Nonequilibrium Thermodynamics and Distributions Time to Achieve a Given Level of a Stochastic Process for Energy of System

Abstract

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.

1. Introduction

In [1] expressions for a nonequilibrium entropy 𝑆, entropy production 𝜎, 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 [1] for this quantity we used the term lifetime. These expressions generalize similar expressions of the extended irreversible thermodynamics (EIT) [2], a theory which takes into account the nonvanishing character of the relaxation time of the heat flux and other thermodynamic fluxes. In [1] 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 [1] was based on the distribution containing lifetime as a thermodynamic parameter of the form 𝜌𝑧;𝐸=𝑣,Γ=𝑦=exp−𝛽𝑣−𝛾𝑦𝑍,𝛽,𝛾 (1) where 𝐸 is energy of the system, 𝑧 are dynamical variables, coordinates and momenta of the system particles, Γ is random variables of lifetime, a first-passage time till the random process 𝑣(𝑡)=𝐸(𝑡) reaches its zero value, 𝛽 and 𝛾 are Lagrange multipliers conjugated to thermodynamic parameters 𝐸 and Γ, and 𝑍(𝛽,𝛾) is the partition function. Reference [1] uses an exponential distribution for the lifetime 𝑝_Γ𝑦=Γ₀⁻¹−𝑦expΓ₀, (2) where Γ₀ is average lifetime. The expression (2) can be obtained from the algorithm of phase coarsening of the complex systems (Appendix 2 [1]). 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 [1] to this more general situation.

2. Generalized Lifetime Thermodynamics with Lifetime Distribution in (3)

Setting the form of the function 𝑝_Γ(𝑦) 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 𝑝_Γ(𝑦) 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 𝑝_Γ(𝑦) can be used.

In [3] the maximum entropy principle for Liouville equations with source for the determination of the function 𝑝_Γ(𝑦) was applied. The result obtained in [3] can be written as 𝑝_Γ=𝑝𝑦,𝑡^Γ(0)𝑒^{−𝐶𝑖𝑦/𝐹𝑖}1+𝑝_Γ(0)/𝐹_𝑖𝑒^{−𝐶𝑖𝑦/𝐹𝑖}𝑅(𝑡)−𝑅𝑡₀,𝑦=𝑡−𝑡₀, (3) where 𝑡₀ is some initial moment of birth of the system, 𝑡 is the current time, 𝑅(𝑡)=_𝑗𝑚𝐹_𝑚𝑡₀𝐹_𝑗𝑡₀×_𝑘𝑃_𝑘𝑃_𝑗𝑃_𝑚−𝑃_𝑗𝑃_𝑚𝑃_𝑘𝑃_𝑖𝑃_𝑘−𝑃_𝑖𝑃_𝑘+𝐹_𝑖ln𝑍_𝛽𝑡₀−−_𝑚𝑗𝐹_𝑗𝑡₀𝑃_𝑗𝑃_𝑚−𝑃_𝑗𝑃_𝑚𝑃_𝑖𝑃_𝑚−𝑃_𝑖𝑃_𝑚. (4)

In (4) it is assumed that, along with the energy 𝐸=𝑃₀ system is described and by 𝑀 other physical quantities; for example, 𝑃₁=𝑝 is density of momentum, 𝑃_𝑖+1=𝑛_𝑖 are the number of particles 𝑖th component, and so forth and conjugate thermodynamic parameters 𝐹_𝑗. If we restrict ourselves, 𝑃₀=𝐸, 𝐹_𝑗=𝛽, 𝑅(𝑡) becomes 𝑅(𝑡)=𝛽²𝑡₀𝐸³−𝐸²⟨𝐸⟩𝐸²−⟨𝐸⟩²+𝛽(𝑡)ln𝑍_𝛽𝑡₀−𝛽𝑡₀,𝑍_𝛽=𝜔(𝐸=𝑣)𝑒^−𝛽𝑣𝑑𝑣. (5)

