# Nonequilibrium Thermodynamics Based on the Distributions

Authors: Vasiliy Vasiliy Ryazanov
Journal: Journal of Thermodynamics
 (2011)

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## Abstract

To describe the nonequilibrium states of a system, we introduce a new thermodynamic parameter—the lifetime of a system. The statistical distributions which can be obtained out of the mesoscopic description characterizing the behaviour of a system by specifying the stochastic processes are written down. The change in the lifetime values by interaction with environment is expressed in terms of fluxes and sources. The expressions for the nonequilibrium entropy, temperature, and entropy production are obtained, which at small values of fluxes coincide with those derived within the frame of extended irreversible thermodynamics. The explicit expressions for the lifetime of a system and its thermodynamic conjugate are obtained.


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## Body

## 1. Introduction

In papers [1–3], the linear nonequilibrium thermodynamics (LIT) was developed which found its application in the variety of physical problems. But the linear irreversible thermodynamics possesses a number of restrictions, which present certain obstacles for adequately describing such phenomena as the propagation and absorption of ultrasound in liquids, density profile of shock waves in gases, and so forth. The attempts to overcome these difficulties led to creating the extended irreversible thermodynamics (EIT) [4]. If in the frame of LIT [1–3] the local entropy density is merely a function of conserving quantities such as local energy and mass densities, in EIT a set of additional variables is added thereto, which are chosen as densities of fluxes in a system (nonconserving dissipative values). Phenomenological laws appear to be nonstationary equations of motions, the entropy flux-containing nonlinear terms; the velocity of propagation of perturbations is finite in the contrast to the infinite velocity (e.g., of propagating heat waves) predicted by classical thermodynamics. The Fourier law q⃗=-λ∇⃗T (where q⃗ is the heat flux, λ is the heat conductivity coefficient, and T is absolute temperature) of the classical LIT is replaced by the Maxwell-Cattaneo equation(1)q⃗=-λ∇⃗T-τq∂q⃗∂t,
where τq is the time of the flux correlation.Characterizing the nonequilibrium state by means of an additional parameter related to the deviation of a system from the equilibrium (field of gravity, electric field for dielectrics, etc.) was used in [5]. In the present paper, it is suggested the new choice of such an additional parameter as the lifetime of a physical system, which is defined as a first-passage time till the random process y(t) describing the behaviour of the macroscopic order parameter of a system (energy, e.g.,) reaches its zero value. The lifetime is thus a random process which is slave (in terms of the definitions of the theory of random processes [6]) with respect to the master process y(t), (2)Γx=inf⁡{t:y(t)=0},y(0)=x>0.The characteristics ofΓ depend on those of  E=y(t). It is important to render firstly a physical interpretation to the definition (2). So, the lifetime is related to the period of stable existence of a system, and its time of dwelling within the homeokinetic plateau whose distribution was related in [7] to the entropic and information parameters of a system, its response to internal and external influences, and its stability and adaptation facilities. Lets show this relation using an illustrative example. Let the behaviour of a system be described by the potential U(q) (Figure 1) having some “potential well” at y>b and a minimum in the point y=a.Figure 1
The shape of the potential of a systemU(y) in the potential well model. The point y=a is the potential well minimum; the barrier in the point point y=b separates the potential well from the zero minimum of y.The major part of the living time the system dwells within the vicinity of the pointa. The time of trespassing by the system value y=b can be considered the finishing of the time of stable existence Tstab. Thus during a small time tb0≪Tstab, the system will roll down to the point y=0. The lifetime Γ=Tstab+tb0≈Tstab. Real systems are characterized by much more complicated behaviour, but the dependence Γ≈Tstab seems to hold (with exception of the cases of phase transitions, chaotic regimes, etc.). The states of a system within the homeokinetic plateau are characterized by the mutual compensation of the entropic effects related to the energy dissipation and by the effects of negative entropy determined by the existence of the negative feedbacks. When the system exits out of the limits of the plateau, unstable structures and sharp qualitative changes in the behaviour of the system arise.The lifetime seems to be a value of more general character than the fluxes are (this fact being the prerequisite of formulating a theory which would include the EIT as its partial case) and, moreover, it seems to be more conservative (that is, the lifetime in this respect stands closer to the ordinary conserving values). The history of a system is the succession of repeating “busy periods”—lifetimes in the sense of the definition (2), and idle periods (like it is the case of the queues systems) which are marked by the observable events of lifetime finishing and system regenerating. The internal time in the system is related to the existence of the nonzero number of elements within. The principal consideration grounding the introduction of the lifetime as a physical parameter is an empirical fact of all systems having limited time of existence. Of interest seems to be applying this notion to the biological objects, for example, investigating the living time of an organism.Very close to the scope of the present work stand the papers [8–12]. In the papers [8, 9], the transition of a system from a given initial state to the final state is supposed to take place within a fixed time gap. In the papers [10–12], a microscopic treatment of the decay rate or escape rate out the repeller which is the inverse of the lifetime of the system is performed, which can be readily related to the first-passage-time problems [6, 13]. But the escape time and escape rate definitions of [10–12] differ from those of the lifetime and its conjugate γ used in our paper.In the present paper, we provide the statistical foundations for the thermodynamic relations and introduce the distribution for the lifetime of a system (Section2). The main assumptions, namely, concerning the explicit lifetime distribution and the relation between its thermodynamic conjugate and the fluxes are given in the Sections 3 and 4. The explicit expressions for the temperature, entropy, and entropy production are written down and the stability analysis is performed in Section 4. In the Conclusion, various aspects of proposed method for describing complex systems are treated.


