Relevance of initial and final conditions for the Fluctuation Relation in Markov processes

TL;DR

This paper investigates how initial and final conditions affect the validity of the fluctuation relation in Markov processes, revealing that boundary terms can be significant even at large times, with similar behaviors observed in Markov chains and granular tracers.

Contribution

It demonstrates that boundary terms influence fluctuation relations in Markov processes and shows the similarity between Markov chains and granular tracers in this context.

Findings

Boundary terms are non-negligible at large times.

Fluctuation relation deviations are linked to initial and final states.

Markov chain and granular tracer exhibit similar boundary effects.

Abstract

Numerical observations on a Markov chain and on the continuous Markov process performed by a granular tracer show that the ``usual'' fluctuation relation for a given observable is not verified for finite (but arbitrarily large) times. This suggests that some terms which are usually expected to be negligible, i.e. ``border terms'' dependent only on initial and final states, in fact cannot be neglected. Furthermore, the Markov chain and the granular tracer behave in a quite similar fashion.