Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes

¹ Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
² Department of Statistics, University of Warwick, Coventry CV4 7AL UK

^⋆ Corresponding author. E-mail: ag153@le.ac.uk

Abstract

The nonlinear Markov processes are 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.