Generalized exchange cluster algorithm to compute efficiently covariances and susceptibilities in Monte Carlo

TL;DR

This paper introduces a generalized exchange cluster Monte Carlo algorithm that efficiently computes covariances and susceptibilities in particle systems, significantly reducing statistical fluctuations and computational complexity.

Contribution

It extends the exchange cluster algorithm to any interacting particle model, providing an improved estimator with near-zero variance for covariances.

Findings

Reduces fluctuation scaling from O(N^2) to O(N)

Demonstrates efficiency on Lennard-Jones model

Provides a general method for any 2-body interactions

Abstract

We present a Monte Carlo method to compute efficiently susceptibilites or covariances of two physical variables. The method relies on a generalization of the exchange cluster algorithm to any model of interacting particles with any $2$-body interactions. The principle is to select clusters of variables belonging to two independent replicas of the system. An improved estimator of the covariance of two physical variables (in one replica) is then proposed. This estimator has the zero-variance property in the limit wh ere these variables are independent.In practice the scaling of the statistical fluctuations as a function of the number of degrees of freedom $N$ is reduced from $O(N ^2)$ to $O(N)$. This lower scaling is illustrated on a Lennard Jones model.