176 Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 On the Classical Energy Equipartition Theorem 1 2 y J. A. S. Lima and A. R. Plastino 1 Universidade Federal do Rio Grande do Norte, Departamento de F sica, Caixa Postal 1641, 59072-970 Natal, RN, Brazil 2 Facultad de Ciencias Astronomicas y Geo sicas Universidad Nacional de La Plata, C.C. 727, 1900 La Plata, Argentina Received 20 April 1999. Revised version received on 17 Septemb er 1999 A general pro of of the energy equipartition theorem is given. Our derivation holds for any distribu- tion function dep ending 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 con- siders the contribution to the total mean energy of homogeneous functions having a more general typ e than the ones usually discussed in the literature. On the other hand, our pro of do es not rely on the assumption of a Boltzmann-Gibbs exp onential distribution. The classical energy equipartition theorem consti- value of g is given by tutes an imp ortant result in thermo dynamics, statisti- 1 cal mechanics and kinetic theory, which has b een ex- kT : 3 hg i = r tensively discussed in the literature [1-9]. Its simplest version deals with the contribution to the mean energy We emphasize that the pro of furnished byTurner is of a system in thermal equilibrium at temp erature T valid only for one homogeneous degree of freedom and due to quadratic terms in the Hamiltonian. More pre- makes explicit use of the exp onential Boltzmann-Gibbs cisely, it attests that any canonical variable x entering distribution. In brief, the main goal of the presentwork the Hamiltonian through an additive term prop ortional is just to relax these twohyp otheses. We b elieve that kT 2 to x has a thermal mean energy equal to , where k the analysis of the basic conditions needed for the valid- 2 is the Boltzmann constant. The most familiar example ity of the equipartition theorem will contribute to the is provided by a three dimensional classical ideal gas understanding of its physical meaning and to the clari - P 2 p with Hamiltonian =2m. The mean value of the i cation of its relationship with other physical principles i kinetic energy asso ciated with the x-comp onent of the and theorems. As it app ears, the version of equipar- velo cityofeach particle is ensemble average is denoted tition theorem presented here have at least twointer- by hi, esting virtues. The rst one is to extend the scop e of the theorem so as to incorp orate the case of an homo- E D E D 1 1 1 geneous comp onent of the Hamiltonian dep ending on 2 2 = = m x_ p kT : 1 x many canonical variables. The second one is to demon- 2 2m 2 strate that the theorem is not an exclusive prop erty Some time ago, the derivation of the equipartition of the exp onential Boltzmann-Gibbs distribution func- theorem was generalized byTurner[7] in order to in- tion. We shall see that the theorem holds true essen- clude the presence of nonquadratic terms. More pre- tially for any distribution function that dep ends on the cisely, assuming that the Hamiltonian H q ;p ;i = i i canonical variables only through the system's Hamil- 1; 2; :::f ; may b e additively separated in the form tonian. As it is well known, this set of distribution functions constitutes a family of legitimate stationary H = g x+ h; 2 solutions to Liouville's equation [3]. That is to say, they are fully consistent with the general principles of dynamics. As we p oint out b elow, these facts may also where x denotes one of the 2f co ordinates or momenta, b e exploited to go o d metho dological advantage in pre- g is an homogeneous function of degree r , and the func- senting the equipartition theorem. tion h is indep endentofx, he shown that the mean lima [email protected] y [email protected] J.A.S. Lima and A.R. Plastino 177 Let us consider a Hamiltonian system with f degrees Z n o of freedom i.e., with a 2f -dimensional phase space r I = A g x ;:::;x +h d : 11 1 L whose Hamiltonian function is of the form Computing the logarithmic derivatives of expressions H = g x ;:::;x + h; 4 1 L 10 and 11 and evaluating them at =1,we obtain, where x ;:::;x constitutes a subset of the 2f phase 1 L resp ectively, space co ordinates, h do es not dep end on any of those L 1 dI variables, and g is an homogeneous function of degree = L; 12 r . The homogeneitycharacter of g means that for any I d =1 >0, and r g x ;:::;x = g x ;:::;x : 5 dI 1 r 1 L 1 L = hg i: 13 I d I 1 =1 What we are going to prove is that the only prop- These last two relations give us a simple expression for erties of g determining its contribution hg i to the mean the mean value of g , namely value of the Hamiltonian are its degree r of homogene- ity and the number L of its arguments. Thus, our gen- L eral form of the energy equipartition theorem can b e hg i = I ; 14 1 r stated as follows: Al l the homogeneous terms of the or equivalently,by identifying with the inverse tem- Hamiltonian that are characterized by the same values p erature 1=k T , of the parameters r and L make the same contribution to the total mean energy. As already mentioned, we L shall assume that our Hamiltonian system is describ ed hg i = kT I : 15 1 r by a phase space probability density f q ;p dep end- i i This expression constitutes the main result of the ing on the canonical variables only through the system's present note. Before discussing the physical meaning Hamiltonian, of equation 15 a few comments on its mathematical derivation are in order. First of all, wehave assumed f q ;p = F [ H q ;p ]; 6 i i i i that b oth the integrals de ning the quantities I that is, equation 11 and dI =d converge. That is to say, equa- where is a dimensionless constant determined by the tion 15 is veri ed provided that the integrals for I and normalization condition and the constant is such that dI =d exist. The convergence of these integrals must the argument of the function F is dimensionless. For be checked in each sp eci c case by recourse to the stan- instance, in the case of Gibbs' canonical ensemble we x dard techniques of the Calculus. This is just the usual have F x= e , is equal to the logarithm of the par- situation with general theorems in theoretical physics, tition function Z , and the constant can b e identi ed as can b e illustrated by the following elementary exam- with the inverse temp erature 1=k T . ple. Given a one dimensional, normalized probability Now, in order to compute the mean value of g ,it distribution pxitiswell-known that proves convenienttointro duce the function D E 2 2 2 x hxi = hx i hxi : 16 Z n o I = A g x ;:::;x +h d ; 7 1 L However, this relation makes sense only if the integral R 2 2 hx i = x pxdx converges. There are imp ortant 2 probability distributions for which the moment hx i di- where d =dq ;:::;dp is the phase space volume el- 1 f verges. As a famous example wehave the Cauchy dis- ement and the function Ax is a primitive function of tribution F x, that is 1 0 A x= F x: 8 px= : 17 2 1+ x By recourse to the change of integration variables In spite of these problematic cases there can b e no doubt that it is b oth interesting and useful to know the 0 = x ; i =1;:::;L; 9 x i i general theorem 16, even if it holds only when the relevantintegrals converge. Another moral that we can it is easy to see that extract from the ab ove little example is that it is not enough to restrict our considerations to the moments L I = I ; 10 1 of p ositive quantities in order to avoid divergence dif- where I = I = 1. On the other hand, making ex- culties. It should b e stressed that in the case of the 1 plicit use of the homogeneityof g one nds equipartition theorem the problem of convergence is by 178 Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 no means a consequence of our generalization. These n o 0 0 g x ;:::;x + h = a ; 19 kind of diculties may arise also within the standard 1;2 1 L case of the Boltzmann-Gibbs statistical mechanics. For and equation 10 follows immediately. As a nal com- instance, in the case of a classical self gravitating N - ment, notice that when one evaluates the derivative b o dy system not even the integral de ning the parti- with resp ect to of the expression 7 for I , a sur- tion function converges. Secondly, let us consider the face term app ears due to the dep endence on of the limits of integration for the integral 11 and the asso- b oundary of integration.