Table of Contents
Journal of Thermodynamics
Volume 2011, Article ID 203203, 10 pages
Research Article

Nonequilibrium Thermodynamics Based on the Distributions Containing Lifetime as a Thermodynamic Parameter

Kiev Institute for Nuclear Research, National Academy of Sciences of Ukraine, prospekt Nauki 47, 252028 Kiev, Ukraine

Received 7 June 2011; Revised 3 August 2011; Accepted 5 August 2011

Academic Editor: L. De Goey

Copyright © 2011 Vasiliy Vasiliy Ryazanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.

1. Introduction

In papers [13], 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 [13] 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 𝑞=𝜆𝑇 (where 𝑞 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 equation𝜏𝑞=𝜆𝑇𝑞𝜕𝑞𝜕𝑡,(1) 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), Γ𝑥=inf{𝑡𝑦(𝑡)=0},𝑦(0)=𝑥>0.(2)

The characteristics of Γ 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 system 𝑈(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 point a. 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 𝑡𝑏0𝑇stab, the system will roll down to the point 𝑦=0. The lifetime Γ=𝑇stab+𝑡𝑏0𝑇stab. Real systems are characterized by much more complicated behaviour, but the dependence Γ𝑇stab 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 [812]. 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 [1012], 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 [1012] 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 (Section 2). 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 observable 𝐸 (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)𝑃(𝐸,Γ)=𝑝𝐸Γ(𝑥,𝑦)=𝛿(𝐸𝐸(𝑧))𝛿(ΓΓ(𝑧))𝜌(𝑧;𝐸,Γ)𝑑𝑧,(3) where𝑃(𝐸,Γ)=𝑝𝐸Γ||(𝑥,𝑦)=𝜌(𝑧;𝐸,Γ)𝜔(𝐸,Γ)𝐸=𝑥,Γ=𝑦.(4) Lets introduce the conditional distribution density 𝑃(Γ𝐸)=𝑝Γ𝑝(𝑦𝑥)=𝐸Γ(𝑥,𝑦)𝑝𝐸(=𝑥)𝑃(𝐸,Γ)𝑃(𝐸),(5) where𝑃(𝐸)=𝑝𝐸(𝑥)=0𝜌(𝑧;𝐸,Γ=𝑦)𝜔(𝐸,𝑦)𝑑𝑦.(6)

Now we make an assumption about the form of 𝜌(𝑧;𝐸,Γ)𝜌(𝑧;𝐸=𝑥,Γ=𝑦)=exp{𝛽𝑥𝛾𝑦}𝑍(𝛽,𝛾),(7) where=𝑍(𝛽,𝛾)=exp{𝛽𝑥𝛾𝑦}𝑑𝑧𝑑𝑥𝑑𝑦𝜔(𝑥,𝑦)exp{𝛽𝑥𝛾𝑦}(8) is the partition function and 𝛽 and 𝛾 are Lagrange multipliers satisfying the equations for the averages𝐸=𝜕ln𝑍||||𝜕𝛽𝛾,Γ=𝜕ln𝑍||||𝜕𝛾𝛽.(9)

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(Γ𝑥), 𝑃(Γ,𝑥)𝐸𝑒𝜃𝑥𝑒𝑄𝜃=𝑒𝜃𝑥𝐸𝑒𝑃(𝑥)𝑑𝑥,(10)𝑠Γ(𝑥)𝑒𝐿(𝑥,𝑠)=𝑠𝑦𝑒𝑃(Γ=𝑦𝑥)𝑑𝑦,(11)𝐺(𝜃,𝑠)𝜃𝑥𝑠𝑦𝜕𝑃(𝐸=𝑥,Γ=𝑦)𝑑𝑥𝑑𝑦=𝐿𝑄𝑒𝜕𝜃,𝑠𝜃,(12) 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 lifetime𝐺(𝜃,𝑠)𝑍(𝛽+𝜃,𝑠+𝛾)𝑍(𝛽,𝛾).(13)

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),𝑍𝛽(𝛽,𝛾)=𝑍0,𝛾0𝐺𝛽𝛽0,𝛾𝛾0.(14)

