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Journal of Thermodynamics
Volume 2012 (2012), Article ID 318032, 5 pages
Nonequilibrium Thermodynamics and Distributions Time to Achieve a Given Level of a Stochastic Process for Energy of System
Institute for Nuclear Research, Prospect Nauki 47, Kiev 252028, Ukraine
Received 21 August 2012; Revised 9 October 2012; Accepted 9 October 2012
Academic Editor: L. De Goey
Copyright © 2012 V. V. 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.
In a previous paper (Ryazanov (2011)) with the joint statistical distribution for the energy and lifetime (time to achieve a given level of a stochastic process for energy of system) to derive thermodynamic relationships, clarifying similar expressions of extended irreversible thermodynamics we used an exponential distribution of lifetime. In this paper, we explore a more realistic expression for the distribution of time to achieve a given level of a stochastic process for energy of system (or relaxation times or lifetimes), and we analyse how such distribution affects the corresponding expressions of nonequilibrium entropy, temperature, and entropy production.
In  expressions for a nonequilibrium entropy , entropy production , nonequilibrium temperature, and generalized transport equations for thermodynamic fluxes were proposed, taking into account a given distribution of internal relaxation times (or lifetimes) of the variables of the system. In the theory of random processes such values are called time to achieve a given level of the variables of the system (or the first-passage-time problems or escape time). In  for this quantity we used the term lifetime. These expressions generalize similar expressions of the extended irreversible thermodynamics (EIT) , a theory which takes into account the nonvanishing character of the relaxation time of the heat flux and other thermodynamic fluxes. In  we proposed to take into consideration not only a single or a few relaxation time, but a whole distribution of relaxation times, generalizing in this way the previous proposals of EIT.
There are many motivations for taking a full distribution of relaxation times; for instance, in heat transfer, the heat flux is the sum of contributions of molecules moving at different speeds, or of phonons having different frequencies, and the relaxation times of the mentioned contributions usually depend on the speed or the frequency, thus yielding a relaxation time distribution. Glassy materials have also a wide distribution of relaxation times. Collisions of heavy nuclei yield different products having widely different decay times. Thus, the consideration of relaxation time distributions seems natural in the analysis of nonequilibrium systems.
Approach  was based on the distribution containing lifetime as a thermodynamic parameter of the form where is energy of the system, are dynamical variables, coordinates and momenta of the system particles, is random variables of lifetime, a first-passage time till the random process reaches its zero value, and are Lagrange multipliers conjugated to thermodynamic parameters and , and is the partition function. Reference  uses an exponential distribution for the lifetime where is average lifetime. The expression (2) can be obtained from the algorithm of phase coarsening of the complex systems (Appendix 2 ). The distribution in the form (2) is valid for the existence of the weak ergodicity in a system. Mixing the system states at long times will lead to the distribution (2). For systems in the vicinity of phase transitions or in the chaotic regime, distribution (2) is no longer valid. The aim of this paper is to generalize the formalism proposed in  to this more general situation.
2. Generalized Lifetime Thermodynamics with Lifetime Distribution in (3)
Setting the form of the function reflects not only the internal properties of a system, but also the influence of the environment on an open system and the particular character of its interaction with the environment. The following physical interpretation of the exponential distribution for the function is given: a system evolves freely like an isolated system governed by the Liouville operator. Besides that the system undergoes random transitions and the phase point representing the system switches from one trajectory to another one with an exponential probability under the influence of the ‘‘thermostat.” The exponential distribution describes completely random systems. The influence of the environment on a system can have organized character as well; for example, this is the case of systems in a nonequilibrium state with input and output nonstationary fluxes. The character of the interaction with the environment can also vary; therefore different forms of the function can be used.
