Energy equipartition and the emergence of damping in lossless systems

The principle of equipartition of energy is usually viewed as a statistical result formulated in terms of the probability distribution of configurations, that is, entropy. In this paper we use deterministic linear systems techniques to analyze the vibrational energy of systems of undamped coupled oscillators with identical coupling. Our approach is based on time averaging of squared outputs of the system and thus avoids both recurrence and statistical arguments. We first consider a single undamped oscillator and show that the time averaged potential energy and the time-averaged kinetic energy converge to the same value. Next, we consider a collection of n identical undamped oscillators with lossless coupling. As in the case of a single oscillator, equipartition of energy holds for the total system kinetic and potential energies. We focus on the equipartition of oscillator energy, that is, the equal distribution of energy among oscillators, regardless of the form of energy. We derive expressions for the transient and steady-state behavior of each oscillator.