1549380323-Statistical Mechanics Theory and Molecular Simulation

Phase space and partition functions 223

N , V , E 2 2 2

H 2 ( x 2 )

N , V , E

H ( x )

1 1 1

1 1

Fig. 5.1Two systems in contact with a common thermal reservoir at temperatureT. System
1 hasN 1 particles in a volumeV 1 ; system 2 hasN 2 particles in a volumeV 2. BothV 1 andV 2
can vary.

volumes, sayV 1. Thus, we write the correct canonical partition function for the total
system as

Q(N,V,T) =g(N,N 1 ,N 2 )

∫V

0

dV 1 Q 1 (N 1 ,V 1 ,T)Q 2 (N 2 ,V−V 1 ,T). (5.3.10)

The canonical phase space distribution functionf(x) of the combined system 1
and 2 is

f(x) =

CNe−βH(x)

Q(N,V,T)

. (5.3.11)

In order to determine the distribution functionf 1 (x 1 ,V 1 ) of system 1, we need to
integrate over the phase space of system 2:

f 1 (x 1 ,V 1 ) =

g(N,N 1 ,N 2 )

Q(N,V,T)

CN 1 e−βH^1 (x^1 )CN 2

∫

dx 2 e−βH^2 (x^2 )

=

Q 2 (N 2 ,V−V 1 ,T)

Q(N,V,T)

g(N,N 1 ,N 2 )CN 1 e−βH^1 (x^1 ). (5.3.12)

The distribution in eqn. (5.3.12) satisfies the normalization condition:

∫V

0

dV 1

∫

dx 1 f 1 (x 1 ,V 1 ) = 1. (5.3.13)

The ratio of partition functions can be expressed in terms of Helmholtz free energies
according to