Fluctuation-Dissipation Relations Far from Equilibrium: A Case Study

TL;DR

This study investigates the validity of fluctuation-dissipation relations in non-equilibrium systems through simulations of active microrheology, revealing that one FDT always holds while the other depends on the system's proximity to equilibrium.

Contribution

The paper demonstrates that the second fluctuation-dissipation theorem always holds in non-equilibrium systems when using a generalized Langevin equation derived from a Volterra equation.

Findings

First FDT valid near equilibrium, breaks down far from equilibrium

Second FDT always valid in the studied systems

Proposes GLE with orthogonality constraint for consistent coarse-grained modeling

Abstract

Fluctuation-dissipation relations or "theorems" (FDTs) are fundamental for statistical physics and can be rigorously derived for equilibrium systems. Their applicability to non-equilibrium systems is, however, debated. Here, we simulate an active microrheology experiment, in which a spherical colloid is pulled with a constant external force through a fluid, creating near-equilibrium and far-from-equilibrium systems. We characterize the structural and dynamical properties of these systems, and reconstruct an effective generalized Langevin equation (GLE) for the colloid dynamics. Specifically, we test the validity of two FDTs: The first FDT relates the non-equilibrium response of a system to equilibrium correlation functions, and the second FDT relates the memory friction kernel in the GLE to the stochastic force. We find that the validity of the first FDT depends strongly on the strength of the external driving: it is fulfilled close to equilibrium and breaks down far from it. In contrast, we observe that the second FDT is always fulfilled. We provide a mathematical argument why this generally holds for memory kernels reconstructed from a deterministic Volterra equation for correlation functions, even for non-stationary non-equilibrium systems. Motivated by the Mori-Zwanzig formalism, we therefore suggest to impose an orthogonality constraint on the stochastic force, which is in fact equivalent to the validity of this Volterra equation. Such GLEs automatically satisfy the second FDT and are unique, which is desirable when using GLEs for coarse-grained modeling.