The value 𝑍(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

If 𝛾=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 process 𝑦(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)1𝛽=𝑘𝐵𝑇,(15) where 𝑇 is the average (over the body volume) local equilibrium temperature. Since at fluxes 𝑞0 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=𝑉1𝑉𝑇(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:𝑝Γ(𝑦)=Γ01𝑦expΓ0.(16)

The values Γ0 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 factor 𝜔(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:𝜔(𝐸,Γ=𝑦)=𝜔(𝐸)Γ01𝑦expΓ0.(17) Substituting into the partition function (8) yields𝑍(𝛽,𝛾)=𝑍(𝛽)1+𝛾Γ01,(18) 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, Γ𝛾=Γ01+𝛾Γ01,𝛾=Γ𝛾1Γ0,(19) 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 𝐽𝜌𝑑𝑢/𝑑𝑡+𝑞=0, 𝐸=𝑉𝜌𝑢𝑑𝑉, 𝜌=𝑀/𝑉 is the mass density, and 𝐽𝑞=𝐽𝑢=𝑞 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 relation𝑆𝑘𝐵=ln𝜌(𝑧;𝐸,Γ)=𝛽𝐸+𝛾Γ+ln𝑍(𝛽,𝛾).(20) 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 have𝑆𝑘𝐵=𝑆𝛽𝑘𝐵+𝑥𝑆(1+𝑥)ln(1+𝑥),𝛽𝑘𝐵=𝛽𝐸+ln𝑍𝛽,𝑥=𝛾Γ0.(21) From (20) treating 𝐸 and Γ as variables,𝑑𝑆=𝑘𝐵𝛽𝑑𝐸+𝑘𝐵𝛾𝑑Γ.(22) 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/𝜕𝑞=0 (since Γ0 by definition corresponds to the state with 𝑞=0 and Γ0 independent on 𝑞). Thats why 𝑑Γ0=0 and 𝑑𝑆=𝑘𝐵𝑘𝛽𝑑𝐸𝐵𝑥𝑑𝑥(1+𝑥)2,(23) which coincides with the differential of (21).

Now we determine the explicit form for 𝛾 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 𝑥𝑞=𝛾𝑞Γ0𝑡0𝑞𝑆𝑎𝑞𝐸=𝑡0𝑞𝑞𝜌𝑢𝑅=𝑡0𝑞𝑦𝑞,𝑦𝑞=𝑞𝜌𝑢𝑅,𝐸𝜌𝑢𝑉,(24) where 𝑞=±|𝑞|=(𝑞𝑞)1/2, 𝑢=𝐸/𝜌𝑉, energy density in a system, 𝑅 its linear size, 𝑉 its volume 𝑅3, and the surface 𝑆𝑎𝑅2; 𝑞+=(𝑞+𝑠)=𝑞=(𝑞𝑞)1/2, where 𝑞 and 𝑞+ are projections of the outcoming and incoming heat flux density vectors on the surface normal vector 𝑠 (ordered chronologically, Figure 3); the signs of 𝑞 and 𝑞+ depend on whether we are heating or cooling the systém. Then 𝑞+=𝑞𝑅𝜌𝜕𝑢/𝜕𝑡, in the stationary state 𝑞+=𝑞.

Figure 2: Geokinetic plateau model.
Figure 3: The volume element of the size 𝑅, incoming 𝑞+, and outcoming 𝑞 fluxes, normal to the surface element 𝑠.

