Virial theorem and energy partitioning in systems with mixed power-law potentials

Abstract

Clausius’ Virial Theorem is often invoked to predict the partitioning of kinetic and potential energies in either classical or quantum systems with simple power-law potentials. Here, the Virial Theorem, and related statistical mechanical identities, are used to investigate energy partitioning in classical systems with mixed power-law potentials, in either one or three dimensions, with either positive or negative mixed power-law exponents. Such classical systems are found to have temperature-dependent heat capacities reminiscent of quantum behaviour, as well as other unusual properties. Closed form analytical expressions are obtained for energy partitioning in a family of one-dimensional generalised asymmetric oscillators. Unlike the corresponding symmetric oscillators, these systems are predicted to undergo a thermally driven migration of energy between spatially separated regions. Analogous behaviour is obtained for three-dimensional systems with mixed inverse power-law potentials, as exemplified by non-ideal gases with Lennard–Jones n 0–n 1 interaction potentials. In this case the Virial Theorem implies that the ratio of the average energy associated with the negative (n 1) and positive (n 0) parts of the potential is exactly equal to −n 0/n 1 at the Boyle temperature (where the second virial coefficient vanishes), and approximately so at other temperatures.