## 2. Lifetime Distributions

Let us consider a macroscopic observableE (it might be something different from energy as well) being distributed with function P(E)=pE(x). Using the maximum-entropy inference [14, 15], one can reconstruct out of the macroscopic distribution P(E)=pE(x) the microscopic probability density in the phase space z. The constructing of Gibbs microcanonic distribution corresponds to imposing the additional condition of the equiprobable distribution of all possible microstates. This “maxent” (i.e., related to the maximum-entropy principle) procedure corresponds [16] to the process of coarsening and means physically the loss of information about further details of the behaviour of a system. Let us suppose that bringing a system in the nonequilibrium state violates the equiprobable distribution of equilibrium. We thus introduce a novel parameter Γ(z) supposing its observability. We can introduce as well the cells of the extended phase space with constant values of the set (E,Γ) within (instead of the cells with constant values of E). The structure factor ω(E) is thus replaced by ω(E,Γ)—the volume of the hyperspace containing given values of E and Γ. If μ(E,Γ) is the number of states in the phase space which have the values of E and Γ less than given numbers, then ω(E,Γ)=d2μ(E,Γ)/dEdΓ. It is evident that ∫ω(E,Γ=y)dy=ω(E). The number of phase points between E, E+dE and Γ, Γ+dΓ equals ω(E,Γ)dEdΓ. We make use now of the principle of equiprobability applied to the extended cells (E,Γ). The coarse-grained density representing the growth of the information entropy [16] is merely one of possible ways of introducing irreversibility into the statistical physics. The same procedure was performed in [17], where the systematic loss of information appears by choosing the appropriate operator and eliminating the initial correlations. This approach corresponds to the subjectivistic interpretation [14, 15] of the statistical physics. Another approach [10, 11, 18, 19] corresponds to the objectivistic interpretation and considers the stochasticity as being the inherent property of the phase space. The lifetime concept is closer to this latter interpretation. From the mathematical viewpoint, one could trace the differences between these two concepts from the differences in constructing the space of elementary events (which in general need not coinciding with the space of coordinates and momenta of all particles as the Gibbs statistics implies).The standard procedure (e.g., [16, 20]) allows one to write down the relation between the distribution density P(E,Γ)=pEΓ(x,y) and microscopic (coarse-grained) density ρ(z;E,Γ)(3)P(E,Γ)=pEΓ(x,y)=∫δ(E-E(z))δ(Γ-Γ(z))ρ(z;E,Γ)dz,
where(4)P(E,Γ)=pEΓ(x,y)=ρ(z;E,Γ)ω(E,Γ)|E=x,Γ=y.
Lets introduce the conditional distribution density (5)P(Γ∣E)=pΓ(y∣x)=pEΓ(x,y)pE(x)=P(E,Γ)P(E),
where(6)P(E)=pE(x)=∫0∞ρ(z;E,Γ=y)ω(E,y)dy.Now we make an assumption about the form ofρ(z;E,Γ)(7)ρ(z;E=x,Γ=y)=exp⁡{-βx-γy}Z(β,γ),
where(8)Z(β,γ)=∫exp⁡{-βx-γy}dz=∬dxdyω(x,y)exp⁡{-βx-γy}
is the partition function and β and γ are Lagrange multipliers satisfying the equations for the averages(9)〈E〉=-∂ln⁡⁡Z∂β|γ,〈Γ〉=-∂ln⁡⁡Z∂γ|β.We did not concretize yet the parameterΓ(z). Now let it be the lifetime of a system (2). Introducing Γ means effective account for more information than merely in linear terms of the canonical distribution.The linearity of the exponent in (7) allows one to relate the expression for the partition function Z(β,γ) with the Lagrange transforms of the macroscopic distributions P(x), P(Γ∣x), P(Γ,x)(10)E(e-θx)≡Q(e-θ)=∫e-θxP(x)dx,(11)E(e-sΓ(x))≡L(x,s)=∫e-syP(Γ=y∣x)dy,(12)G(θ,s)≡∫e-θx-syP(E=x,Γ=y)dxdy=L(-∂∂θ,s)Q(e-θ),
where E(⋯) means averaging. The expressions (10)–(13) can be easily obtained explicitly using the known model of the process y(t) (see Appendix A).From the expressions (4), (12), and (7)-(8), it is easy to derive the relation between microdynamics (containing in the partition function Z(β,γ)) and the Laplace transforms of the macrovariable and its lifetime(13)G(θ,s)≡Z(β+θ,s+γ)Z(β,γ).Let us choose some reference point(β0,γ0) in the space of the Lagrange parameters. If the macroscopic model holds for this point, one can get (using (13)) the value of the nonequilibrium partition function for the other point (β,γ),(14)Z(β,γ)=Z(β0,γ0)G(β-β0,γ-γ0).The valueZ(β0,γ0) is merely a number characterizing the “bound part” of the entropy of the microscopic motion. The multiplier G corresponds to the free information derivable when concretizing the stochastic process [21].


## 3. Generalized Lifetime Thermodynamics

Ifγ=0 and β=β0=(kBTeq)-1, where kB is the Boltzmann constant and Teq  is the equilibrium temperature, then the expressions (7)-(8) yield the equilibrium Gibbs distribution. One can thus consider (7)-(8) as a generalization of the Gibbs statistics to cover the nonequilibrium situation. Such physical phenomena as the metastability, phase transitions, and stationary nonequilibrium states are known to violate the equiprobability of the phase space points. The value γ can be regarded as a measure of the deviation from the equiprobability hypothesis. In general, one might choose the value Γ as a subprocess of some other kind as chosen above.Let us suppose that the processy(t)=E is the energy of a system (equivalently one could choose the particle number, pulse, and so forth but for the present paper we shall restrict ourselves with the heat conductivity problem). Further detalization will require the concretization of the Γ distribution and the interpretation of the Lagrange parameters β and γ. The Lagrange parameter β is supposed to be (likewise the equilibrium Gibbs statistics)(15)β=1kBT,
where T is the average (over the body volume) local equilibrium temperature. Since at fluxes q⃗≠0 the temperature is not the same all over the bulk of the body, one can define T in a system with volume V as the volume average, that is, T=V-1∫VT(r,t)dr (the same definition was used in [22]). Of course, the thermodynamic description itself is supposed to be already coarse-grained. To get the explicit form of the Γ distribution, we shall use the general results of the mathematical theory of phase coarsening of the complex systems [23] (Appendix B), which imply the following distribution of the lifetime for coarsened random process:(16)pΓ(y)=Γ0-1exp⁡⁡{-yΓ0}.The valuesΓ0 are averaging of the residence times and the degeneracy probabilities over stationary ergodic distributions (in our case, Gibbs distributions). The physical reason for the realization of the distribution in the form (16) is the existence of the weak ergodicity in a system. Mixing the system states at big times will lead to the distributions (16).As we note in Conclusion, the structure factorω(E,Γ) has a meaning of the joint probability density of values E, Γ. For the distribution (16), the function ω(E,Γ) from (4), (6), (8) takes on the form:(17)ω(E,Γ=y)=ω(E)Γ0-1exp⁡⁡{-yΓ0}.
Substituting into the partition function (8) yields(18)Z(β,γ)=Z(β)(1+γΓ0)-1,
where Z(β)=∫ω(E=x)exp{-βx}dx is the Gibbs partition function.We have from (9) and (18) when Γγ=-∂lnZ(β,γ)/∂γ, Γ0(V)=Γγ(V)|γ=0, (19)Γγ=Γ01+γΓ0,γ=1Γγ-1Γ0,
that is, γ is the difference between the inverse lifetimes of the open system 1/Γγ and the system without external influences 1/Γ0 which can degenerate only because of its internal fluctuations. The value γ is thus responsible for describing the interaction with the environment, and its existence is the consequence of the open character of a system. When defining γ, one should take into account all factors which contribute to the interaction between the system and the environment. If one denotes in (19) x=γΓ0, then x=x1+x2+⋯+xn, where the value  xi is determined by the flux labelled by the index i. Below we shall treat only the case of one flux without conservative external fields and velocity gradients and the equation for the specific energy u has the form   ρdu/dt+∇⃗J⃗q=0, E=∫VρudV, ρ=M/V is the mass density, and J⃗q=J⃗u=q⃗ is the heat flux. We consider the case of the constant mass density.Let us introduce the nonequilibrium entropy corresponding to the distribution (7) in the analogy to [17] by the relation(20)SkB=-〈ln⁡⁡ρ(z;E,Γ)〉=β〈E〉+γ〈Γ〉+ln⁡⁡Z(β,γ).
For the case of several potential wells (in contrast to Figure 1) with the same minima positions (which means the case of a system with several ergodic states), one can expect the appearance of the Erlang-type distributions instead of the exponential one (16) [23]. More precise approximations can also be used.


## 4. One-Phase Systems (Systems Possessing One Class of the Stable States), Heat Conductivity, Mass Transfer, and Chemical Reactions

Substituting into (20) the expressions (18) and (19), we have(21)SkB=SβkB+x(1+x)-ln⁡⁡(1+x),SβkB=β〈E〉+ln⁡⁡Zβ,x=γΓ0.
From (20) treating E and Γ as variables,(22)dS=kBβdE+kBγdΓ.
From (19), dΓ=-Γ2dγ+(Γ/Γ0)2dΓ0. Substituting into (22), we find from the Maxwell relations that ∂Γ0/∂E and ∂Γ0/∂R are proportional to ∂Γ0/∂q⃗=0 (since Γ0 by definition corresponds to the state with q⃗=0 and Γ0 independent on q⃗). Thats why dΓ0=0 and (23)dS=kBβdE-kBxdx(1+x)2,
which coincides with the differential of (21).Now we determine the explicit form forγ and x. Considering the simplest model, for example, that of the geokinetic plateau [7], Figure 2, which features a system in the stationary state, which degenerates during a short-time interval t0q, changing from a stationary state to zero state, we can obtain for the heat conductivity in this case (24)xq=γqΓ0≈t0qSaqE=t0qqρuR=t0qyq,yq=qρuR,E≈ρuV,
where q=±|q⃗|=(q⃗q⃗)1/2,  u=E/ρV, energy density in a system, R its linear size, V its volume ~R3, and the surface Sa~R2; q+=(q⃗+s⃗)=q=(q⃗q⃗)1/2, where q- and q+ are projections of the outcoming and incoming heat flux density vectors on the surface normal vector s⃗ (ordered chronologically, Figure 3); the signs of  q- and q+ depend on whether we are heating or cooling the systém. Then q+=-q--Rρ∂u/∂t, in the stationary state q+=-q-.Figure 2
Geokinetic plateau model.Figure 3
The volume element of the sizeR, incoming q⃗+, and outcoming q⃗- fluxes, normal to the surface element s⃗.Outcoming fluxq⃗- and incoming flux q⃗+ differ in their times: we consider q⃗+ to be the first flux. In (24) we suppose the finite velocity of the heat propagation in the system. Thus the linear dimensions enter to the characteristics of the “point” system. For continuous systems R=L, where L is the size of the “point” of the continuous medium in the continuous description [20, 22]. In the frame of the kinetic theory L=lph=l1(ε1)1/2≪l1,ε1=nr03, where n is the density, r0 is the diameter of the atom, l1 is the free path length. In the gasodynamic description L=lphG~LG/N1/5, where LG is the size of the system, N is the particle number. One should take into account the source density σ when describing the reaction systems. For example, in the nuclear reactors one must take into account the terms responsible for the fuel sources. But even in the equations for the internal energy one should account for the sources if external fields and/or inner heterogeneities are present [3]. Let the system be open with respect to the quantity A=∫VρadV, for which the balance equation           ∂(ρa)/∂t+∇⃗J⃗a0=σa;  J⃗a0=ρava⃗ [3] holds (v⃗ is the velocity, with ρ=M/V being the mass density); divJ⃗a0≈(Ja-0+Ja+0)/R, where R is the “size” of the point in the continuous description [22]. Then the value Ja+0-Rσa=-Ja-0-R∂(ρa)/∂t is proportional to γ from (19). And the value ya=(Ja+0-Rσa)/Ra, multiplied by the time parameter t0a (from (28)), coincides with x=γΓ0. The equation for the specific energy u has the form ρdu/dt+∇⃗J⃗q=0; E=∫VρudV;  J⃗q=J⃗u=q⃗ is the heat flux. We consider the case of the constant mass density.The same result (24) can be obtained if we compare the entropy of macroobservables E and q⃗ of EIT [4] with the entropy of macroobservables E and Γ.For this purpose, we will recall the expressions of the EIT [4]:(25)S=Sβ-∫Vραq(q⃗q⃗)dr⃗2,αq=τqρλT2
