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

For a system in contact with several reservoirs r at different inverse-temperatures , we describe how the Markov jump dynamics with the generalized detailed balance condition can be analyzed via a statistical physics approach of dynamical trajectories over a long time interval . The relevant intensive variables are the time-empirical density , that measures the fractions of time spent in the various configurations , and the time-empirical jump densities , that measure the frequencies of jumps from configuration to configuration when it is the reservoir r that furnishes or absorbs the corresponding energy difference ().