Sum rule for response function in nonequilibrium Langevin systems

TL;DR

This paper explores the properties of response functions in nonequilibrium Langevin systems, highlighting the validity of the sum rule in underdamped cases and its violation in overdamped cases, with implications for modeling accuracy.

Contribution

It extends the understanding of response functions in nonequilibrium Langevin systems, especially regarding the sum rule's applicability in different damping regimes.

Findings

Sum rule holds for velocity response in underdamped systems.

Sum rule is violated in overdamped Langevin models.

Relation between sum rule and energy dissipation in nonequilibrium systems.

Abstract

We derive general properties of the linear response functions of nonequilibrium steady states in Langevin systems. These correspond to extension of the results which were recently found in Hamiltonian systems [A. Shimizu and T. Yuge, J. Phys. Soc. Jpn. {\bf 79}, 013002 (2010)]. We discuss one of the properties, the sum rule for the response function, in particular detail. We show that the sum rule for the response function of the velocity holds in the underdamped case, whereas it is violated in the overdamped case. This implies that the overdamped Langevin models should be used with great care. We also investigate the relation of the sum rule to an equality on the energy dissipation in nonequilibrium Langevin systems, which was derived by Harada and Sasa.