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

Abstract

For a system in contact with several reservoirs r at different inverse-temperatures βr, we describe how the Markov jump dynamics with the generalized detailed balance condition can be analyzed via a statistical physics approach of dynamical trajectories [C(t)] 0≤t≤T over a long time interval T → +∞. The relevant intensive variables are the time-empirical density ρ(C), that measures the fractions of time spent in the various configurations C, and the time-empirical jump densities kr(C , C), that measure the frequencies of jumps from configuration C to configuration C when it is the reservoir r that furnishes or absorbs the corresponding energy difference (E(C ) − E(C)). I.arXiv:1906.03606v1 [cond-mat.stat-mech]

Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

“…measure the density of jumps from one configuration y to another configuration x. The joint probability to see the empirical density ρ(x) and the empirical flows q(x, y) follows the large deviation form for large T [25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]]…”

“…Indeed, the initial classification of large deviations into three nested levels (see the reviews [22][23][24] and references therein), with Level 1 for empirical observables, Level 2 for the empirical density, and Level 3 for the empirical process, has been more recently supplemented by the Level 2.5 concerning the joint distribution of the empirical density and of the empirical flows. The rate functions at Level 2.5 are explicit for various types of Markov processes, including discrete-time Markov chains [24][25][26][27][28][29], continuous-time Markov jump processes [25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] and Diffusion processes [28,29,33,34,37,[47][48][49]. As a consequence, the explicit Level 2.5 can be considered as a starting point from which many other large deviations properties can be derived via contraction.…”

Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps

“…(ii)The second contribution involving the jump rate λ(y) and the jump probability Π(x|y) corresponds to the usual rate function for Markov jump processes [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]…”

“…On one hand, the fluctuations of the empirical time-averaged density, of the empirical time-averaged current and of the empirical time-averaged jump-flow of a long dynamical trajectory can be analyzed using the explicit large deviations at Level 2.5 for both continuous-time Markov Jump processes [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and for Diffusion processes [18,19,22,[30][31][32][33][34]. This Level 2.5 can be then contracted to obtain the large deviations properties of any time-additive observable of the dynamical trajectory.…”

Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model

“…While the initial classification involved only three nested levels (see the reviews [17][18][19] and references therein), with Level 1 for empirical observables, Level 2 for the empirical density, and Level 3 for the empirical process, the introduction of the Level 2.5 has been a major progress in order to characterize the joint distribution of the empirical density and of the empirical flows. Its essential advantage is that the rate functions at Level 2.5 can be written explicitly for general Markov processes, including discrete-time Markov chains [19][20][21][22][23][24], continuous-time Markov jump processes [20,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] and Diffusion processes [23,24,28,29,32,[42][43][44][45]. As a consequence, the explicit Level 2.5 can be considered as the central starting point from which many other large deviations properties can be derived via the appropriate contraction.…”