Stochastic Measures and Modular Evolution in Non-equilibrium Thermodynamics

TL;DR

This paper applies stochastic process theory to model non-equilibrium thermodynamics, introducing hierarchical structures of correlation functions and thermodynamical potentials, and proposing a formalism for mesoscopic stochastic dynamics influenced by non-equilibrium fluctuations.

Contribution

It introduces a novel hierarchical framework for non-equilibrium thermodynamics using stochastic measures and modular evolution, linking thermodynamic potentials to stationary measures.

Findings

Hierarchical structure of correlation functions derived from stochastic measures.

Non-equilibrium thermodynamical potentials linked to stationary probability measures.

Formalism describing mesoscopic stochastic dynamics influenced by fluctuations.

Abstract

We present an application of the theory of stochastic processes to model and categorize non-equilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical structure for (physical) correlation functions and non-equilibrium thermodynamical potentials. It is proposed that macroscopic evolution equations (such as dynamic correlation functions) may be obtained from a non-equilibrium thermodynamical description, by using the fact that extended thermodynamical potentials belong to a certain class of statistical systems whose probability distribution functions are defined by a stationary measure; although a measure which is, in general, different from the equilibrium Gibbs measure. These probability measures obey a certain hierarchy on its stochastic evolution towards the most probable (stationary) measure. This in turns defines a convergence sequence. We propose a formalism which considers the mesoscopic stage (typical of non-local dissipative processes such as the ones described by extended irreversible thermodynamics) as being governed by stochastic dynamics due to the effect of non-equilibrium fluctuations. Some applications of the formalism are described.