82 Microcanonical ensemble
〈
xi∂H
∂xj〉
=kTδij, (3.3.1)where the average is taken with respect to a microcanonical ensemble.
To prove this theorem, we begin with the following ensemble average:
〈
xi
∂H
∂xj〉
=
MN
Ω(N,V,E)
∫
dxxi∂H
∂xjδ(E−H(x)). (3.3.2)In eqn. (3.3.2), we have used the fact thatδ(x) =δ(−x) to write the energy conserving
δ-function asδ(E−H(x)). It is convenient to express eqn. (3.3.2) in an alternate way
using the fact that the Diracδ-functionδ(x) = dθ(x)/dx, whereθ(x) is the Heaviside
step function,θ(x) = 1 forx≥0 andθ(x) = 0 forx <0. Using these relations, eqn.
(3.3.2) becomes
〈
xi
∂H
∂xj〉
=
MN
Ω(N,V,E)
∂
∂E
∫
dxxi∂H
∂xjθ(E−H(x)). (3.3.3)The step function restricts the phase space integral to those microstates for which
H(x)< E. Thus, eqn. (3.3.3) can be expressed equivalently as
〈
xi
∂H
∂xj〉
= =
MN
Ω(N,V,E)
∂
∂E
∫
H(x)<Edxxi∂H
∂xj=
MN
Ω(N,V,E)
∂
∂E
∫
H(x)<Edxxi∂(H(x)−E)
∂xj, (3.3.4)
where the last line follows from the fact thatE is a constant. Recognizing that
∂xi/∂xj =δij since all phase space components are independent, we can express
the phase space derivative as
xi∂(H(x)−E)
∂xj=
∂
∂xj[xi(H(x)−E)]−δij(H(x)−E). (3.3.5)Substituting eqn. (3.3.5) into eqn. (3.3.4) gives
〈
xi
∂H
∂xj〉
=
MN
Ω(N,V,E)
×
∂
∂E
∫
H(x)<Edx{
∂
∂xj[xi(H(x)−E)] +δij(E−H(x))}
.(3.3.6)
The first integral is over a pure derivative. Hence, when the integral overxjis per-
formed, the integrandxi(H(x)−E) must be evaluated at the limits ofxjwhich lie
on the constant energy hypersurfaceH(x) =E. SinceH(x)−E= 0 at the limits, the
first term vanishes leaving
〈
xi
∂H
∂xj〉
=
MN
Ω(N,V,E)
δij∂
∂E
∫
H(x)<Edx (E−H(x)). (3.3.7)If we now carry out the energy derivative in eqn. (3.3.7), we obtain