On Equilibrium Fluctuations

Abstract

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.

Keywords

• low-frequency internal variability
• equilibrium variance and its decomposition in frequency domain
• spectrum at frequency zero
• differential forcing