Outcoming flux 𝑞 and incoming flux 𝑞+ differ in their times: we consider 𝑞+ 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)1/2𝑙1,𝜀1=𝑛𝑟30, where 𝑛 is the density, 𝑟0 is the diameter of the atom, 𝑙1 is the free path length. In the gasodynamic description 𝐿=𝑙𝐺𝑝𝐿𝐺/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 𝐽𝜕(𝜌𝑎)/𝜕𝑡+0𝑎=𝜎𝑎; 𝐽0𝑎=𝜌𝑎𝑣𝑎 [3] holds (𝑣 is the velocity, with 𝜌=𝑀/𝑉 being the mass density); 𝐽div0𝑎(𝐽0𝑎+𝐽0𝑎+)/𝑅, where 𝑅 is the “size” of the point in the continuous description [22]. Then the value 𝐽0𝑎+𝑅𝜎𝑎=𝐽0𝑎𝑅𝜕(𝜌𝑎)/𝜕𝑡 is proportional to 𝛾 from (19). And the value 𝑦𝑎=(𝐽0𝑎+𝑅𝜎𝑎)/𝑅𝑎, multiplied by the time parameter 𝑡0𝑎 (from (28)), coincides with 𝑥=𝛾Γ0. The equation for the specific energy 𝑢 has the form 𝐽𝜌𝑑𝑢/𝑑𝑡+𝑞=0; 𝐸=𝑉𝜌𝑢𝑑𝑉; 𝐽𝑞=𝐽𝑢=𝑞 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 𝑞 of EIT [4] with the entropy of macroobservables 𝐸 and Γ.

For this purpose, we will recall the expressions of the EIT [4]:𝑆=𝑆𝛽𝑉𝜌𝛼𝑞𝑞𝑞𝑑𝑟2,𝛼𝑞=𝜏𝑞𝜌𝜆𝑇2(25) (𝑞 and 𝜏𝑞 are values from (1)). Comparing with (21) at small 𝑞: 𝑆=𝑆𝛽𝑘𝐵𝑥2/2, 𝑘𝐵𝑥2𝑞=𝑉𝜌𝛼𝑞(𝑞𝑞)𝑑𝑟, 𝑥=±𝑉𝜌𝛼𝑞(𝑞𝑞)𝑑𝑟𝑘𝐵1/2=𝛾Γ0.(26) The sign “−” in (26) is chosen if the incoming flux is directed towards the system (the system is being heated). The derivative 𝜕𝑥/𝜕𝑞𝐸,𝑅 is written as 𝑉[𝛿𝑥/𝛿𝑞(𝑟)𝐸,𝑅]𝑑𝑟=𝑉𝜌𝛼𝑞𝑞𝑑𝑟/𝑘𝐵𝑥, where 𝑅 is the linear size of the system. The expressions for 𝜕𝑥/𝜕𝐸𝑞,𝑅, 𝜕𝑥/𝜕𝑅𝐸,𝑞 can be obtained in the same fashion. Then from (23) 𝑘𝑑𝑆=𝐵𝛽𝑉𝜕𝜌𝛼𝑞/𝜕𝑢𝑞𝑞𝑑𝑟𝜌𝑉2(1+𝑥)2𝑑𝐸𝜌𝛼𝑞𝑞𝑑𝑟𝑑𝑞(1+𝑥)22𝑥2𝑘𝐵𝑑𝑅𝑅(1+𝑥)2.(27) If the fluxes 𝑞 depend only weakly on the spatial variables, then (26) is cast as𝑥𝑞±𝑉𝜌𝛼𝑞𝑑𝑟𝑘𝐵1/2𝑞𝑞1/2𝑡0𝑞𝑆𝑎𝑞𝐸=𝑡0𝑞𝑞𝜌𝑢𝑅=𝑡0𝑞𝑦𝑞,𝑡0𝑞𝐸𝑉𝜌𝛼𝑞𝑑𝑟/𝑘𝐵1/2𝑆𝑎,𝑦𝑞=𝑞𝜌𝑢𝑅,𝐸𝜌𝑢𝑉,(28) which coincides with (24).

