# Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs

## TL;DR

This paper analyzes the long-time behavior of Markov jump systems in contact with multiple thermal reservoirs using a statistical physics approach focused on dynamical trajectories and empirical measures.

## Contribution

It introduces a framework for studying dynamical trajectories of systems with multiple reservoirs via empirical densities and jump frequencies, extending the analysis of detailed balance conditions.

## Key findings

- Characterization of empirical density and jump distributions in long trajectories.
- Application of statistical physics methods to systems with multiple reservoirs.
- Insights into energy exchange and configuration dynamics in non-equilibrium systems.

## Abstract

For a system in contact with several reservoirs $r$ at different inverse-temperatures $\beta_r$, we describe how the Markov jump dynamics with the generalized detailed balance condition can be analyzed via a statistical physics approach of dynamical trajectories $[{\cal C}(t)]_{0 \leq t \leq T} $ over a long time interval $T \to + \infty$. The relevant intensive variables are the time-empirical density $\rho(\cal C)$, that measures the fractions of time spent in the various configurations ${\cal C}$, and the time-empirical jump densities $k_r ({\cal C', \cal C}) $, that measure the frequencies of jumps from configuration ${\cal C} $ to configuration ${\cal C '} $ when it is the reservoir $r$ that furnishes or absorbs the corresponding energy difference ($E({\cal C '})- E({\cal C })$).