138 Canonical ensemble
N , V , E 2 2 2
H 2 ( x 2 )N , V , E 1 1 1H 1 ( x 1 )Fig. 4.1A system (system 1) in contact with a thermal reservoir (system 2). System 1 has
N 1 particles in a volumeV 1 ; system 2 hasN 2 particles in a volumeV 2.
between the two systems. Since the two systems can exchange energy, we do not expect
H(x 1 ) andH(x 2 ) to be separately conserved.
The microcanonical partition function of this thermodynamic universe is
Ω(N,V,E) =MN
∫
dxδ(H(x)−E)=MN
∫
dx 1 dx 2 δ(H 1 (x 1 ) +H 2 (x 2 )−E). (4.3.1)The corresponding phase space distribution functionf(x 1 ) is obtained by integrating
only over the phase space variables of system 2, yielding
f(x 1 ) =∫
dx 2 δ(H 1 (x 1 ) +H 2 (x 2 )−E), (4.3.2)which is unnormalized. Because thermodynamic quantities are obtained from deriva-
tives of the logarithm of the partition function, it is preferable to work with the
logarithm of the distribution:
lnf(x 1 ) = ln∫
dx 2 δ(H 1 (x 1 ) +H 2 (x 2 )−E). (4.3.3)We now exploit the fact that system 1 is small compared to system 2.SinceE 2 ≫E 1 ,
it follows thatH 2 (x 2 )≫H 1 (x 1 ). Thus, we expand eqn. (4.3.3) aboutH 1 (x 1 ) = 0 at
an arbitrary phase space point x 1. Carrying out the expansion to first order inH 1
gives