From (23) we have𝑑𝑆=𝑘𝐵𝑥𝛽+2𝐸(1+𝑥)2𝑘𝑑𝐸𝐵𝑥2𝑞(1+𝑥)2𝑘𝑑𝑞𝐵2𝑥2𝑅(1+𝑥)2𝑑𝑅.(29)

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 condition 𝛿2𝑆0 should satisfy. We have 𝑘𝐵1𝜕2𝑆𝜕𝐸2𝑞,𝑅=𝜕𝛽𝑥𝜕𝐸2(3+𝑥)𝐸2(1+𝑥)3,𝑘𝐵1𝜕2𝑆𝜕𝑞2𝐸,𝑅=𝑥2(𝑥1)𝑞2(1+𝑥)3,𝑘𝐵1𝜕2𝑆𝜕𝑅2𝐸,𝑞=2𝑥2(𝑥3)𝑅2(1+𝑥)3,𝑘𝐵1𝜕2𝑆=𝜕𝐸𝜕𝑞2𝑥2𝑞𝐸(1+𝑥)3,𝑘𝐵1𝜕2𝑆=𝜕𝑞𝜕𝑅4𝑥2𝑞𝑅(1+𝑥)3,𝑘𝐵1𝜕2𝑆=𝜕𝐸𝜕𝑅4𝑥2𝐸𝑅(1+𝑥)3,𝜕𝛽=𝜕𝐸1𝑘𝐵𝑇2𝑐,(30)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 𝛿2𝑆0 holds for 𝜕𝑆2/𝜕𝐸2𝑞,𝑅0, 𝜕2𝑆/𝜕𝑞2𝐸,𝑅0. These conditions satisfy when 1/𝑇2𝑐𝑘𝐵𝑥2(3+𝑥)/𝐸2(1+𝑥)20 and 𝑘𝐵𝑥2(𝑥1)/𝑞2(1+𝑥)30. One more condition satisfies if (𝜕𝑆2/𝜕𝐸2𝑞,𝑅)(𝜕2𝑆/𝜕𝑞2𝐸,𝑅)(𝜕2𝑆/𝜕𝐸𝜕𝑞)20. Hence𝑘|𝑥|1+𝐵𝑇2𝑐𝐸21/2.(31) 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 𝑗(𝑑𝑆/𝑑𝑡)/𝑉=𝑆+𝜎𝑆,𝑆=𝑉𝜌𝑠𝑑𝑟, where for 𝜌=const, 𝑗𝑆=𝜃1𝑞, 𝜎𝑆=𝑞𝜃1𝑘𝐵𝑥2(𝑑𝑞/𝑑𝑡)𝑞𝑉(1+𝑥)2+𝑘𝐵(𝑑𝑅/𝑑𝑡)3𝑢𝛽𝜌+𝑥2𝑉1(1+𝑥)2𝑅.(32)

