On Equilibrium Fluctuations

This paper considers a dynamical system described by a multidimensional state vector x. A component x of x evolves according to dx/dt = f(x). Equilibrium fluctuations are fluctuations of an equilibrium solution x(t) obtained when the system is in its equilibrium state reached under a constant external forcing. The frequencies of these fluctuations range from the major frequencies of the underlying dynamics to the lowest possible frequency, the frequency zero. For such a system, the known feature of the differential operator d(·)/dt as a high-pass filter makes the spectrum of f to vanish not only at frequency zero, but de facto over an entire frequency range centered at frequency zero (when considering both positive and negative frequencies). Consequently, there is a non-zero portion of the total equilibrium variance of x that cannot be determined by the differential forcing f. Instead, this portion of variance arises from many impulse-like interactions of x with other components of x, which are received by x along an equilibrium solution over time. The effect of many impulse-like interactions can only be realized by integrating the evolution equations in form of dx/dt = f(x) forward in time. This integral effect is not contained in, and can hence not be explained by, a differential forcing f defined at individual time instances.