Macroscopic fluctuation theory

In the last fifteen years different tools have been developed to extend thermodynamic formalism out of the scope of equilibrium statistical mechanics. Following seminal ideas from A. Einstein, some coherent theory emerged to give a rigorous mathematical treatment of "non-equilibrium stationary states". I will briefly review this approach, sometimes called "Macroscopic fluctuation theory". I will discuss some random dynamics (e.g. exclusion processes and weakly coupled systems) that fit into this framework, with emphasis on critical behaviors that may appear as a mark of dynamical phase transitions.