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Phenomenological Boltzmann formula for currents

by Matteo Polettini, Izaak Neri

Submission summary

Authors (as Contributors):

Izaak Neri · Matteo Polettini

Abstract

Processes far from equilibrium defy a statistical characterization in terms of simple thermodynamic quantities, such as Boltzmann's formula at equilibrium. Here, for continuous-time Markov chains on a finite state space, we find that the probability $\mathfrak{f}_-$ of the current along a transition to ever become negative can be expressed in terms of the effective affinity $F (> 0)$, an entropic measure of dissipation as estimated by an observer that only monitors that specific transition. In particular for cyclic processes we find $\mathfrak{f}_- = \exp - F$, which generalizes the concept of mesoscopic noria and is reminiscent of Boltzmann's formula. We then compare different estimators of the effective affinity, arguing that stopping problems may be best in assessing the nonequilibrium nature of a system. The results are based on a constructive first-transition time approach.

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