首页|Markov trajectories: Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators
Markov trajectories: Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators
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NSTL
Springer Nature
Abstract The Ensemble of trajectories x(0≤t≤T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(0 \le t \le T)$$\end{document} 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≤t≤T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(0 \le t \le T)$$\end{document} can be rewritten as the exponential of a linear combination of its relevant empirical time-averaged observables En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document}, where the coefficients involving the Markov generator are their fixed conjugate parameters; (C2) the large deviations properties of these empirical observables En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document} for large T are governed by the explicit rate function IM[2.5](E.)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{[2.5]}_M (E_.) $$\end{document} at Level 2.5, while in the thermodynamic limit T=+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=+\infty $$\end{document}, they concentrate on their typical values Entyp[M]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n^{typ[M]}$$\end{document} determined by the Markov generator M. This concentration property in the thermodynamic limit T=+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=+\infty $$\end{document} suggests to introduce the notion of the ‘Microcanonical Ensemble’ at Level 2.5 for stochastic trajectories x(0≤t≤T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(0 \le t \le T)$$\end{document}, where all the relevant empirical variables En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document} are fixed to some values En?\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^*_n$$\end{document} 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≤t≤T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}
NSTL
Springer Nature
