A Statistical Mechanical Approximation for the Calculation of Time Auto-Correlation Functions

TL;DR

This paper introduces a statistical mechanical approximation to estimate time auto-correlation functions in configuration space, especially useful for viscous liquids where direct simulation is challenging.

Contribution

It develops a novel approximation reducing auto-correlation calculations to a double canonical average, applicable to Langevin dynamics.

Findings

Good agreement with exact auto-correlation functions in simulations

Applicable to viscous liquids with long time scales

Uses double partition function for mean-square displacement

Abstract

This paper considers the problem of estimating the time auto-correlation function for a quantity that is defined in configuration space, given a knowledge of the mean-square displacement as function of time in configuration space. The problem is particularly relevant for viscous liquids, where the interesting time-scales are often beyond those reachable by computer simulation. An approximate formula is derived which reduces the calculation of the time auto- correlation function to a "double canonical" average. In this approximation, in the case of Langevin dynamics the mean-square displacement itself may be evaluated from the "double partition function" . The scheme developed is illustrated by computer simulations of a simple one-dimensional system at different temperatures, showing good agreement between the exact time auto-correlation functions and those found by the approximation.