Energy, entropy, and the principle of maximum caliber as a framework for non equilibrium statistical mechanics

Equilibrium statistical mechanics, based in the method of Gibbsian ensemble, is a well established and unified theory that can be derived completely from a single fundamental principle, namely the principle of Maximum Entropy (MaxEnt). Non-equilibrium statistical mechanics, in contrast, lacks this kind of axiomatic formulation. Rather, it is presented as a set of equations derived by different approaches, like BBGKY, master equation, Fokker-Planck, each of them with a restricted application's domains. Much of the on-going research looks for formalisms based on a few as possible principles and consistent with MaxEnt. One very promising candidates is the formalism based on the Maximum Caliber principle, which consists in the application of MaxEnt to the space of trajectories, and therefore the concept of paths probabilities occupies a central place. Here we will presents how by means of the principle of Maximum Caliber (or Maximum Entropy of Trajectories) and what we call inference on trajectories it is possible to put non-equilibrium statistical mechanics on a firm foundation, deriving its main characteristics. In this way, we are able to establish a systematic and orderly method to understand the origin of the relations and differential equations that make up this theory. (author)