On the classical energy equipartition theorem

Abstract

A general proof of the energy equipartition theorem is given. Our derivation holds for any distribution function depending on the phase space variables only through the Hamiltonian of the system. This approach generalizes the standard theorem in two main directions. On the one hand, it considers the contribution to the total mean energy of homogeneous functions having a more general type than the ones usually discussed in the literature. On the other hand, our proof does not rely on the assumption of a Boltzmann-Gibbs exponential distribution.The classical energy equipartition theorem constitutes an important result in thermodynamics, statistical mechanics and kinetic theory, which has been extensively discussed in the literature 1-9 . Its simplest version deals with the contribution to the mean energy of a system in thermal equilibrium at temperature T due to quadratic terms in the Hamiltonian. More precisely, it attests that any canonical variable x entering the Hamiltonian through an additive term proportional to x 2 has a thermal mean energy equal to kT 2 , where k is the Boltzmann constant. The most familiar example is provided by a three dimensional classical ideal gas with Hamiltonian Some time ago, the derivation of the equipartition theorem was generalized by T urner 7 in order to include the presence of nonquadratic terms. More precisely, assuming that the Hamiltonian Hq i ; p i ; i = 1; 2; :::f; may be additively separated in the form H = gx + h; 2 where x denotes one of the 2f coordinates or momenta, g is an homogeneous function of degree r, and the function h is independent o f x, he shown that the mean value of g is given by hgi = 1 r kT: 3We emphasize that the proof furnished by T urner is valid only for one homogeneous degree of freedom and makes explicit use of the exponential Boltzmann-Gibbs distribution. In brief, the main goal of the present w ork is just to relax these two h ypotheses. We believe that the analysis of the basic conditions needed for the validity of the equipartition theorem will contribute to the understanding of its physical meaning and to the clarication of its relationship with other physical principles and theorems. As it appears, the version of equipartition theorem presented here have at least two i n teresting virtues. The rst one is to extend the scope of the theorem so as to incorporate the case of an homogeneous component of the Hamiltonian depending on many canonical variables. The second one is to demonstrate that the theorem is not an exclusive property of the exponential Boltzmann-Gibbs distribution function. We shall see that the theorem holds true essentially for any distribution function that depends on the canonical variables only through the system's Hamiltonian. As it is well known, this set of distribution functions constitutes a family of legitimate stationary solutions to Liouville's equation 3 . That is to say, they are fully consistent with the general principles of dynamics. As we point out below, these facts may also be exploited to good methodologi...

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