A review of linear response theory for general differentiable dynamical systems

TL;DR

This paper reviews how linear response theory extends to general differentiable dynamical systems away from equilibrium, highlighting conditions under which classical relations hold or break down, especially in non-hyperbolic systems.

Contribution

It provides a comprehensive overview of linear response in nonequilibrium steady states, emphasizing differences between hyperbolic and non-hyperbolic systems and introducing new phenomena when the chaotic hypothesis fails.

Findings

Linear response similar to equilibrium in hyperbolic systems

Violation of linear response and dispersion relations in non-hyperbolic systems

Presence of 'energy nonconservation' leading to 'active' nonequilibrium states

Abstract

The classical theory of linear response applies to statistical mechanics close to equilibrium. Away from equilibrium, one may describe the microscopic time evolution by a general differentiable dynamical system, identify nonequilibrium steady states (NESS), and study how these vary under perturbations of the dynamics. Remarkably, it turns out that for uniformly hyperbolic dynamical systems (those satisfying the "chaotic hypothesis"), the linear response away from equilibrium is very similar to the linear response close to equilibrium: the Kramers-Kronig dispersion relations hold, and the fluctuation-dispersion theorem survives in a modified form (which takes into account the oscillations around the "attractor" corresponding to the NESS). If the chaotic hypothesis does not hold, two new phenomena may arise. The first is a violation of linear response in the sense that the NESS does not depend differentiably on parameters (but this nondifferentiability may be hard to see experimentally). The second phenomenon is a violation of the dispersion relations: the susceptibility has singularities in the upper half complex plane. These "acausal" singularities are actually due to "energy nonconservation": for a small periodic perturbation of the system, the amplitude of the linear response is arbitrarily large. This means that the NESS of the dynamical system under study is not "inert" but can give energy to the outside world. An "active" NESS of this sort is very different from an equilibrium state, and it would be interesting to see what happens for active states to the Gallavotti-Cohen fluctuation theorem.