Comparing entropy production 𝜎𝑆 and 𝜎𝑆=𝑞𝑞/𝜆𝑇2 [3], we shall find𝜆𝑇2𝑘𝐵𝑥2𝑑𝑞𝑞𝑑𝑡2𝑉1(1+𝑥)2+𝑞=𝜆𝑇2𝜃1+𝑑𝑅𝑑𝑡𝜆𝑇2𝑘𝐵3𝑢𝛽𝜌+𝑥2𝑉1(1+𝑥)2.𝑞𝑅(33) The inverse nonequilibrium temperature𝜃1=𝜕𝑆|||𝜕𝐸𝑞,𝑅=𝑘𝐵𝑥𝛽+2𝐸(1+𝑥)2=1𝑇+𝑞2𝛼𝑞𝑢1+𝑞𝜌𝛼𝑞𝑉/𝑘𝐵1/22(34) coincides with the value 1/𝜃 of the extended thermodynamics [4] at small 𝑞 (when (1+𝑥)21). The value entropy production 𝜎𝑆 and the expression (33) when substituting therein (28) take on the form𝜎𝑆=𝑞𝜃1𝜌𝑞𝛼𝑞(𝑑𝑞/𝑑𝑡)(1+𝑥)2+(𝑑𝑅/𝑑𝑡)𝜌3𝑢𝑇1+𝑞2𝛼𝑞(1+𝑥)2𝑅,𝑑𝑞𝜏𝑑𝑡𝑞(1+𝑥)2+𝑞=𝜆𝑇2𝜃1+(𝑑𝑅/𝑑𝑡)3𝑢𝜆𝑇𝜌/𝑞+𝑞𝜏𝑞(1+𝑥)2𝑅,(35) 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 for 𝑥 must have (similar to [24]) the form𝑥=±𝑉𝑑𝑟𝑘𝐵𝜌𝜏𝑞𝜌𝜆𝑇2𝑞𝑞+(𝜎𝑣𝜎𝑣)𝜏𝑣𝜌𝜇𝑇1/2,(36) 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 𝑗𝑆, 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 for 𝛽𝐸 the values 𝑑𝑟𝛽(𝑟,𝑡)̂𝑢(𝑟), where ̂𝑢(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)[̂𝑢(𝑟)𝑘𝜇𝑘(𝑟,𝑡)̂𝑛𝑘(𝑟)], where ̂𝑛𝑘(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 𝑟𝑗=1𝜈+𝑗𝑋𝑗𝑘+𝑟𝑗=1𝜈𝑗𝑋𝑗, we get 𝑥𝜌=±𝑟𝑘=1𝑉𝑑𝑟𝑘𝐵𝜏𝑘di𝐿𝑘𝑉𝑗𝑘𝑗𝑘+𝜔𝑘𝜔𝑘𝜏𝑘chem𝐿𝑘chem𝑉𝑇1/2,(37) where 𝑥𝑘=𝑡0𝑘(𝐽𝑘𝜔𝑘𝑅)/𝑅𝑐𝑘, 𝑡0𝑘=𝑡0𝑘di=𝑐𝑘𝑅(𝜏𝜌di/𝑘𝐿𝑘)1/2, where 𝜏𝜌di is the correlation time for the fluxes 𝐽𝑘, 𝐽𝑘 is the normal projection of 𝐽𝑘 (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 𝐽𝑖=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; 𝑟𝑘=1𝜌𝑘=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 𝜕𝑐𝑗𝐽/𝜕𝑡=𝑗+𝜔𝑗, where 𝐽𝑗=𝐷𝑗𝑐𝑗 is the diffusive flux of the 𝑗-th component, 𝐷𝑗—its diffusion coefficient, 𝜔𝑗=𝑘+𝑟𝑛=1𝑐𝜈+𝑛𝑛𝑘𝑟𝑛=1𝑐𝜈𝑛𝑛 the rate of the 𝑗-th reaction per unit volume, 𝑘+ and 𝑘 are reaction constants, 𝑣𝑛=𝑣𝑛𝑣+𝑛 are stechiometric coefficients [3, 25]. Besides the value 𝑡0𝑘di, one can also introduce the time 𝑡0kchem=(𝛼𝜌𝑘chem𝑉/𝑘)1/2, 𝛼𝜌𝑘chem=𝜏𝑘chem/(𝑉𝑇𝐿𝑘chev), 𝐿𝑘chem=𝑐𝑘𝜔eq+/𝑘𝑇, 𝜔eq+=𝑘+𝑟𝑗=1(𝑐eq𝑗)𝜈+𝑗 (such choice of kinetic coefficients for chemical reactions where 𝜔=𝐿𝐴(𝐴=𝑟𝑛=1𝑣𝑛𝜇𝑛=𝑘𝑇ln𝑟𝑗=1(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 𝑡0𝑘di enter the same balance equation for 𝑐𝑘, 𝑡0𝑘di=𝑡0kchem. From this, we get the relation between 𝜏𝑘chem and 𝜏𝑘di: 𝜏𝑘chem/(𝑇𝐿𝑘chem)=𝑅2𝑐2𝑘𝜏𝜌di/𝐿𝑘. As thermodynamic quantities, let us take 𝑐𝑘, 𝐽𝑘, 𝑚𝑘, 𝑚𝑘=𝑟𝑗=1𝛾𝑘𝑗𝜔𝑗,𝜔𝑗=(𝜕𝑗𝜌𝑘/𝜕𝑡)/𝛾𝑘𝑗 in the 𝑗-th chemical reaction [3], 𝐾𝑘=1𝑚𝑘=0. Since 𝜕𝑐𝑘/𝜕𝜌𝑘=(1𝑐𝑘)/𝜌, 𝜕𝑥𝑘/𝜕𝑐𝑘𝐽𝑘,𝑚𝑘=𝑥𝑘(𝜕𝜌𝑘/𝜕𝑐𝑘)/𝜌𝑘=𝑥𝑘/𝑐𝑘(𝑐𝑘1), 𝜕𝑥𝑘/𝜕𝐽𝑘𝑐𝑘,𝑚𝑘=𝑥𝑘𝑚𝑘𝑅/𝐽𝑘(𝐽𝑘𝑅𝑚𝑘), 𝜕𝑥𝑘/𝜕𝑚𝑘𝑐𝑘,𝐽𝑘=𝑅𝑥𝑘/(𝐽𝑘𝑅𝑚𝑘). Substituting these expressions into (3), we get 𝑘𝐵1𝑘𝑑𝑆=𝐵1𝑑𝑆𝛽(𝛾Γ)2𝑟𝑘=1𝛾𝑘[𝔄]𝛾,(38) where 𝔄 denotes 𝑑𝑐𝑘/𝑐𝑘(𝑐𝑘1)+𝑅𝑚𝑘𝑑𝐽𝑘/𝐽𝑘(𝐽𝑘𝑅𝑚𝑘)𝑅𝑑𝑚𝑘/(𝐽𝑘𝑅𝑚𝑘).

