Second-order fluctuation theory and time autocorrelation function for currents

TL;DR

This paper extends the Onsager-Machlup fluctuation theory to second order in time for systems described by Langevin dynamics, deriving an analytical autocorrelation function that accurately fits molecular dynamics data in equilibrium and nonequilibrium states.

Contribution

It generalizes the fluctuation theory to second order, providing a new analytical expression for autocorrelation functions applicable to hydrodynamic currents in various states.

Findings

The derived autocorrelation function fits simulation data well.

It accurately captures short-time correlations unlike first-order theories.

The autocorrelation function is independent of shear force in linear regimes.

Abstract

By using recent developments for the Langevin dynamics of spatially asymmetric systems, we routinely generalize the Onsager-Machlup fluctuation theory of the second order in time. In this form, it becomes applicable to fluctuating variables, including hydrodynamic currents, in equilibrium as well as nonequilibrium steady states. From the solution of the obtained stochastic equations we derive an analytical expression for the time autocorrelation function of a general fluctuating quantity. This theoretical result is then tested in a study of a shear flow by molecular dynamics simulations. The proposed form of the time autocorrelation function yields an excellent fit to our computational data for both equilibrium and nonequilibrium steady states. Unlike the analogous result of the first-order Onsager-Machlup theory, our expression correctly describes the short-time correlations. Its utility is demonstrated in an application of the Green-Kubo formula for the transport coefficient. Curiously, the normalized time autocorrelation function for the shear flow, which only depends on the deterministic part of the fluctuation dynamics, appears independent of the external shear force in the linear nonequilibrium regime.