fluctuation-relations

Fluctuations play a vital role in systems characterized by a limited number of degrees of freedom, significantly influencing their measurable physical attributes. This observation suggests that fluctuations may also hold significance for macroscopic systems. By appropriately considering these fluctuations, commonly referred to as noise, it becomes possible to extract information regarding the transition between two equilibrium states from a collection of non-equilibrium states. In this paper, we revisit two fundamental fluctuation relations, originally discovered in recent decades, that establish connections between equilibrium properties, specifically the Helmholtz free energy, and non-equilibrium trajectories in classical and quantum systems: the Jarzynski equality and the Crooks fluctuation theorem. Notably, the Jarzynski equality asserts a direct relationship between the difference in free energy of an initial and final equilibrium state and the average of the irreversible work performed along a set of trajectories, which are predominantly non-equilibrium, connecting these states. In other words, Jarzynski equality gives an exact expression relating the normalizing constants (or equivalently free energy) of two distributions linked by out-of-equilibrium continuous-time dynamics [Carbone et al., NeurIPS 2023]. This article also delves into potential applications of these relations.