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We present a spectral-theoretic approach to time-average statisticalmechanics for general, non-equilibrium initial conditions. We consider thestatistics of bounded, local additive functionals of reversible as well asirreversible ergodic stochastic dynamics with continuous or discretestate-space. We derive exact results for the mean, fluctuations andcorrelations of time average observables from the eigenspectrum of theunderlying generator of Fokker-Planck or master equation dynamics, and discussthe results from a physical perspective. Feynman-Kac formulas are re-derivedusing It\^o calculus and combined with non-Hermitian perturbation theory. Theemergence of the universal central limit law in a spectral representation isshown explicitly on large deviation time-scales. For reversible dynamics withequilibrated initial conditions we derive a general upper bound to fluctuationsof occupation measures in terms of an integral of the return probability.Simple, exactly solvable examples are analyzed to demonstrate how to apply thetheory. As a biophysical example we revisit the Berg-Purcell problem on theprecision of concentration measurements by a single receptor. Our results aredirectly applicable to a diverse range of phenomena underpinned by time-averageobservables and additive functionals in physical, chemical, biological, andeconomical systems.

Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems