Thermodynamic Formalism Out of Equilibrium, and Gibbs Processes

TL;DR

This work extends thermodynamic formalism to settings where the potential depends on an exterior system, introducing the pressure out of equilibrium (POE) and a variational principle that optimizes over invariant measures. By embedding the dynamics in a Hölder skew-product and analyzing a random family of potentials, the authors define a maximal instability potential $\widetilde{\phi}$ and prove $\check{P}(\{\phi_t\}_t)=\max_{\nu\ text{ T-inv}}\{ h_\nu(T)+\int\widetilde{\phi}\,d\nu\}$, with a constructive lower bound via an instability kernel ensuring the maximum is attained. The POE framework is applied to Gibbs-driven random dynamical systems, leading to hyperbolicity estimates under uniform expansion on average, and these estimates are supported by a perturbative Ruelle operator analysis showing a spectral gap and exponential decay. The combination of POE, variational principle, and perturbative Ruelle theory provides a robust pathway to quantify instability and hyperbolicity in non-i.i.d. random dynamics, with potential implications for understanding mixing and stability in physically interacting systems.

Abstract

We study the thermodynamic formalism of systems where the potential depends randomly on an exterior system. We define the {\em pressure out of equilibrium} for such a family of potentials, and prove a corresponding variational principle. We present an application to random dynamical systems. In particular, we study an open condition for random dynamical systems where the randomness is driven by a Gibbs process, and prove hyperbolicity estimates that were previously only known in the i.i.d setting.

Theorems & Definitions (27)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 3.1: Maximal asymptotic instability
• Definition 3.2
• Theorem 3.3
• proof
• Proposition 3.4
• proof
• Lemma 3.5