Normal and Anomalous Fluctuation Relations for Gaussian Stochastic Dynamics

TL;DR

This paper investigates how Gaussian stochastic systems with anomalous diffusion exhibit different fluctuation relations depending on whether internal or external noise is present, revealing normal and anomalous behaviors.

Contribution

It demonstrates the connection between fluctuation-dissipation relations and fluctuation relations in Gaussian systems, distinguishing between normal and anomalous transient work FRs.

Findings

Internal noise with FDR II leads to normal FRs.

External noise causes violations, resulting in anomalous FRs.

Logarithmic factors appear in FRs at intermediate times.

Abstract

We study transient work Fluctuation Relations (FRs) for Gaussian stochastic systems generating anomalous diffusion. For this purpose we use a Langevin approach by employing two different types of additive noise: (i) internal noise where the Fluctuation-Dissipation Relation of the second kind (FDR II) holds, and (ii) external noise without FDR II. For internal noise we demonstrate that the existence of FDR II implies the existence of the Fluctuation-Dissipation Relation of the first kind (FDR I), which in turn leads to conventional (normal) forms of transient work FRs. For systems driven by external noise we obtain violations of normal FRs, which we call anomalous FRs. We derive them in the long-time limit and demonstrate the existence of logarithmic factors in FRs for intermediate times. We also outline possible experimental verifications.