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The nonlinear Markov processes are the measure-valued dynamical systems whichpreserve positivity. They can be represented as the law of large numbers limitsof general Markov models of interacting particles. In physics, the kineticequations allow Lyapunov functionals (entropy, free energy, etc.). This may beconsidered as a sort of inheritance of the Lyapunov functionals from themicroscopic master equations. We study nonlinear Markov processes that inheritthermodynamic properties from the microscopic linear Markov processes. Wedevelop the thermodynamics of nonlinear Markov processes and analyze theasymptotic assumption, which are sufficient for this inheritance.

Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes