Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models

The Ruelle thermodynamic formalism for dynamical trajectories over the large time T corresponds to the large deviation theory for the information per unit time of the trajectories probabilities. The microcanonical analysis consists in evaluating the exponential growth in T of the number of trajectories with a given information per unit time, while the canonical analysis amounts to analyze the appropriate non-conserved β-deformed dynamics in order to obtain the scaled cumulant generating function of the information, the first cumulant being the famous Kolmogorov–Sinai entropy. This framework is described in detail for discrete-time Markov chains and for continuous-time Markov jump processes converging towards some steady-state, where one can also construct the Doob generator of the associated β-conditioned process. The application to the directed random trap model on a ring of L sites allows to illustrate this general framework via explicit results for all the introduced notions. In particular, the glassy phase is characterized by anomalous scaling laws with the size L and by non-self-averaging properties of the Kolmogorov–Sinai entropy and of the higher cumulants of the trajectory information. (paper)