In deriving (3) we used the Zubarev-Peletminsky rule [4, 5] 𝑑^→𝑧=𝑑𝑡^→𝑤^→𝑧,_→𝑤^→∇𝑃_𝑖=^𝑀_𝑗=1𝐶_𝑖𝑗𝑃_𝑗𝐶;𝑖=1,…,𝑀,_𝑖=_𝑗𝐶_𝑗𝑖𝐹_𝑗𝑡₀, (6) where 𝐶_𝑖𝑗 are 𝑐-numbers. When the local density of dynamic variables is considered, the value of 𝑃_𝑚 may depend on the spatial variables. Since then the value 𝐶_𝑖𝑗 may also depend on the spatial variables or may be differential operators. The equation for the specific energy 𝑢 has the form 𝜌𝑑𝑢/𝑑𝑡+^→∇^→𝐽𝑞=0; 𝐸=∫_𝑉𝜌𝑢𝑑𝑉; ^→𝐽_𝑞=^→𝐽_𝑢=^→𝑞 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 [1], the function of the form (3).

From the normalization of the distribution (3) it is seen that 𝑝_Γ(0)=𝛽1−𝑒^{−𝑟𝐶𝑖/𝛽2}𝑟,𝑟=𝑅𝑡₀−𝑅(𝑡). (7) The value of (𝐶_𝑖/𝛽)⁻¹ is close to the average lifetime Γ₀ of the relation (2). It may be shown that in the linear approximation to 𝑟𝑝_Γ0,𝑡₀𝐶=𝑎=^𝑖𝐹_𝑖,𝑝_Γ𝑦,𝑡=𝑎𝑒^−𝑎𝑦1+𝑎𝑟𝑒^−𝑎𝑦𝛽. (8)

In the expression for the partition function 𝑍(𝛽,𝛾) of (1), 𝑍𝛽,𝛾=exp−𝛽𝑣−𝛾𝑦𝑑𝑧=𝑑𝑣𝑑𝑦𝜔𝑣,𝑦exp−𝛽𝑣−𝛾𝑦 (9) is the structure factor 𝜔(𝐸,Γ) which has a meaning of the joint probability density of values 𝐸, Γ as it was shown in [1]. It may be shown that for the distribution (3) the function 𝜔(𝐸,Γ) takes on the form: 𝜔(𝐸)𝑝𝐸,Γ=𝑦=𝜔^Γ(0)𝑒^{−(𝐶𝑖𝑦/𝐹𝑖)}1+𝑝_Γ(0)/𝐹_𝑖𝑒^{−(𝐶𝑖𝑦/𝐹𝑖)}𝑅(𝑡)−𝑅𝑡₀. (10)

Substituting (10) into the partition function (9) yields 𝑍𝛽𝑍𝛾,𝑍𝛾=𝛽,𝛾=𝑍_∞0𝑒^−𝛾𝑦𝑝_Γ=𝑝𝑦,𝑡𝑑𝑦_Γ(0)𝑎1+𝛾/𝑎²𝐹₁𝛾1,1+𝑎𝛾,2+𝑎,𝑝,𝑤𝑤=^Γ(0)𝑟𝛽, (11) where 𝑍(𝛽)=∫𝜔(𝐸=𝑣)exp{−𝛽𝑣}𝑑𝑣 is the Gibbs partition function, ₂𝐹₁(.,.,.,.) is ordinary hypergeometric function [6].

We have from (11) when Γ𝛾=−𝜕ln𝑍(𝛽,𝛾)/𝜕𝛾; Γ₀(𝑉)=Γ_𝛾(𝑉)/𝛾=0;𝑥𝛾Γ𝛾=𝑤()1+𝑥−𝑥1/2+𝑥²+⋯+𝑤^𝑛𝑛/(𝑥+𝑛+1)²+⋯1+𝑤((𝑥+1)/(𝑥+2))+⋯+𝑤^𝑛)/(𝛾((𝑥+1𝑥+𝑛+1))+⋯,𝑥=𝑎≈𝛾Γ₀,₂𝐹₁(1,𝑥+1,𝑥+2,𝑤)=1+𝑤𝑥+1𝑥+2+⋯+𝑤^𝑛𝑥+1𝑥+𝑛+1+⋯. (12)The expression for 𝑥=𝛾/𝑎 is close to that considered in [1], the expression 𝑥=𝛾Γ₀, but not identical to it. At 𝑍(𝛽,𝛾)=𝑍(𝛽)𝑍(𝛾) the entropy is equal to 𝑆𝑘_𝐵(=𝑆=−ln𝜌𝑧;𝐸,Γ)^𝛽𝑘_𝐵+𝑆_𝛾𝑘_𝐵, (13) where𝑆_𝛽=𝛽⟨𝐸⟩+ln𝑍𝛽,𝑆_𝛾⟨Γ⟩𝛾,𝑆=𝛾+ln𝑍_𝛾=𝑥()𝑝1+𝑥−ln1+𝑥+ln^Γ(0)𝑎𝑤()−𝑥1/2+𝑥²+⋯+𝑤^𝑛𝑛/(𝑥+𝑛+1)²+⋯1+𝑤((1+𝑥)/(2+𝑥))+⋯+𝑤^𝑛)/(+()((1+𝑥1+𝑛+𝑥))+⋯+ln1+𝑤1+𝑥(2+𝑥)+⋯+𝑤^𝑛(1+𝑥)().1+𝑛+𝑥+⋯ (14)The first two terms on the right side of (14) coincide with the expression for 𝑆_𝛾 of [1]. The remaining terms are a supplement to this expression. They depend on 𝑤 and tend to 0 if a value 𝑤 (or a value 𝑟 in (11), (3), (7)) tends to 0.

From (14) treating 𝐸 and Γ as variables, 𝑑𝑆=𝑘_𝐵𝛽𝑑𝐸+𝑘_𝐵𝛾𝑑Γ. (15)

From (14)𝜕𝑆_𝛾𝐷𝜕𝑥=−𝑥,𝑥𝐷=𝑥(1+𝑥)²−𝑥1+𝑤((1+𝑥)/(2+𝑥))+⋯+𝑤^𝑛)/(×2𝑤((1+𝑥1+𝑛+𝑥))+⋯(2+𝑥)³𝑤+⋯+^𝑛𝑛(1+𝑛+𝑥)³++()+⋯𝑤/2+𝑥²+⋯+𝑤^𝑛𝑛/(1+𝑛+𝑥)²+⋯²1+𝑤((1+𝑥)/(2+𝑥))+⋯+𝑤^𝑛)/(.((1+𝑥1+𝑛+𝑥))+⋯ (16)

In [1] for the case of thermal conductivity obtained the expression 𝑥_𝑞=𝛾_𝑞Γ₀≈𝑡_0𝑞𝑆_𝑎𝑞𝐸=𝑡_0𝑞𝑞𝜌𝑢𝐿=𝑡_0𝑞𝑦_𝑞,𝑦_𝑞=𝑞𝜌𝑢𝐿,𝐸≈𝜌𝑢𝑉, (17) where 𝑞=±|^→𝑞|=(^→𝑞^→𝑞)^1/2, 𝑢=𝐸/𝜌𝑉 is energy density in a system, 𝐿 is its linear size, 𝑉 is its volume ~𝐿³, and the surface 𝑆_𝑎∼𝐿²; 𝑞₊=(^→𝑞₊^→𝑠)=𝑞=(^→𝑞^→𝑞)^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 [1]); the signs of 𝑞₋ and 𝑞₊ depend on whether we are heating or cooling the system. Then 𝑞₊=−𝑞₋−𝑅𝜌𝜕𝑢/𝜕𝑡; in the stationary state 𝑞₊=−𝑞₋.

Thermodynamic parameters that determine the value of 𝑥 will be the energy 𝐸, the heat flux 𝑞, and the size of 𝐿. From (17) we obtain 𝑑𝑥=𝑥−𝑑𝐸𝐸+𝑑𝑞𝑞+2𝑑𝐿𝐿. (18)

Expression (15) can be rewritten as 𝑑𝑆=𝑘_𝐵𝛽𝑑𝐸+𝜕𝑆_𝛾𝜕𝑥𝑑𝑥. (19)

The value 𝑟 in (11) is small, although it is possible to specify a physical situation where the value of 𝑟 is not small. In the linear approximation to 𝑟 (and 𝑤) the relationship (16) takes the form 𝜕𝑆_𝛾𝑥𝜕𝑥≈−(1+𝑥)²+𝑤2𝑥(2+𝑥)³. (20)

In [1] value of 𝑥 was determined by comparing the obtained expression for the nonequilibrium entropy to the same expression of the EIT [2]. It is possible to determine the value of 𝑥 in the other ratios, for example, by comparing the expression for the inverse nonequilibrium temperature with corresponding expression of EIT in the form 𝜃⁻¹=1𝑇+𝑞²𝛼_𝑞𝑢,𝛼_𝑞=𝜏_𝑞𝜌𝜆𝑇², (21) where 𝜏_𝑞 is the time of the flux correlation, 𝑇 is equilibrium temperature, and 𝜆 is the heat conductivity coefficient.

