Abstract
We present a spectral-theoretic approach to time-average statistical mechanics for general, non-equilibrium initial conditions. We consider the statistics of bounded, local additive functionals of reversible as well as irreversible ergodic stochastic dynamics with continuous or discrete state-space. We derive exact results for the mean, fluctuations and correlations of time average observables from the eigenspectrum of the underlying generator of Fokker-Planck or master equation dynamics, and discuss the results from a physical perspective. Feynman-Kac formulas are re-derived using It\^o calculus and combined with non-Hermitian perturbation theory. The emergence of the universal central limit law in a spectral representation is shown explicitly on large deviation time-scales. For reversible dynamics with equilibrated initial conditions we derive a general upper bound to fluctuations of occupation measures in terms of an integral of the return probability. Simple, exactly solvable examples are analyzed to demonstrate how to apply the theory. As a biophysical example we revisit the Berg-Purcell problem on the precision of concentration measurements by a single receptor. Our results are directly applicable to a diverse range of phenomena underpinned by time-average observables and additive functionals in physical, chemical, biological, and economical systems.
Highlights
Many experiments on soft and biological matter, such as single-particle tracking [1,2,3,4] and single-molecule spectroscopy [5,6,7,8,9,10,11,12,13], probe individual trajectories
We focus on the statistics of bounded, local, additive functionals of normal ergodic Markovian stochastic processes with continuous and discrete statespaces, including functionals of their lowerdimensional projections
We developed a general spectral-theoretic approach to time-average statistical mechanics, i.e., to the statistics of bounded, local additive functionals of ergodic stochastic processes with continuous and discrete statespaces
Summary
Many experiments on soft and biological matter, such as single-particle tracking [1,2,3,4] and single-molecule spectroscopy [5,6,7,8,9,10,11,12,13], probe individual trajectories. Except for (ergodically) long observations, time-averages inferred from individual trajectories are random with nontrivial statistics. This naturally leads to the study of statistical properties of time-averages which formally represent functionals of stochastic processes. Exact results were obtained for occupation time statistics for a general class of Markov processes [56] and a discrete stationary non-Markovian sequence [57]. Exact results were recently obtained on local times for projected observables in stochastic many-body systems [59,60], which provided insight into the emergence of memory on the level of individual non-Markovian trajectories.
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