Arcsine Laws in Stochastic Thermodynamics

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.