The expression of 𝜃⁻¹=(𝜕𝑆/𝜕𝐸_|𝑞,𝑅), obtained from (14), (16), (18) in the linear approximation to 𝑤 is given by 𝜃⁻¹=𝜕𝑆𝜕𝐸_|𝑞,𝑅=𝑘_𝐵𝑥𝛽+𝐸𝑥(1+𝑥)²−𝑤2𝑥(2+𝑥)³. (22)

Equating the expressions (21) and (22), we obtain the equation for 𝑥. Neglecting the powers of 𝑥 higher than the second, we find the solution of the resulting quadratic equation ((𝑥=−28𝑐±16𝑐30𝑐+4−𝑤))^1/22()𝑞38𝑐−8−2𝑤,𝑐=²𝛼_𝑞𝐸𝑢𝑘_𝐵. (23) The exact expression of 𝜃⁻¹ is more cumbersome ((20) than linear approximation of (16)) 𝜃⁻¹=𝑘_𝐵𝐷𝛽+𝐸, (24) where 𝐷 is from (16). Here you can substitute the value of 𝑥 from (17). From (16)–(19) we obtain 𝑑𝑆=𝑘_𝐵𝐷𝛽+𝐸𝑑𝐸−𝑘_𝐵𝐷𝑞𝑑𝑞−𝑘_𝐵2𝐷𝐿𝑑𝐿. (25)

Determining in (25) the value of 𝑑𝑆/𝑑𝑡, we find the entropy balance equation (1/𝑉)(𝑑𝑆/𝑑𝑡)=−^→∇^→𝑗_𝑆+𝜎_𝑆, 𝑆=∫_𝑉𝜌𝑠𝑑^→𝑟, where for 𝜌=const, ^→𝑗_𝑆=𝜃^→−1𝑞, 𝜎_𝑆=^→𝑞^→∇𝜃⁻¹−𝑘_𝐵𝐷𝑑𝑞/𝑑𝑡+𝑘𝑞𝑉_𝐵(𝑑𝐿/𝑑𝑡)3𝑢𝛽𝜌+𝑉⁻¹𝐷𝐿. (26)

Comparing entropy production 𝜎_𝑆 (26) and 𝜎_𝑆=^→𝑞^→𝑞/𝜆𝑇² of EIT [2], we shall find 𝜆𝑇²𝑘_𝐵𝑑^→𝑞𝑞𝑑𝑡⁻²𝑉⁻¹𝐷+^→𝑞=𝜆𝑇²∇𝜃⁻¹++𝑑𝐿𝑑𝑡𝜆𝑇²𝑘_𝐵3𝑢𝛽𝜌+𝐷𝑉⁻¹_→𝑞𝐿. (27) The first two terms of the expressions (24)–(27) coincide with the expressions of [1]. The remaining terms are a supplement to these expressions. They depend on 𝑤 and tends to 0 if a value 𝑤 tends to 0.

Just as in [1], 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.

3. Conclusion

In this paper instead of the exponential distribution for the lifetime of [1] 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 [1]. Expressions have the following structure: the terms of [1] and supplements, depending on the value of 𝑤 (11).

There are other possibilities to generalize the results of [1]. Thus, in [7] the linearised dual-phase-lag model of heat transfer is based on the equation 𝜏𝜕𝑞𝜕𝑡+𝑞=−𝜆∇𝑇−𝜀𝜕∇𝑇,𝜕𝑡 (28) where 𝑇 is the equilibrium temperature, 𝜆 the heat conductivity, 𝑞 the heat flux, 𝜏 and 𝜀 are relaxation times. In [7] it was supposed that the entropy density 𝑠 depends not only on the internal energy density 𝑢 but also on 𝑢̇ and 𝑢̈=(𝜕²𝑢)/(𝜕𝑡²),𝑠=𝑠(𝑢,𝑢̇,𝑢̈). 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.