Continuous time random walk for open systems: Fluctuation theorems and counting statistics

TL;DR

This paper develops a theoretical framework for continuous time random walks in open systems, deriving fluctuation theorems and counting statistics that depend on the waiting time distribution's decay properties.

Contribution

It introduces a generalized fluctuation theorem for CTRWs with various waiting time distributions and clarifies conditions under which entropy change can be identified.

Findings

Fluctuation theorem holds for any CTRW, but entropy interpretation depends on WTD properties.

Long-time moments of energy and matter transfer grow as t^{n a} for WTDs with decay parameter a.

No meaningful fluctuation theorem exists for WTDs with decay parameter a<1.

Abstract

We consider continuous time random walks (CTRW) for open systems that exchange energy and matter with multiple reservoirs. Each waiting time distribution (WTD) for times between steps is characterized by a positive parameter a, which is set to a=1 if it decays at least as fast as t^{-2} at long times and therefore has a finite first moment. A WTD with a<1 decays as t^{-a-1}. A fluctuation theorem for the trajectory quantity R, defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory, holds for any CTRW. However, R can be identified as a trajectory entropy change only if the WTDs have a=1 and satisfy separability (also called "direction time independence"). For nonseparable WTDs with a=1, R can only be identified as a trajectory entropy change at long times, and a fluctuation theorem for the entropy change then only holds at long times. For WTDs with 0<a<1 no meaningful fluctuation theorem can be derived. We also show that the (experimentally accessible) n'th moments of the energy and matter transfers between the system and a given reservoir grow as t^{n a} at long times.