mschossler/fluctuationRelations

Fluctuations are important for systems with a small number of degrees of freedom and have a strong effect on their measurable physical properties. This points us in the direction that fluctuations could be important for macroscopic systems as well. By properly accounting for the fluctuations (noise), one can extract information between two equilibrium states out of an ensemble of non-equilibrium states. In this paper we rederive two important fluctuation relations discovered in the past decades that relate an equilibrium property, the Helmholtz free energy, with non-equilibrium state trajectories for both classical and quantum systems: the Jarzynski equality and the Crooks fluctuation theorem. Especially, the Jarzynski equality states that the difference of the free energy between an initial and final equilibrium state is directly related to the average of the irreversible work along an ensemble of, mostly non-equilibrium, trajectories joining these two states. Applications are discussed.