Mathematical Background of 1/f Fluctuations

Energy of harmonic oscillators in equilibrium decays exponentially in time when they are coupled in quadratic forms in amplitudes. In reality, however, their Hamiltonian includes higher-order coupling terms. Not all of the higher-order coupling terms contribute to the energy decay of oscillators after averaging over reservoir oscillators, and we find that one of the lowest higher-order terms makes a finite contribution to the energy decay. This effect is equivalently represented by a modified coupling coefficient of quadratic coupling terms. This modification works as a positive feedback to the action-reaction process between oscillators. Eventually the modified coupling terms generate 1/f fluctuations in energy partition among oscillators in equilibrium. It is concluded that 1/f type of energy partition is observable with harmonic oscillators if they obey the Bose-Einstein statistics regardless of whether the collective system is classical or quantum mechanical regime.