If we substitute 𝑑𝑐𝑘/𝑑𝑡 from the balance equation and use the transformation 𝜇𝐽𝑘/𝑘𝐵𝑇=(𝜇𝐽𝑘/𝑘𝐵𝑇)𝐽𝑘(𝜇/𝑘𝐵𝑇), we get the entropy balance equation with the entropy flux 𝑗𝑆=𝑘𝐽𝑘𝜇𝑘/𝑇, 𝜇𝑘/𝑘𝐵𝑇=𝜇0𝑘/𝑘𝐵𝑇+(𝛾Γ)2𝛾𝑘/𝛾𝑐𝑘(𝑐𝑘1), and entropy production 𝜎𝑆=𝑟𝑘=1𝐽{𝑘(𝜇𝑘/𝑇)𝜇𝑘𝑚𝑘/𝑇𝜌𝛾Γ2𝑘𝐵𝛾𝑘[𝑅𝑚𝑘(𝑑𝐽𝑘/𝑑𝑡)/𝐽𝑘(𝐽𝑘𝑅𝑚𝑘)𝑅(𝑑𝑚𝑘/𝑑𝑡)/(𝐽𝑘𝑅𝑚𝑘)]}. Setting this value 𝜎𝑆 equal to 𝜎𝑆=𝑟𝑘=1𝐽2𝑘/𝐿𝑘𝑘+𝑠𝑟=1𝑠𝑙=1𝑅𝑟𝑙𝜔𝑟𝜔𝑙/𝑇, where 𝑅𝑟𝑙 are resistivity matrices [3] from the expression 𝐴𝑟=𝑠𝑙=1𝑅𝑟𝑙𝜔𝑙, (𝑠, number of chemical reactions in a system) we get the general expression relating 𝑑𝐽𝑘/𝑑𝑡, 𝑑𝑚𝑘/𝑑𝑡 with 𝐽𝑘, 𝑚𝑘. Further, considering 𝑚𝑘𝑅𝐽𝑘 (almost pure diffusion), one can get an equation for 𝐽𝑘, similar to (9) for 𝑞. Vice versa, when 𝐽𝑘𝑚𝑘𝑅 (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 form𝑆𝑘𝐵=𝑆𝛽𝑘𝐵+𝑥01+𝑥0ln1+𝑥0𝑥𝑥0𝑥01+𝑥02+𝑥𝑥02𝑥0121+𝑥03+.(39) 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 as𝑆𝑘𝐵=𝑆𝛽𝑘𝐵+𝑥1+ln1𝑥1𝑥𝑥0𝑥11𝑥1+𝑥𝑥022𝑥111𝑥122+.(40)

