Martingale theory for housekeeping heat

¹ Université Côte d'Azur, CNRS, LJAD - Parc Valrose, 06108 Nice Cedex 02, France
² Department of Physics, Ramakrishna Mission Vivekananda University - Belur Math, Howrah 711 202, India
³ Department of Mathematics, King's College London - Strand, London WC2R 2LS, UK
⁴ Max Planck Institute for the Physics of Complex Systems - Nöthnitzer Strasse 38, 01187 Dresden, Germany
⁵ ICTP - The Abdus Salam International Centre for Theoretical Physics - Strada Costiera 11, 34151 Trieste, Italy

Received: 23 October 2018
Accepted: 7 December 2018

Abstract

The housekeeping heat is the energy exchanged between a system and its environment in a nonequilibrium process that results from the violation of detailed balance. We describe fluctuations of the housekeeping heat in mesoscopic systems using the theory of martingales, a mathematical framework widely used in probability theory and finance. We show that the exponentiated housekeeping heat (in units of k_BT, with k_B the Boltzmann constant and T the temperature) of a Markovian nonequilibrium process under arbitrary time-dependent driving is a martingale process. From this result, we derive universal equalities and inequalities for the statistics of stopping times and suprema of the housekeeping heat. We test our results with numerical simulations of a system driven out of equilibrium and described by Langevin dynamics.