Role of the sampling weight in evaluating classical time autocorrelation functions

TL;DR

This paper investigates how different sampling weights impact the efficiency of Monte Carlo methods for computing classical time autocorrelation functions, proposing an optimal weight that reduces computational cost.

Contribution

It introduces a sampling weight that makes the Monte Carlo evaluation cost independent of system size and dynamics, improving efficiency over standard methods.

Findings

Optimal sampling weight reduces convergence cost

Standard sampling may scale exponentially with system size

For harmonic systems, standard sampling cost is dimension-independent

Abstract

We analyze how the choice of the sampling weight affects the efficiency of the Monte Carlo evaluation of classical time autocorrelation functions. Assuming uncorrelated sampling or sampling with constant correlation length, we propose a sampling weight for which the number of trajectories needed for convergence is independent of the correlated quantity, dimensionality, dynamics, and phase-space density. In contrast, it is shown that the computational cost of the "standard" intuitive algorithm which samples directly from the phase-space density may scale exponentially with the number of degrees of freedom. Yet, for the stationary Gaussian distribution of harmonic systems and for the autocorrelation function of a linear function of phase-space coordinates, the computational cost of this standard algorithm is also independent of dimensionality.