Second-order fluctuation theory and time autocorrelation function for currents

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.

Nonequilibrium Langevin dynamics: A demonstration study of shear flow fluctuations in a simple fluid

“…Ref. [11]. This introduces essentially a nuisance parameter a, if we are interested merely in the behavior of α(t).…”

“…In this section we continue to study the same Molecular Dynamics (MD) model of a shear flow in a simple fluid, as we did in Ref. [11]. Here we briefly summarize, that we consider N particles of equal mass m, interacting through the Weeks-Chandler-Anderson (WCA) potential [12] in three dimensions.…”

“…[11], by using the operator splitting formalism [20][21][22]; cf. [11,Appendix C]. Let us rewrite Eq.…”

“…In our most recent paper [11] we showed, that a secondorder Langevin equation, suggested in Ref. [4], provides excellent means for quantitative studies of fluctuations at mesoscopic scales in both, equilibrium and nonequilibrium, steady-state systems.…”

“…In the equilibrium regime [2,3,10], f = 0, the stochastic term r(t) represents effects of the thermal fluctuations and is commonly given by r(t) = Aω(t), where A > 0 is a constant proportional to the square root of the system's temperature, while ω(t) is a Gaussian white noise with zero mean and unit variance parameters. In general, this model yields a dynamics, which is accurate up to the third-order statistics, due to symmetry considerations [10,11].…”