Dynamics of energy partition among coupled harmonic oscillators in equilibrium

Abstract

Energy partition among many weakly coupled harmonic oscillators in equilibrium is found to be subject to 1/ f fluctuations, and that the power spectral density (PSD) of fractional energy fluctuations of each oscillator is equal to 1/ ω for frequency ω ⪡ γ , where γ is the simple relaxation frequency, while it is equal to γ/ ω 2 for ω ⪢ γ . The PSD of resistance fluctuations of semiconductors is empirically given by P R ( ω ) / R 2 = α H / ( N e ω ) , where α H is a dimensionless constant, and N e is the number of mobile carriers in the specimen. The present theory derives that α H = ( d / λ e ) , where d is the lattice constant, and λ e is the mean free path of a mobile carrier for the case when the scattering of carriers is dominated by phonons rather than impurities, lattice defects, etc. The present result was applied to carefully prepared semiconductor heterogeneous junctions, and a satisfactory agreement with observations has been achieved. Since the 2-sample variance of fractional fluctuations with 1/ f PSD equals 2ln2, which is larger than unity, the classical equipartition law of energy does not hold for a small component of a complex system. It is probable that the precision measurement of micro- or nano-scale systems will uncover 1/ f fluctuations in the energy partition.