(q⃗ and τq are values from (1)). Comparing with (21) at small q⃗: S=Sβ-kBx2/2,   kBxq2=∫Vραq(q⃗q⃗)dr⃗, (26)x=±(∫Vραq(q⃗q⃗)dr⃗kB)1/2=γΓ0.
The sign “−” in (26) is chosen if the incoming flux is directed towards the system (the system is being heated). The derivative ∂x/∂q⃗∣E,R is written as   ∫V[δx/δq⃗(r⃗)∣E,R]dr⃗=∫Vραqq⃗dr⃗/kBx, where R is  the linear size of the system. The expressions for ∂x/∂E∣q,R, ∂x/∂R∣E,q can be obtained in the same fashion. Then from (23) (27)dS={kBβ-∫V[∂(ραq)/∂u](q⃗q⃗)dr⃗ρV2(1+x)2}dE-∫ραqq⃗dr⃗dq⃗(1+x)2-2x2kBdRR(1+x)2.
If the fluxes q depend only weakly on the spatial variables, then (26) is cast as(28)xq≈±(∫Vραqdr⃗kB)1/2(q⃗q⃗)1/2≈t0qSaqE=t0qqρuR=t0qyq,t0q≈E(∫Vραqdr⃗/kB)1/2Sa,yq=qρuR,E≈ρuV,
which coincides with (24).From (23) we have(29)dS=kB[β+x2E(1+x)2]dE-[kBx2q(1+x)2]dq-[kB2x2R(1+x)2]dR.The behaviour of a system depends on its finite size, since we consider the systems with finite lifetime and finite volumes without performing the thermodynamic limit transition.Consider now the stability of the thermodynamic system. To ensure its stability, the conditionδ2S≤0 should satisfy. We have
(30)kB-1∂2S∂E2∣q,R=∂β∂E-x2(3+x)E2(1+x)3,kB-1∂2S∂q2∣E,R=x2(x-1)q2(1+x)3,kB-1∂2S∂R2∣E,q=2x2(x-3)R2(1+x)3,kB-1∂2S∂E∂q=2x2qE(1+x)3,kB-1∂2S∂q∂R=-4x2qR(1+x)3,kB-1∂2S∂E∂R=4x2ER(1+x)3,∂β∂E=-1kBT2c,c=∂E/∂T is the heat capacity. The nonequilibrium heat capacity cN=∂E/∂θ is likewise defined. If R=const (variables are E and q), the condition δ2S≤0 holds for ∂S2/∂E2∣q,R≤0,  ∂2S/∂q2∣E,R≤0. These conditions satisfy when -1/T2c-kBx2(3+x)/E2(1+x)2≤0 and kBx2(x-1)/q2(1+x)3≤0. One more condition satisfies if (∂S2/∂E2∣q,R)(∂2S/∂q2∣E,R)-(∂2S/∂E∂q)2≥0. Hence(31)|x|≤(1+kBT2cE2)-1/2.
It is seen from (31) that at q<0, x<0, c>0, |x|<1 that is the denominator in (19), (29) does not equal zero in the domain of the thermodynamic stability and the expression for Γ converge. The same relations are written for E=const in the variables q, R and for q=const in terms of E, R.Determining in (29) dS/dt, we find the entropy balance equation (dS/dt)/V=-∇⃗j⃗S+σS,S=∫Vρsdr⃗, where for ρ=const, j⃗S=θ-1q⃗, (32)σS=q⃗∇⃗θ-1-kBx2(dq/dt)qV(1+x)2+kB(dR/dt)[3uβρ+x2V-1(1+x)-2]R.Comparing entropy productionσS and σS=q⃗q⃗/λT2 [3], we shall find(33)λT2kBx2(dq⃗dt)q-2V-1(1+x)-2+q⃗=λT2∇θ-1+(dRdt)λT2kB[3uβρ+x2V-1(1+x)-2]q⃗R.
The inverse nonequilibrium temperature(34)θ-1=∂S∂E|q,R=kB[β+x2E(1+x)2]=1T+q2αqu[1+q(ραqV/kB)1/2]2
coincides with the value 1/θ of the extended thermodynamics [4] at small q (when (1+x)-2≈1). The value entropy production σS and the expression (33) when substituting therein (28) take on the form(35)σS=q⃗∇⃗θ-1-ρqαq(dq/dt)(1+x)2+(dR/dt)ρ[3uT-1+q2αq(1+x)-2]R,(dq⃗dt)τq(1+x)-2+q⃗=λT2∇⃗θ-1+(dR/dt)[3uλTρ/q⃗+q⃗τq(1+x)-2]R,
and coincide with the corresponding expressions of EIT and Maxwell-Cattaneo equation (1) at dR/dt=0 and small q. The same can be derived from (27).Above we considered the lifetime as a quantity related to the heat in a system. Since the heat transfer in a body is accompanied by the processes of deformation of a continuous medium, the energy dissipation is conditioned not only by the heat transfer, but by the internal friction of a system which is represented by the dissipative part of the stress tensor; thus the full expression forx must have (similar to [24]) the form(36)x=±(∫Vdr⃗kBρ[τqρλT2(q⃗q⃗)+(σv:σv)τvρμT])1/2,
