Abstract

A physical background is derived for the generation of 1/f fluctuations that are observed ubiquitously in energy partition in quasi-harmonic oscillators that exchange energy through weak nonlinear interactions. Weak three-mode coupling terms in the unperturbed Hamiltonian bring no net transfer of energy between oscillators, but four-mode coupling terms bring about irreversible energy transfer between unperturbed oscillators as well as 1/f fluctuations in energy partition in which the reaction of energy transfer plays a crucial role. Oscillation (vibration) modes in material media are accompanied by nonlinear coupling of modes, and 1/f energy fluctuations are always observed in each oscillation mode. This is the main reason for the ubiquitous existence of this kind of fluctuation. The power spectral density of fractional 1/f energy fluctuations is simply given by (1/n)(1/f), where n is the number of quasi-oscillators involved in the generation of 1/f fluctuations. The present theoretical formula allows estimation of the number of quasi-harmonic oscillators involved in the generation of observed 1/f spectra, and this approach is applicable to oscillation phenomena from the nanometer to the astronomical scale. The origin of 1/ffluctuations in biological rhythm is possibly attributed to vibrations of ion channel conductance.

Key words: Heart rate, 1/f fluctuations, nonlinearity, harmonic oscillators, biological rhythm.