The Green–Kubo formula, autocorrelation function and fluctuation spectrum for finite Markov chains with continuous time

Abstract

A general form of the Green–Kubo formula, which describes the fluctuations pertaining to all the steady states whether equilibrium or non-equilibrium, for a system driven by a finite Markov chain with continuous time (briefly, MC) , is shown. The equivalence of different forms of the Green–Kubo formula is exploited. We also look at the differences in terms of the autocorrelation function and the fluctuation spectrum between the equilibrium state and the non-equilibrium steady state. Also, if the MC is in the non-equilibrium steady state, we can always find a complex function , such that the fluctuation spectrum of is non-monotonous in [0, + ∞).