On the Classical Energy Equipartition Theorem

Home , Boltzmann constant

176 Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000

1 2 y

J. A. S. Lima and A. R. Plastino

Universidade Federal do Rio Grande do Norte,

Departamento de F sica, Caixa Postal 1641,

59072-970 Natal, RN, Brazil

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-

cal mechanics and kinetic theory, which has b een ex-

kT : 3 hg i =

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

to x has a thermal mean energy equal to , where k

the analysis of the basic conditions needed for the valid-

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 with Hamiltonian =2m. The mean value of the

i cation of its relationship with other physical principles

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

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]

[email protected]

J.A.S. Lima and A.R. Plastino 177

Let us consider a Hamiltonian system with f degrees

n o

of freedom i.e., with a 2f -dimensional phase space

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

>0,

and

g x ;:::;x = g x ;:::;x : 5

dI 1 r

1 L 1 L

= hg i: 13

I d I

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-

eral form of the energy equipartition theorem can b e

hg i = I ; 14

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

shall assume that our Hamiltonian system is describ ed

hg i = kT I : 15

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

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

x hxi = hx i hxi : 16

n o

I = A g x ;:::;x +h d ; 7

1 L

However, this relation makes sense only if the integral

2 2

hx i = x pxdx converges. There are imp ortant

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

A x= F x: 8

px= : 17

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

= x ; i =1;:::;L; 9 x

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

I = I ; 10

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

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. This term, however, makes no

ciated integral for dI =d. If those limits go to 1 in

contribution to dI =d b ecause the integrand A vanishes

all the variables x ;:::x then our derivation is clearly

1 L

at the integration b oundary.Taking into account the

correct. Let us consider the case when those limits do

ab ove remarks and caveats, equations 12 and 13 are

not go to 1. In principle, this may happ en bytwo

easily veri ed.

di erent reasons:

Equation 15 generalizes the standard energy equipar-

tition theorem. For a system in equilibrium at a given

1 The system has intrinsic constraints. For in-

temp erature, it means that the contribution to the

stance, if wehave a particle moving inside the one

mean energy due to an homogeneous comp onent g of

dimensional b ox[L; L] it is plain that the limits

the Hamiltonian is only a function of the temp erature

of integration for the co ordinate x do not go to

T , the degree of homogeneity r , and the total num-

ber L of arguments entering the function g . The mean

2 The limits of integration do not go to 1 be-

value of g does not depend upon its detailed structure.

cause the distribution function f q ;p given by

In particular, each quadratic term in the Hamiltonian i i

6 has a cut-o as happ ens, for example, with

dep ending on a single canonical variable for example,

Tsallis distribution.

each comp onent of the kinetic energy makes the same

contribution to the total mean energy of the system,

Our generalized version of the equipartition theorem

i.e.,

do es not hold in the case 1, and the same happ ens

with the standard pro of of the theorem. The equipar-

kT I : 20

tition theorem as applied to the mean value of an ad-

ditive term in the p otential function is not veri ed

It is imp ortant to stress that the expression 15 was ob-

under this condition b ecause a system endowed with

tained without any assumption ab out the sp eci c form

those kind of constraints is, strictly sp eaking, outside

of the phase space distribution function. However, our

the hyp othesis of the theorem. Those constraints are

general result do es dep end on the distribution function

describ ed by \rigid wall" terms in the p otential func-

through the quantity see 7

tion asso ciated with the system's Hamiltonian. The

equipartition theorem is not applicable b ecause p oten-

I = A H d : 21

tial functions exhibiting rigid walls are not, in general,

homogeneous functions of the co ordinates. It is clear

In the case of Gibbs canonical ensemble, the distribu-

that our generalization must not b e blamed for this

tion function adopts the form

diculty.

On the other hand, our theorem still holds true in

f q ;p = exp H q ;p ; 22

i i i i

the case 2. Let us assume that the function F x

where the parameter is equal to the logarithm of the

app earing in the de nition 6 of the phase space dis-

partition function Z and has a value such that the dis-

tribution function vanishes outside the interval [a ;a ],

1 2

tribution f q ;p is appropriately normalized, that is

i i

but is continuous at the p oints a .We can then cho ose

1;2

Ax see equation 8 in suchaway that it vanishes

Z Z

to o outside that interval, while b eing itself continuous

f q ;p d = exp H q ;p d = 1:

i i i i

and endowed with continuous derivatives at the p oints

a . The b oundary of the integration region asso ciated

1;2

Hence, wehave F x = exp x and, since the only re-

with I and dI =d is then given by the two phase

quirement on the function Ax is to b e a primitive

space hip ersurfaces de ned by the equations

1;2

function of F x, we can take

n o

g x ;:::;x + h = a : 18

Ax= F x= e : 24

1 L 1;2

Thus, b ecause of the normalization of the canonical dis-

When we p erform the change 9 of integration vari-

tribution,

ables it is easy to realize, according to elementary calcu-

the b oundary lus, that in terms of the new variables x

I = F H d = 1: 25

of the integration region is now given by the equations

J.A.S. Lima and A.R. Plastino 179

Hence, within the Boltzmann-Gibbs formalism, our between brackets b ecomes negative. We shall refer to

generalized theorem 15 leads to this last requirement as the \cut-o " condition.

In this context, the functions F x and Ax pre-

viously intro duced now assume, resp ectively, the forms

hg i = kT : 26

b elow

Turner's result reduces to the particular L = 1 case

1=q 1

of the ab ove expression, which describ es the mean value

F x= [1+q 1x] ; 32

of an homogeneous comp onent of the Hamiltonian de-

p ending on one single canonical variable [7]. On the

and

other hand, if g is an homogeneous quadratic func-

tion, like the kinetic energy of the N particles in a

q=q 1

[1+ q 1x] : 33 Ax=

nonrelativistic classical gas L =3N; r = 2, we re-

cover the standard version of the energy equipartition

The contribution of one velo city comp onenttothe

theorem usually app earing in textb o oks of statistical

mean kinetic energy is

mechanics[1-5]

E D

1 1

mv = kT I ; 34

NkT: 27 hg i =

2 2

with

Notice, however, that the relation 26 is more gen-

eral than the one usually found in textb o oks, since it

deals with homogeneous functions of arbitrary degree of

q=q 1

1 1

2 3

homogeneity and arbitrary numb er of arguments. For

I = 1 q 1 mv d v: 35

qZ 2

instance, consider the case where the p otential func-

tion dep ends on two generalized co ordinates q and q

1 2

By taking q>1, and making the change of variables

through a term of the form

de ned by

i h

1=2

g q ;q = q q : 28

1 2 2

m v ; 36 u = q 1

This is an homogeneous function of degree 3 in the vari-

ables q and q and, consequently,wehave r = 3 and

one maycheek that I can b e put under the guise

1 2

L =2.Itthus follows that the mean value of g evalu-

1 C

ated on the Gibbs canonical ensemble is

I = ; 37

q C

hg i = kT : 29

where

Nowwe are going to illustrate the application of

q=q 1

2 2

C = u 1 u du; 38

the general equipartition theorem to a nonexp onen-

tial phase space distribution. Among the several avail-

and

able p ossibilities, let us consider the case of a distri-

bution function exhibiting a p ower law dep endence on

1=q 1

2 2

the Hamiltonian. These kind of distribution functions

C = u 1 u du: 39

app ear naturally in manyphysical scenarios and are

nowadays b eing intensively studied see [10, 11] and ref-

The ab ovetwointegrals are tabulated see, for instance,

erences therein. For example, these p ower-law mo dels

[13]. Inserting their values into 37 one obtains

have b een successfully applied to describ e the observed

velo city distribution of clusters of galaxies [12]. The

; 40 I =

5q 3

one-particle p ower-law distribution is

and from equation 34,

1=q 1

1 1

E D

kT 1

f H = ; 30 1 q 1 mv

= mv : 41

Z 2

2 5q 3

where

Although evaluating I under assumption q>1, it is

not hard to prove that the expression 40 still holds

1=q 1

true if 3=5

2 3

1 q 1 Z = mv d v; 31

the critical value q =3=5, the mean values of v and

v diverge. For q<1=3, nor even the probability dis-

tribution itself is normalizable. Like in the standard and b oth f and the integrand app earing in the de ni-

Boltzmann-Gibbs approach, it should b e noticed that tion of Z are set equal to zero whenever the quantity q

180 Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000

the quantity I in this example do es not dep end on the nol ogico - CNPq and CAPES Brazilian Research

temp erature T . It is also worth remarking that, in the Agencies, and by CONICET Argentina Research

case of Tsallis statistics, wehave illustrated our gen- Agency.

eral theorem with the mean value of the kinetic energy

which is always an extensive quantity. That is, the total

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selho Nacional de Desenvolvimento Cienti co e Tec-