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 [

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 [

Setting the form of the function

In [

In (

In deriving (

From the normalization of the distribution (

In the expression for the partition function

Substituting (

We have from (

From (

From (

In [

Thermodynamic parameters that determine the value of

Expression (

The value

In [

The expression of

Equating the expressions (

Determining in (

Comparing entropy production

Just as in [

In this paper instead of the exponential distribution for the lifetime of [

There are other possibilities to generalize the results of [

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.