Fluctuations of two-time quantities and non-linear response functions

TL;DR

This paper investigates the fluctuations and variances of two-time autocorrelation and autoresponse functions, establishing a fluctuation-dissipation relation beyond linear order and applying it to ferromagnets in equilibrium and during coarsening.

Contribution

It introduces a formalism linking response variances to second-order susceptibilities and demonstrates their scaling behavior in ferromagnetic systems.

Findings

Variance of response function equals second-order susceptibility.

Variances and non-linear susceptibility scale with coherence length and growing length.

Numerical results confirm scaling laws in ferromagnetic systems.

Abstract

We study the fluctuations of the autocorrelation and autoresponse functions and, in particular, their variances and co-variance. In a first general part of the Article, we show the equivalence of the variance of the response function with the second-order susceptibility of a composite operator, and we derive an equilibrium fluctuation-dissipation theorem beyond-linear order relating it to the other variances. In a second part of the paper we apply the formalism to the study to non-disordered ferromagnets, in equilibrium or in the coarsening kinetics following a critical or sub-critical quench. We show numerically that the variances and the non-linear susceptibility obey scaling with respect to the coherence length $\xi$ in equilibrium, and with respect to the growing length $L(t)$ after a quench, similarly to what is known for the autocorrelation and the autoresponse functions.