where σv is the viscous stress tensor, τv is the correlation time of viscous stresses, and μ is the shear viscosity. Correspondingly, the expressions for j⃗S, σS,… acquire more cumbersome form. Similar expressions can be written down for the density changes (if one considers the lifetime for the full mass), for velocity and for other factors contributing to the expression for σu from the equation for the specific energy density [3], and so forth.In the examples considered above, one took forβE the values ∫drβ(r,t)û(r), where û(r) is the dynamical variable of the energy density, β(r,t)=1/kBT(r,t); if we consider the processes with variable mass, we should take as βE the quantities ∫drβ(r,t)[û(r)-∑kμk(r,t)n̂k(r)], where n̂k(r) is dynamical variable of the particle density for the particles of k-th kind, μk(r,t) is the chemical potential of the k-th particles. Taking into account the chemical reactions like ∑j=1rνj+Xj⟷k+∑j=1rνj-Xj, we get (37)xρ=±∑k=1r(∫Vdr⃗kB[τkdiffLkV(j⃗kj⃗k)+(ωkωk)τkchemLkchemVT])1/2,
where xk=t0k(Jk-ωkR)/Rck, t0k=t0kdiff=ckR(τρdiff/kLk)1/2, where τρdiff is the correlation time for the fluxes J⃗k, Jk is the normal projection of J⃗k (the incoming flux into the system of the size R) to the surface, Lk=DkT/(∂μk/∂ck)T,P, μk is the chemical potential (=μ0k without perturbation), and Lk is Onsager coefficient from J⃗i=-Li∇⃗(μi/T)T,P. We consider a multicomponent mixture with mass densities of the components ρk and concentrations ck=ρk/ρ, k=1,…,r; ∑k=1rρk=ρ, cn=Nn/V, where r is the total number of particle types involved into reactions, Nn is the number of particles of the n-th type. The balance equation for cj has the form ∂cj/∂t=-∇⃗J⃗j+ωj, where J⃗j=-Dj∇⃗cj is the diffusive flux of the j-th component, Dj—its diffusion coefficient, ωj=k+∏n=1rcnνn+-k-∏n=1rcnνn- the rate of the j-th reaction per unit volume, k+ and k- are reaction constants, vn=vn--vn+ are stechiometric coefficients [3, 25]. Besides the value t0kdiff, one can also introduce the time t0kchem=(αρkchemV/k)1/2, αρkchem=τkchem/(VTLkchev), Lkchem=ckω+eq/kT, ω+eq=k+∏j=1r(cjeq)νj+ (such choice of kinetic coefficients for chemical reactions where ω=LA(A=-∑n=1rvnμn=-kTln∏j=1r(cj/cjeq)vj is chemical affinity) is performed, e.g., in [25]), and τkchem is the correlation time of the fluxes ωk caused by chemical reactions. Since t0kchem and t0kdiff enter the same balance equation for ck, t0kdiff=t0kchem. From this, we get the relation between τkchem and τkdiff: τkchem/(TLkchem)=R2ck2τρdiff/Lk. As thermodynamic quantities, let us take ck, Jk, mk, mk=∑j=1rγkjωj,ωj=(∂jρk/∂t)/γkj in the j-th chemical reaction [3], ∑k=1Kmk=0. Since ∂ck/∂ρk=(1-ck)/ρ, ∂xk/∂ck∣Jk,mk=-xk(∂ρk/∂ck)/ρk=xk/ck(ck-1), ∂xk/∂Jk∣ck,mk=xkmkR/Jk(Jk-Rmk),  ∂xk/∂mk∣ck,Jk=-Rxk/(Jk-Rmk). Substituting these expressions into (3), we get
(38)(kB)-1dS=(kB)-1dSβ-(γΓ)2∑k=1rγk[A]γ,
where 𝔄 denotes dck/ck(ck-1)+RmkdJk/Jk(Jk-Rmk)-Rdmk/(Jk-Rmk).If we substitutedck/dt from the balance equation and use the transformation μ∇⃗Jk/kBT=∇⃗(μJk/kBT)-Jk∇⃗(μ/kBT), we get the entropy balance equation with the entropy flux j⃗S=-∑kJ⃗kμk/T, μk/kBT=μ0k/kBT+(γΓ)2γk/γck(ck-1), and entropy production   σS=∑k=1r{-J⃗k∇⃗(μk/T)-μkmk/T-ργΓ2kBγk[Rmk(dJk/dt)/Jk(Jk-Rmk)-R(dmk/dt)/(Jk-Rmk)]}. Setting this value σS equal to σS=∑k=1rJk2/Lkk+∑r=1s∑l=1sRrlωrωl/T, where Rrl are resistivity matrices [3] from the expression  Ar=∑l=1sRrlωl, (s, number of chemical reactions in a system) we get the general expression relating dJk/dt, dmk/dt with Jk, mk. Further, considering mkR≪Jk (almost pure diffusion), one can get an equation for Jk, similar to (9) for q. Vice versa, when Jk≪mkR (that is, chemical reactions only), one can get an equation for mk and ωj. The stability conditions are written similarly to the heat conductivity equation.


