Martingale theory for housekeeping heat (Vol. 50, No. 2)

Traces of fluctuating housekeeping heat (grey lines) behave like downtrend stocks. Our work investigates statistics of extrema (black arrows) against the average tendency

Which universal thermodynamic properties emerge in a nonequilibrium process, in isothermal conditions at temperature T, that result from the violation of detailed balance, and how they may be quantified? The housekeeping heat is the fluctuating heat exchanged between a mesoscopic system and its environment due to the violation of detailed balance. Using the framework of martingale theory widely used in probability theory and finance, we derive a number of universal equalities and inequalities for extreme-value and stopping-time statistics of the housekeeping heat. Our theory provides a quantitative link between minimal models of gambling and financial markets (martingales) and heat fluctuations. The housekeeping heat behaves like a gambler’s fortune in a casino: its expected value in the future is always smaller or equal regardless of its past values. The super-martingale structure of the housekeeping heat implies that certain statistical properties of the housekeeping heat are system-independent, i.e. universal. A particular result of our theory is that the average value of the maximum housekeeping heat that a system absorbs from its environment cannot exceed kB T, with kB Boltzmann’s constant.

R. Chétrite, S. Gupta, I. Neri and E. Roldán, Martingale theory for housekeeping heat, EPL 124, 60006 (2018),