In  the maximum entropy principle for Liouville equations with source for the determination of the function was applied. The result obtained in  can be written as where is some initial moment of birth of the system, is the current time,
In (4) it is assumed that, along with the energy system is described and by other physical quantities; for example, is density of momentum, are the number of particles th component, and so forth and conjugate thermodynamic parameters . If we restrict ourselves, , , becomes
In deriving (3) we used the Zubarev-Peletminsky rule [4, 5] where are -numbers. When the local density of dynamic variables is considered, the value of may depend on the spatial variables. Since then the value may also depend on the spatial variables or may be differential operators. The equation for the specific energy has the form ; ; is the heat flux. In this paper we consider what changes in the thermodynamic behavior of the system results with replacement of the form (2), used in , the function of the form (3).
In the expression for the partition function of (1), is the structure factor which has a meaning of the joint probability density of values , as it was shown in . It may be shown that for the distribution (3) the function takes on the form:
We have from (11) when ; ;The expression for is close to that considered in , the expression , but not identical to it. At the entropy is equal to whereThe first two terms on the right side of (14) coincide with the expression for of . The remaining terms are a supplement to this expression. They depend on and tend to 0 if a value (or a value in (11), (3), (7)) tends to 0.
From (14) treating and as variables,
In  for the case of thermal conductivity obtained the expression where , is energy density in a system, is its linear size, is its volume ~, and the surface ; , where and are projections of the outcoming and incoming heat flux density vectors on the surface normal vector (ordered chronologically ); the signs of and depend on whether we are heating or cooling the system. Then ; in the stationary state .
Thermodynamic parameters that determine the value of will be the energy , the heat flux , and the size of . From (17) we obtain
Expression (15) can be rewritten as
In  value of was determined by comparing the obtained expression for the nonequilibrium entropy to the same expression of the EIT . It is possible to determine the value of in the other ratios, for example, by comparing the expression for the inverse nonequilibrium temperature with corresponding expression of EIT in the form where is the time of the flux correlation, is equilibrium temperature, and is the heat conductivity coefficient.
Equating the expressions (21) and (22), we obtain the equation for . Neglecting the powers of higher than the second, we find the solution of the resulting quadratic equation The exact expression of is more cumbersome ((20) than linear approximation of (16)) where is from (16). Here you can substitute the value of from (17). From (16)–(19) we obtain
Determining in (25) the value of , we find the entropy balance equation , , where for , ,
Comparing entropy production (26) and of EIT , we shall find The first two terms of the expressions (24)–(27) coincide with the expressions of . The remaining terms are a supplement to these expressions. They depend on and tends to 0 if a value tends to 0.
Just as in , we can write the conditions of thermodynamic stability and also consider not only by heat transfer, but by the internal friction of the system which is represented by the dissipative part of the stress tensor, the processes with variable mass, and so forth.
In this paper instead of the exponential distribution for the lifetime of  we examined the lifetime distribution of the form (3). The expressions derived here generalize expressions for the nonequilibrium entropy, entropy production, inverse nonequilibrium temperature, and Maxwell-Cattaneo equation derived in . Expressions have the following structure: the terms of  and supplements, depending on the value of (11).
There are other possibilities to generalize the results of . Thus, in  the linearised dual-phase-lag model of heat transfer is based on the equation where is the equilibrium temperature, the heat conductivity, the heat flux, and are relaxation times. In  it was supposed that the entropy density depends not only on the internal energy density but also on and . The use of these variables could be extended to higher-order time derivatives; such formalism is compatible with our proposal of taking a distribution of relaxation times, because it allows for rather general dynamical decays, much more complicated than an exponential decay with a single relaxation time.
It is possible to suggest some actual physical systems to which these ideas may be applicable. Those are systems which take into account the nonvanishing character of the relaxation time of the heat flux and other thermodynamic fluxes. Examples include solids with phonons having different frequencies and the relaxation times of the mentioned contributions usually depend on the speed or the frequency. A more detailed study of glass materials with a wide range of relaxation times, different times of the decay in collisions of heavy nuclei—those and other examples—can serve as objects of description of the proposed theory. The proposed description also applies to the processes of deformation of a continuous medium, to the chemical reactions, and so forth. The proposed description characterizes open systems describing the interaction with the environment.
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