## 5. Stationary Nonequilibrium States

We suppose that the expressions (20), (21) which are results of the present paper are satisfied for arbitrary values of fluxes. Thats why they are believed to be capable of describing the stationary nonequilibrium states far from equilibrium in a more accurate fashion than EIT does. Linear deviations should thus be calculated starting from some reference states determined in this case by the stationary values of fluxes. For the distribution (16), one should generate the expansion round about the value x0=xst=q0R2t0/E0,   E0=Est,   q0=qst,   γ0=γst=εq0R2/E0.The expression for the entropy thus has a form(39)SkB=SβkB+x0(1+x0)-ln⁡⁡(1+x0)-(x-x0)x0(1+x0)2+(x-x0)2(x0-1)2(1+x0)3+⋯.
If one introduces the lifetime in a stationary nonequilibrium state Γ1(γ0)=Γ0/(1+γ0Γ0),x1=γ0Γ1(γ0)=x0/(1+x0), then the latter expression is cast as(40)SkB=SβkB+x1+ln⁡⁡(1-x1)-(x-x0)x1(1-x1)+(x-x0)2(2x1-1)(1-x1)22+⋯.


## 6. Conclusion

The assumption about the physical systems living for a finite period of time which was the starting point of the exposed here theory allows one to get the mesoscopic theory of the stationary nonequilibrium states at any deviation from the equilibrium. For the method applied, it is essential to have the relation ofΓ as a slave process to the master process E(t). But the concept of the lifetime has a more profound physical sense to cast in one fashion the Newtonian approach to the absolute time and the concepts of time-generating matter. The lifetime parameters is a compromising concept uniting in itself the properties of ordinary dynamical values like energy and the particle number are and the coordinate variables like the time variable is. Mathematically introducing lifetime means yielding an additional information on the stochastic process besides its stationary distribution leaning upon the stationary properties of the slave process. Distribution of the form (3)–(9) studied in [26, 27]. Statistical justification for introducing such distributions is given in [28].Let us underline the principal features of the suggested approach.(1) We introduce a novel variableΓ which can be used to derive additional information about a system in the stationary nonequilibrium state. We suppose that  Γ is a measurable quantity at macroscopic level, thus values like entropy which are related to the order parameter (principal macroscopic variable) can be defined. At the mesoscopic level, the variable Γ is introduced as a variable with operational characteristics of a random process slave with respect to the process describing the order parameter.(2) We suppose that thermodynamic forcesγ related to the novel variable can be defined. One can introduce the “equations of state” β(〈E〉〈Γ〉), γ(〈E〉,〈Γ〉). Thus we introduce the mapping (at least approximate) of the external restrictions on the point in the plane β, γ.(3) We suppose that a “refined” structure factorω(E,Γ)  can be introduced which satisfies the condition ∫ω(E,Γ=y)dy=ω(E) (ordinary structure factor). This function (like ω(E)) is the internal (inherent) property of a system. At the mesoscopic level, we can ascribe to this function some inherent to the system (at given restrictions (β0,γ0)) random process. The structure factor has the meaning of the joint probability density for the values E,Γ understood as the stationary distribution of this process. Provided the “reper” random process for the point (β0,γ0), one can derive therefrom the shape of the structure function. If we model the dependence of the system potential of the order parameter by some potential well, the lifetime distribution within one busy period and probabilities ω(E,Γ) can be viewed as distributions of the transition times between the subset of the phase space (possibly of the fractal character) corresponding to the potential well, and the subset corresponding to the domain between the “zero” and the “hill” of the potential wherefrom the system will roll down to the zero state (i.e., from the domain y>b towards y<b, Figure 1). To determine the explicit form of Γ (at (β0,γ0)), the algorithm of the asymptotic phase coarsening of complex system is used.