Stationarity of time correlation functions for globally linear classical systems and its consequences

A necessary and sufficient condition for the stationarity of all time correlation functions associated with a given globally linear classical dynamical system is rigorously established from basic principles. Since stationarity of time correlation functions is a physical requirement that must be satisfied, the necessary and sufficient condition obtained for its realization represents a universal dynamical constraint on globally linear classical dynamical models intended to describe the execution of spontaneous fluctuations about a stationary state. This dynamical constraint is shown to (i) impose restrictions on the symmetry properties of the transition operator appearing in the global propagator for a system; (ii) represent a universal operator relation that embodies detailed balance and microscopic reversibility, giving rise to their traditional formulations; and (iii) imply the existence of certain generalized symmetry relations for time correlation functions and their Laplace and Fourier transforms. Apart from elucidating some fundamental symmetries of classical dynamical systems, the reported theory has the advantage of providing a simple model independent framework for treating classical time correlation functions via the extraction and utilization of dynamically embedded information. This is demonstrated in a concrete way by exploiting the mathematical apparatus of dual Lanczos transformation theory to determine the advanced and retarded components of the elements of the correlation matrices for first and second moment coordinate and momentum fluctuations for the Brownian harmonic oscillator. The Laplace transforms of the retarded components of the time correlation functions and the Fourier transforms of the full-time correlation functions are also obtained.