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The Markov process admits a consistent steady-state thermodynamic formalism. (English) Zbl 1383.82044

Summary: The search for a unified formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for the master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that (1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; (2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; (3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that general Markov processes admit a unified and self-consistent non-equilibrium steady-state thermodynamic formalism, regardless of underlying detailed models.{
©2018 American Institute of Physics}

MSC:

82C35 Irreversible thermodynamics, including Onsager-Machlup theory
82B30 Statistical thermodynamics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
35Q82 PDEs in connection with statistical mechanics
35Q84 Fokker-Planck equations
60G05 Foundations of stochastic processes
60J25 Continuous-time Markov processes on general state spaces
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