(4) It is supposed that at least for certain classes of influences the resulting distribution has the form (4), (7), that is, the change of the principal random process belongs to some class of the invariance leading to this distribution, which explains how one can pass from the process in the reper point (β0,γ0) (e.g., in equilibrium when γ=0 and β=1/kBT) to a system in an arbitrary nonequilibrium stationary. The thermodynamic forces should be chosen so that the distribution leads to new (measurable) values of (〈E〉,〈Γ〉).(5) The valuesγ and x are determined from the comparison with EIT.If one integrates (4) and (7) over Γ, one gets the distribution P(E) depending on E as well as on β and γ as parameters:(41)P(E)=∫P(E,Γ=y)dy=exp⁡{-βE-γτγ}ω(E)Z(β,γ),
where γτγ=-lnL(γ,E) and L(γ,E)=∫e-γyω(E,Γ=y)dy/ω(E). Averaging the expression for lnL(γ,E), we shall get for τ=-〈lnL(γ,E)〉/γ,(42)〈ln⁡⁡L(γ,E)〉=∫exp⁡⁡{-βx-γy}ω(x,y)ln⁡⁡L(γ,x)dxdyZ(β,γ)=∫exp⁡⁡{-βx}ω(x)L(γ,x)ln⁡⁡L(γ,x)dxZ(β,γ).
Let us introduce the value τ0=〈Γ〉-τ=∫exp{-βx}ω(x)[LlnL-γ∂L/∂γ]dx/γZ(β,γ), which vanishes at γ=0. The value S-Sβ=ΔS=γτ0(<0) is the loss of information when passing from the distribution (7), (4) to (41).Essential assumption is that the valuesΓ0 and lnL(γ,E) do not depend upon the initial value of E=x, that is, the value Γ0 can be understood as an average on 〈E〉 rather than of dynamical random variable E. This fact is the direct consequence of the Markoff chain subject to the unperturbed process being equally ergodic and possessing a single (for the distribution (16)) class of the ergodic states (Appendix B). Then the entropy S〈E〉=-〈lnρ(z;E)〉=β〈E〉+γτ+lnZ(β,γ) as an average of the distribution (42) coincides with Sβ; that is, the correspondence principle holds: the distribution (42) coincides with the Gibbs one. In Section 2 (expressions (3)–(9)), we performed the splitting of the phase space into the cells (E,Γ) giving up the equiprobable distribution of points in ω(E). But although we are now ascribing different weights to different points, collecting them together (summation over Γ) yields the Gibbs distribution.The lifetime (or escape time if one refers to the terminology of [10–12]) is described by smooth differentiable distributions (16)-(17) rather than by singular measures of the fractal repeller object (like it was done in [10–12]). This transition is possible because of use of the complex system coarse-graining algorithm [23]. But in the very general case, the lifetime in the phase space should be of pronounced fractal character. The distribution density (16) can be obtained out of the maximum-entropy principle at given Γ0 value (when Γ0 does not depend on the random value E). The entropy (20) behaves for γ<0 and γ>0 in a different fashion, with this property corresponding to the irreversibility since different signs of γ mean different signs of the fluxes q. The Gibbs theorem and H-theorem [22] hold: the entropy maximum indeed is attained at zero deviations from the equilibrium γ=0. The irreversibility in this approach appears as a consequence of the lifetime finiteness hypothesis.If one compares the exposed thermodynamics with EIT, following differences can be outlined.(1) Different expressions for the nonequilibrium temperature, entropyS, entropy production (which yield the EIT expressions as a particular case for small q⃗).(2) A new variable of the system size is introduced which should play certain part in the nonequilibrium case. For the continuous description, this might be the size of the “continuous medium point” [22].(3) Explicit expressions for the lifetimeΓ and its thermodynamic conjugate γ are obtained.



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*Source: 203203-2011-11-10.xml*