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 of Γ 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 variable Γ 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 forces 𝛾 related to the novel variable can be defined. One can introduce the “equations of state” 𝛽(𝐸Γ), 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 factor 𝜔(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 values 𝛾 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:𝑃(𝐸)=𝑃(𝐸,Γ=𝑦)𝑑𝑦=exp𝛽𝐸𝛾𝜏𝛾𝜔(𝐸)𝑍(𝛽,𝛾),(41) where 𝛾𝜏𝛾=ln𝐿(𝛾,𝐸) and 𝑒𝐿(𝛾,𝐸)=𝛾𝑦𝜔(𝐸,Γ=𝑦)𝑑𝑦/𝜔(𝐸). Averaging the expression for ln𝐿(𝛾,𝐸), we shall get for 𝜏=ln𝐿(𝛾,𝐸)/𝛾,ln𝐿(𝛾,𝐸)=exp{𝛽𝑥𝛾𝑦}𝜔(𝑥,𝑦)ln𝐿(𝛾,𝑥)𝑑𝑥𝑑𝑦=𝑍(𝛽,𝛾)exp{𝛽𝑥}𝜔(𝑥)𝐿(𝛾,𝑥)ln𝐿(𝛾,𝑥)𝑑𝑥.𝑍(𝛽,𝛾)(42) Let us introduce the value 𝜏0=Γ𝜏=exp{𝛽𝑥}𝜔(𝑥)[𝐿ln𝐿𝛾𝜕𝐿/𝜕𝛾]𝑑𝑥/𝛾𝑍(𝛽,𝛾), which vanishes at 𝛾=0. The value 𝑆𝑆𝛽=Δ𝑆=𝛾𝜏0(<0) is the loss of information when passing from the distribution (7), (4) to (41).

Essential assumption is that the values Γ0 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 𝑆𝐸=ln𝜌(𝑧;𝐸)=𝛽𝐸+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 [1012]) is described by smooth differentiable distributions (16)-(17) rather than by singular measures of the fractal repeller object (like it was done in [1012]). 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, entropy 𝑆, entropy production (which yield the EIT expressions as a particular case for small 𝑞).

(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 lifetime Γ and its thermodynamic conjugate 𝛾 are obtained.


A. Markovian Stochastic Model

Let the process 𝑦(𝑡) be Markovian. For the kinetic coefficients𝐾𝛼1𝛼𝑚(𝑦)=lim𝜏0𝜏1Δ𝑦𝛼1Δ𝑦𝛼𝑚𝑦,Δ𝑦=𝑦(𝑡+𝜏)𝑦(𝑡),(A.1) the potential [25] is written𝑉(𝜃,𝑦)𝑚=1𝛽𝑚11𝑚!𝛼1𝛼𝑚𝐾𝛼1𝛼𝑚(𝑦)𝜃𝛼1𝜃𝛼𝑚,(A.2) where 𝛽1 is the small parameter; for the equilibrium Gibbs system 𝛽=1/𝑘𝐵𝑇eq, 𝑘𝐵 being the Boltzmann constant, 𝑇 is the temperature.

The forward kinetic equation for the distribution density𝜕𝑝(𝑦,𝑡)𝜕𝑡=𝑁𝜕,𝑦Φ𝜕𝜕𝑦,𝑦𝑝(𝑦,𝑡),(A.3) where Φ is the stochastic potential, Φ(𝜃,𝑦)=𝛽𝑉(𝜃/𝛽,𝑦), the operator 𝑁𝜕,𝑦 defines the order of operations (differentiating on y goes the last).

