Phase space and partition function 139lnf(x 1 )≈ln∫
dx 2 δ(H 2 (x 2 )−E)+
∂
∂H(x 1 )ln∫
dx 2 δ(H 1 (x 1 ) +H 2 (x 2 )−E)∣
∣
∣
∣
H 1 (x 1 )=0H 1 (x 1 ).(4.3.4)Since theδ-function requiresH 1 (x 1 ) +H 2 (x 1 )−E= 0, we may differentiate with
respect toEinstead, using the fact that
∂
∂H 1 (x 1 )δ(H 1 (x 1 ) +H 2 (x 2 )−E) =−∂
∂E
δ(H 1 (x 1 ) +H 2 (x 2 )−E). (4.3.5)Then, eqn. (4.3.4) becomes
lnf(x 1 )≈ln∫
dx 2 δ(H 2 (x 2 )−E)−
∂
∂E
ln∫
dx 2 δ(H 1 (x 1 ) +H 2 (x 2 )−E)∣
∣
∣
∣
H 1 (x 1 )=0H 1 (x 1 ). (4.3.6)Now,H 1 (x 1 ) can be set to 0 in the second term of eqn. (4.3.6) yielding
lnf(x 1 )≈ln∫
dx 2 δ(H 2 (x 2 )−E)−∂
∂E
ln∫
dx 2 δ(H 2 (x 2 )−E)H 1 (x 1 ). (4.3.7)Recognizing that ∫
dx 2 δ(H 2 (x 2 )−E)∝Ω 2 (N 2 ,V 2 ,E), (4.3.8)where Ω 2 (N 2 ,V 2 ,E) is the microcanonical partition function of system 2 at energy
E. Since ln Ω 2 (N 2 ,V 2 ,E) =S 2 (N 2 ,V 2 ,E)/k, eqn. (4.3.7) can be written (apart from
overall normalization) as
lnf(x 1 ) =S 2 (N 2 ,V 2 ,E)
k−H 1 (x 1 )∂
∂E
S 2 (N 2 ,V 2 ,E)
k. (4.3.9)
Moreover, because∂S 2 /∂E= 1/T, whereTis the common temperature of systems 1
and 2, it follows that
lnf(x 1 ) =S 2 (N 2 ,V 2 ,E)
k−
H 1 (x 1 )
kT. (4.3.10)
Exponentiating both sides, and recognizing that exp(S 2 /k) is just an overall constant,
we obtain
f(x 1 )∝e−H^1 (x^1 )/kT. (4.3.11)
At this point, the “1” subscript is no longer necessary. In other words, we can conclude
that the phase space distribution of a system with HamiltonianH(x) in equilibrium
with a thermal reservoir at temperatureTis