Non-equilibrium stochastic dynamics

Abstract

In this paper, we mainly review the basic concepts in statistical mechanics of stochastic processes, including entropy production, flux, potential, equilibrium state, temperature in stochastic processes and their mathematical definitions. Moreover, we review some basic properties of the processes derived from these concepts, including fluctuation theorems, Green-Kubo formula, and the monotonicity of power spectrum density. Especially, we point out that we can introduce the temperature parameter in a Markov process to get the multi-layer clustering of the system modeled by the process. This multi-layer clustering shows the interaction network structure among the states. Finally, we give some examples of application problems, in which one can model by stochastic processes to get the answer, explanation, and understanding of concerned problems in natural sciences.