Gallavotti-Cohen theorem, Chaotic Hypothesis and the zero-noise limit

TL;DR

This paper demonstrates that the Gallavotti-Cohen fluctuation relation holds under finite energy-conserving noise and explores the conditions under which the relation is recovered as noise vanishes, linking it to stochastic stability.

Contribution

It shows that the Gallavotti-Cohen relation is trivially valid with energy-conserving noise and relates the zero-noise limit to stochastic stability of the measure.

Findings

Gallavotti-Cohen relation holds at finite noise amplitude

Crossover time depends on instanton orbit action

Chaotic Hypothesis relates to stochastic stability

Abstract

The Fluctuation Relation for a stationary state, kept at constant energy by a deterministic thermostat - the Gallavotti-Cohen Theorem -- relies on the ergodic properties of the system considered. We show that when perturbed by an energy-conserving random noise, the relation follows trivially for any system at finite noise amplitude. The time needed to achieve stationarity may stay finite as the noise tends to zero, or it may diverge. In the former case the Gallavotti-Cohen result is recovered, while in the latter case, the crossover time may be computed from the action of `instanton' orbits that bridge attractors and repellors. We suggest that the `Chaotic Hypothesis' of Gallavotti can thus be reformulated as a matter of stochastic stability of the measure in trajectory space. In this form this hypothesis may be directly tested.