Fluctuation-Response Theory for Nonequilibrium Langevin Dynamics

TL;DR

This paper develops a unified fluctuation-response framework for nonequilibrium Langevin dynamics, extending the fluctuation-dissipation theorem and deriving bounds that connect fluctuations, responses, and thermodynamic uncertainty relations.

Contribution

It introduces a generalized fluctuation-response relation for nonequilibrium Langevin systems, linking fluctuations, responses, and thermodynamic bounds in a unified framework.

Findings

Derived a generalized fluctuation-response identity for nonequilibrium systems.

Established finite-time fluctuation-response inequalities and response uncertainty relations.

Applied the theory to a molecular motor model to constrain diffusion coefficients.

Abstract

We establish a unified fluctuation-response relation for Langevin dynamics. By exploiting the common mathematical structures underlying fluctuations and responses of empirical density and current, we derive a unified identity that generalizes the fluctuation-dissipation theorem from equilibrium to nonequilibrium settings. This relation connects global fluctuations of observables with their local responses to perturbations in force, mobility, and temperature. We further derive finite-time fluctuation-response inequalities, leading to response uncertainty relations that complement the identity by providing more practical bounds. These derivations establish a unified theoretical framework linking the fluctuation-dissipation theorem and thermodynamic uncertainty relations. Using the $F_1$-ATPase molecular motor model, we illustrate how these response-based bounds constrain the long-time diffusion coefficient.