Time-translational symmetry in statistical dynamics dictates Einstein relation, Green-Kubo formula, and their generalizations

A stochastic dynamics has a natural decomposition into a drift capturing mean rate of change and a martingale increment capturing randomness. They are two statistically uncorrelated, but not necessarily independent, components contributing to the overall fluctuations of the dynamics, representing the uncertainties in the past and in the future. We show that fluctuation-dissipation relations of the two aforementioned components, such as the Einstein relation and the Green-Kubo formula, can be formulated for any stochastic process with a steady state, without additional supposition of the process being Markovian, reversible, or linear. Further, by considering the adjoint process defined by the time reversal at the steady state, we show that reversibility in equilibrium leads to an additional symmetry in the covariance between system's state and drift. Potential directions of further generalizing our results to processes without steady states is briefly discussed.

Automated summary

A stochastic dynamics consists of a drift and a martingale increment, representing the mean rate of change and randomness, respectively. These two components, although statistically uncorrelated, contribute to the overall fluctuations of the dynamics, capturing uncertainties in the past and future. We demonstrate that fluctuation-dissipation relations like the Einstein relation and the Green-Kubo formula can be formulated for any stochastic process with a steady state, without assuming the process to be Markovian, reversible, or linear. Furthermore, we show that reversibility in equilibrium leads to an additional symmetry in the covariance between the system's state and drift. Potential directions for generalizing these results to processes without steady states are briefly discussed.