Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes

TL;DR

This paper develops a thermodynamic framework for nonlinear Markov processes, showing how they inherit properties like entropy from microscopic linear models, and analyzes conditions for this inheritance.

Contribution

It introduces a thermodynamic theory for nonlinear Markov processes and identifies conditions under which they inherit thermodynamic properties from microscopic models.

Findings

Established thermodynamic structure for nonlinear Markov processes.

Derived conditions for inheritance of Lyapunov functionals.

Analyzed asymptotic assumptions ensuring thermodynamic inheritance.

Abstract

The nonlinear Markov processes are the measure-valued dynamical systems which preserve positivity. They can be represented as the law of large numbers limits of general Markov models of interacting particles. In physics, the kinetic equations allow Lyapunov functionals (entropy, free energy, etc.). This may be considered as a sort of inheritance of the Lyapunov functionals from the microscopic master equations. We study nonlinear Markov processes that inherit thermodynamic properties from the microscopic linear Markov processes. We develop the thermodynamics of nonlinear Markov processes and analyze the asymptotic assumption, which are sufficient for this inheritance.