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

TL;DR

This paper revisits the Ruelle thermodynamic formalism for Markov trajectories, applying it to analyze the glassy phase of random trap models, revealing anomalous scaling laws and non-self-averaging properties.

Contribution

It extends the Ruelle formalism to Markov trajectories and applies it explicitly to the glassy phase of the directed random trap model, providing detailed analytical results.

Findings

Characterization of glassy phase by anomalous scaling laws

Identification of non-self-averaging properties of trajectory information

Explicit results for Kolmogorov-Sinai entropy and cumulants

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.