论文标题
马尔可夫轨迹：基于经验可观察物的微型合奏，与基于马尔可夫发电机的规范合奏相比
Markov trajectories : Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators
论文作者
论文摘要
由于以下原因，马尔可夫生成器$ m $ $ x（0 \ leq t \ leq t）$可以被视为“规范”：（c1）轨迹$ x（0 \ leq t \ leq t \ leq t）$的概率可以作为$ $ a $的实用性，以便于其相关的时代的实质性组合。涉及马尔可夫发电机的系数是其固定的共轭参数。 （c2）这些经验可观察的$ e_n $的大偏差属性由显式速率函数$ i^{[2.5]} _ m（e_。）$在2.5级时，而热力学限制$ t $ t $ t =+\ hyt =+\ iffty $，它们集中于典型的典型值$ e_n^$ $ $ $} $} $} $} $} $} typpe pyptant $} typpe typpem} typpe [m] typpe [m]热力学限制中的这种集中属性$ t =+\ infty $建议在2.5级的2.5级介绍“微跨式集合”的概念，用于随机轨迹$ x（0 \ leq t \ leq t）$，其中所有相关的经验变量$ e_n $都固定在某些相关的经验变量$ e_n $上都无法固定$ e e^*n $ $ e^*_ n $，并for folducuited $ foldifuare $ foldifutiation $ foldifutiation $ foldifutiand $ foldifutiand $ foldifutiand $ foldifutiate $ foldifutiand。本文的目的是讨论其主要属性：（MC1）当长轨迹$ x（0 \ leq t \ leq t）$属于微型统一的合奏，并带有固定的经验观察值$ e_n^*$，其子量的统计数据是其子量的$ x（0 \ leq t \ leq t \ leq iS $ 1 \ for $ 1 \ for $ 1 \。与Markov Generator $ M^*$相关的合奏，这将使经验可观察到$ e_n^*$典型; （MC2）在微型典型的集合中，核心角色由数字$ω^{[2.5]} _ t（E^*_。）$的持续时间$ t $的随机轨迹，给定的经验观察值$ e^*_ n $，以及相应的boltzmann nockmann n $ ω^{[2.5]} _ t（e^*_。）]/t $。该一般框架应用于连续的马尔可夫跳跃过程，并通过说明性示例离散时间马尔可夫链。
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 τ) $ for $1 \ll τ\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 $Ω^{[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 Ω^{[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.