The Laplace transform gives the equation for 𝑄 (10):𝜕𝑄(exp{𝜃},𝑡)𝜕𝑡=𝑁𝜃,𝜕/𝜕𝜃Φ𝜕𝜃,𝑄𝜕𝜃,𝑡(exp{𝜃},𝑡).(A.4) One can show that the lifetime (2) is governed by the Hermitian conjugate operator and for 𝐿 (11) the following holds: 𝑁𝑥,𝜕Φ𝜕𝜕𝑥,𝑥𝐿(𝑥,𝑠)=𝑠𝐿(𝑥,𝑠)(A.5) with the condition 𝐿(0,𝑠)=𝐿(𝑥,0)=1. For the nonhomogeneous Markoff systems if 𝐿 depends on the initial time 𝑡0, equation (A.5) is generalized𝑠𝐿𝑡0(𝑥,𝑠)=𝜕𝐿𝑡0(𝑥,𝑠)𝜕𝑡0+𝑠𝑁𝑥,𝜕Φ𝜕𝜕𝑥,𝑥,𝑡𝑡0𝜕𝐿𝜕𝑠𝑡0(𝑥,𝑠)1𝑠.(A.6) From these expressions, the precise solutions for the Markoff stochastic models of the physical systems can be found. We note bypassing that in (A.2A.6) the coefficients are unperturbed equilibrium values independent on 𝛽0, 𝛾0.

B. Algorithms of the Phase Coarsening of the Complex Systems

One can represent an open thermodynamic system as evolving in a random medium whose mathematical model will be either Markoff renewal processes or semi-Markoff processes. The local characteristics of the system depend on the random medium state. The scheme of the asymptotic phase coarsening present the simplified description of the system evolution in a random medium which can be performed based upon a simple set of heuristic rules [23]. The semi-Markovian process to describe the evolution of a random system is considered in the enlarged time scale.

In our case, the stationary distributions are described by the Gibbs distributions. Absorbing state is the degenerated state of the system with 𝐸=0, and the degeneracy probability is 𝑃0=1/𝑍(𝛽). The ergodic Markoff chain has the distribution 𝜌. The residence times in the states 𝑥, 𝜃𝑥, are given by the distribution functions 𝐺𝑥(𝑡)=𝑃{𝜃𝑥𝑡}=𝑃{𝜃𝑛+1𝑡𝜉𝑛=𝑥} (𝜉𝑛 are the states of the Markoff chain) and by the average residence times 𝑚(𝑥)=0(1𝐺𝑥(𝑡))𝑑𝑡=𝐸𝜃𝑥, which are limited: 0𝑚(𝑥)<. Let us assume that a real system has besides the class of the ergodic states the class of the trapping states where the absorbing of the Markoff chain is possible. The phase space of a real system is then 𝑋=𝑋0𝑋𝑋0, where 𝑋0 is the class of the ergodic states of the reference enclosed Markoff chain (without taking into account the absorption into 𝑋0), 𝑋0 are trapping states of an enclosed Markoff chain, and 𝑋 is the finite set of the states. In [23] it was shown that the coarsened random process is Markovian with two states 𝑋0coars and 𝑋0coars. The residence time in the stable state is distributed exponentially with the parameter 𝜆1=1𝑚,𝑚=𝑋0𝜌(𝑑𝑥)𝑚(𝑥)𝑃10,𝑃10𝜉=𝑃𝑛+1,coars=𝑋0coars𝜉𝑛,coars=𝑋0coars.(B.1) Thus the expression (16) is obtained in which 𝑚=Γ0. The Markoff processes as objects modelling a complex system appear not as a a priori hypothesis but rather as a result of splitting phase space of states and “gluing up” together the states which belong to one and the same class. The algorithm of the phase coarsening shows the natural property of the complex systems: the transitions in a complex system between the classes of states are rather governed by the Markoff property. If one neglects the detailed description of the system evolution (neglecting the transitions within a class), then (at the condition of big enough residence time within a class) the system losses the dependence of the interclass transitions on the behaviour inside of a class, and the residence time in a class is set as a sum of a big (random) number of the random values which are random times of residence in the states; under certain conditions, this residence time in the class can be considered to have an exponential distribution. We have𝑃Γ𝐺𝑡>𝑡=exp𝑚Γ,𝑃𝑥𝑡>𝑡exp𝑚,(B.2) where Γ𝐺 is the residence time of a coarsened system before absorption and Γ𝑥 is the residence time within the class 𝑋0 with the initial state 𝑥. The error in the approximation (B.2) is proportional to the degeneracy probability of the Gibbs system 𝑃0=1/𝑍𝛽.


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