Fluctuations in nonequilibrium statistical mechanics:: models, mathematical theory, physical mechanisms

The fluctuations in nonequilibrium systems are under intense theoretical and experimental investigation. Topical 'fluctuation relations' describe symmetries of the statistical properties of certain observables, in a variety of models and phenomena. They have been derived in deterministic and, later, in stochastic frameworks. Other results first obtained for stochastic processes, and later considered in deterministic dynamics, describe the temporal evolution of fluctuations. The field has grown beyond expectation: research works and different perspectives are proposed at an ever faster pace. Indeed, understanding fluctuations is important for the emerging theory of nonequilibrium phenomena, as well as for applications, such as those of nanotechnological and biophysical interest. However, the links among the different approaches and the limitations of these approaches are not fully understood. We focus on these issues, providing: ( a) analysis of the theoretical models; ( b) discussion of the rigorous mathematical results; ( c) identification of the physical mechanisms underlying the validity of the theoretical predictions, for a wide range of phenomena.