Equilibrium statistics of a slave estimator in Langevin processes

TL;DR

This paper investigates the statistical properties of a slave estimator used to measure equilibrium susceptibility in Langevin processes, revealing conditions under which its moments may diverge and confirming findings with numerical simulations.

Contribution

It provides a detailed analysis of the slave estimator's probability distribution, especially the emergence of power law tails, and discusses implications for numerical measurements of susceptibility.

Findings

Slave estimator's distribution can have power law tails with temperature-dependent exponents.

Higher moments of the estimator, including variance, may diverge despite finite mean.

Numerical simulations confirm theoretical predictions and highlight potential measurement issues.

Abstract

We analyze the statistics of an estimator, denoted by xi_t and referred to as the slave, for the equilibrium susceptibility of a one dimensional Langevin process x_t in a potential phi(x). The susceptibility can be measured by evolving the slave equation in conjunction with the original Langevin process. This procedure yields a direct estimate of the susceptibility and avoids the need, when performing numerical simulations, to include applied external fields explicitly. The success of the method however depends on the statistical properties of the slave estimator. The joint probability density function for x_t and xi_t is analyzed. In the case where the potential of the system has a concave component the probability density function of the slave acquires a power law tail characterized by a temperature dependent exponent. Thus we show that while the average value of the slave, in the equilibrium state, is always finite and given by the fluctuation dissipation relation, higher moments and indeed the variance may show divergences. The behavior of the power law exponent is analyzed in a general context and it is calculated explicitly in some specific examples. Our results are confirmed by numerical simulations and we discuss possible measurement discrepancies in the fluctuation dissipation relation which could arise due to this behavior.