Monte Carlo estimates of thermal averages and analytic continuation

TL;DR

This paper develops a theoretical framework linking the differential properties of thermal averages to higher-order cumulants and demonstrates its application through numerical tests on classical Lennard-Jones systems.

Contribution

It introduces a theorem connecting thermal average derivatives to cumulants and provides formulas for analytic continuation of Monte Carlo data.

Findings

Theorem linking derivatives of thermal averages to cumulants.

Analytic continuation formulas derived and tested.

Numerical validation on Lennard-Jones clusters.

Abstract

The Monte Carlo (MC) estimates of thermal averages are usually functions of system control parameters $\lambda $, such as temperature, volume, interaction couplings, etc. Given the MC average at a set of prescribed control parameters $\lambda_{0}$, the problem of analytic continuation of the MC data to $\lambda $-values in the neighborhood of $\lambda_{0}$ is considered in both classic and quantum domains. The key result is the theorem that links the differential properties of thermal averages to the higher-order cumulants. The theorem and analytic continuation formulas expressed via higher-order cumulants are numerically tested on the classical Lennard-Jones cluster system of N=13, 55, and 147 neon particles.