.7)

stems from (B.4) and (B.1). Equation (9.3) specified with (B.6) and (B.7) is

1 + [llpll/(mc)] 2 - [ 4 K T / ( m c 2 ) ] v/1 + [Npll/(mc)] 2 f 2 K T M ( x , p ) ~; = - ~(x , u(x , p) ; dt + V ~ dW(¢, t),

where matrix M(x,p) has entries (B.4).


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