Stochastic Thermodynamics with Martingales: Gambling Demons and Extreme Fluctuations

Abstract: Maxwell’s demon may use information about the evolution of a thermodynamic system to extract work using feedback control. However, the implementation of feedback control is challenging in many situations, similarly to what happens in casino games and financial markets, where agents cannot modify the system dynamics at will. This raises the question of whether a Maxwell-like demon would be able to gamble on fluctuations without using feedback control and what the nature of fluctuations would need to be. I will present a recent approach to handle these questions, based on the identification of functionals of stochastic entropy production as martingale processes. This martingale approach allows to introduce and analyze “gambling demons” who, following a customary gambling strategy to stop a nonequilibrium process at a stochastic time, are able to extract more average work than the free energy change. In this context, I will discuss refinements of the second law in the presence of gambling, including a set of universal stopping-time fluctuation relations for the work done in non-stationary processes, and their experimental test in a nanoelectronic system. As a second step I will also discuss how the properties of these martingale processes can be exploited to bound extreme fluctuations of entropy production within time-intervals with applications in thermal machines and molecular motors.