Markov trajectories : Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators

TL;DR

This paper introduces a microcanonical ensemble framework for Markov trajectories based on fixed empirical observables, contrasting with the traditional canonical ensemble, and explores its properties and applications to jump processes and chains.

Contribution

It develops a novel microcanonical ensemble approach for Markov trajectories based on empirical observables, providing insights into their statistical properties and subtrajectory behavior.

Findings

Microcanonical ensemble characterized by fixed empirical observables.

Subtrajectory statistics governed by associated canonical ensemble.

Explicit entropy formula for trajectory counts.

Abstract

The Ensemble of trajectories $x(0 \leq t \leq T)$ produced by the Markov generator $M$ can be considered as 'Canonical' for the following reasons : (C1) the probability of the trajectory $x(0 \leq t \leq T)$ can be rewritten as the exponential of a linear combination of its relevant empirical time-averaged observables $E_n$, where the coefficients involving the Markov generator are their fixed conjugate parameters; (C2) the large deviations properties of these empirical observables $E_n$ for large $T$ are governed by the explicit rate function $I^{[2.5]}_M (E_.) $ at Level 2.5, while in the thermodynamic limit $T=+\infty$, they concentrate on their typical values $E_n^{typ[M]}$ determined by the Markov generator $M$. This concentration property in the thermodynamic limit $T=+\infty$ suggests to introduce the notion of the 'Microcanonical Ensemble' at Level 2.5 for stochastic trajectories $x(0 \leq t \leq T)$, where all the relevant empirical variables $E_n$ are fixed to some values $E^*_n$ and cannot fluctuate anymore for finite $T$. The goal of the present paper is to discuss its main properties : (MC1) when the long trajectory $x(0 \leq t \leq T) $ belongs the Microcanonical Ensemble with the fixed empirical observables $E_n^*$, the statistics of its subtrajectory $x(0 \leq t \leq \tau) $ for $1 \ll \tau \ll T $ is governed by the Canonical Ensemble associated to the Markov generator $M^*$ that would make the empirical observables $E_n^*$ typical ; (MC2) in the Microcanonical Ensemble, the central role is played by the number $\Omega^{[2.5]}_T(E^*_.) $ of stochastic trajectories of duration $T$ with the given empirical observables $E^*_n$, and by the corresponding explicit Boltzmann entropy $S^{[2.5]}( E^*_. ) = [\ln \Omega^{[2.5]}_T(E^*_.)]/T $. This general framework is applied to continuous-time Markov Jump processes and to discrete-time Markov chains with illustrative examples.