An introduction to weak noise large deviations and macroscopic fluctuation theory

Fluctuations are sometimes very important in stochastic systems, for example when large fluctuations can cause dramatic changes. When a system is described by equilibrium statistical mechanics, all the information is encoded in thermodynamical potentials (entropy, free energy...). Out of equilibrium the situation is more complex and, often, no reduced description at the level of thermodynamical potentials is available.

There is, however, an important case where thermodynamic potentials can be defined in a fully consistent way. This is the case, discovered in the '80s of stochastic systems in the limit of weak noise; the corresponding mathematical theory goes under the name of "weak-noise large deviations" or "Freidlin-Wentzell-Graham theory". We will review this theory and comment on recent developments.

In particular, in last 15 years, Jona-Lasinio and collaborators rigorously proved that some (simple!) classes of many-body systems, when analysed at macroscopic level, can be described by an infinite-dimensional generalisation of weak-noise large deviations (Macroscopic Fluctuation Theory). This heated up some research (and confusion) that we will discuss..