Arcsine Laws in Stochastic Thermodynamics

TL;DR

This paper demonstrates that the fraction of time a thermodynamic current exceeds its average follows the arcsine law, revealing long streaks above or below the mean are more probable, with experimental confirmation from a Brownian engine.

Contribution

It introduces the application of the arcsine law to stochastic thermodynamic currents, providing new insights into their temporal fluctuations and confirming with experimental data.

Findings

Fraction of time above average follows arcsine law

Long streaks above/below average are more likely

Experimental validation with Brownian Carnot engine

Abstract

We show that the fraction of time a thermodynamic current spends above its average value follows the arcsine law, a prominent result obtained by L\'evy for Brownian motion. Stochastic currents with long streaks above or below their average are much more likely than those that spend similar fractions of time above and below their average. Our result is confirmed with experimental data from a Brownian Carnot engine. We also conjecture that two other random times associated with currents obey the arcsine law: the time a current reaches its maximum value and the last time a current crosses its average value. These results apply to, inter alia, molecular motors, quantum dots and colloidal systems.