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

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