A Strongly Coupled Open System with Non-linear Bath: Fluctuation-Dissipation and Langevin Dynamics

TL;DR

This paper investigates how the classical fluctuation-dissipation relation (FDR) is affected when a probe system is coupled strongly to an anharmonic bath, revealing complex non-linear effects and breakdown of time-translational invariance.

Contribution

It provides a perturbative analysis of the FDR in a strongly coupled, non-linear bath, extending the understanding beyond harmonic environments.

Findings

FDR holds only at leading order in perturbation.

Multiple time scales and temperature dependence emerge beyond leading order.

Time-translational invariance of noise correlations is broken in the non-linear regime.

Abstract

Study of Langevin dynamics and the fluctuation-dissipation relation (FDR) for a generic probe system (represented by a mass $M$), bilinearly coupled to a bath of harmonic oscillators, has been a standard paradigm for a microscopic theory of stochastic processes for several decades. The question that we probe in this paper is, how far the structure of the classical FDR is robust, when one replaces the harmonic bath by an anharmonic one in the limit of strong system-bath coupling? Such a picture carries the signature of the probe system in the zeroth order through a nonlocal time kernel. We observe that the two-time noise correlations hold out a rich structure from which the usual FDR emerges only in the leading order of perturbation. Beyond this order, multiple time scales and nontrivial dependence on the temperature starts manifesting. These new aspects conspire to break the time-translational invariance of the noise-correlations. Several other interesting features show up and we discuss them methodically through rigorous calculations order-by-order in perturbation. This formalistic derivation along with a specific example of non-linearity can be easily applied to a huge range of processes and statistical observables that fall under the purview of system-reservoir theory.