On the Origin and the Use of Fluctuation Relations for the Entropy

Since about a decade various fluctuation relations for the entropy production have been derived and analyzed. These relations deal with symmetries of the entropy production under time-reversal and have been proposed as a non-perturbative generalization of fluctuation–dissipation relations. I describe a unifying framework for understanding these relations and I present an algorithm to derive them. The fluctuation relations all follow from the main observation that in great generality the pathdependent entropy production is the source-term of time-reversal breaking in the Lagrangian over space-time histories. That is illustrated via a number of examples as well as via a general theoretical argument. I move these relations away from the strict dynamical background in which they originated and take them back to the context of statistical mechanics where entropy is understood in the sense of Boltzmann, as measuring the typicality of a manifest condition. I discuss how a relation between work and free energy is naturally put in that framework and how the transient and steady state fluctuation theorems are simple consequences. The fact that fluctuation symmetries for the entropy production are in general only valid asymptotically for large times, makes them mostly inaccessible for experimental verification, in contrast with a recent claim that they would usefully quantify second law violations. Part of the interest in the resulting fluctuation symmetries is that they are so universally valid, a rare occasion in nonequilibrium statistical mechanics. However they do not provide a systematic perturbation expansion for response functions. For that one needs to go back to the full Lagrangian and also consider the nonequilibrium modifications to its time-symmetric part.