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

Abstract

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.

32 pages

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