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 beta-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 beta-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.

Automated summary

The Ruelle thermodynamic formalism is applied to dynamical trajectories and large deviation theory, with microcanonical and canonical analysis evaluating exponential growth and non-conserved dynamics, respectively. It is described in the context of discrete-time and continuous-time Markov processes, with application to the directed random trap model. Anomalous scaling laws and non-self-averaging properties characterize the glassy phase.