1549380323-Statistical Mechanics Theory and Molecular Simulation

Phase space and partition function 271

f 1 (x 1 ,N 1 ) =

(

e−βH^1 (x^1 ,N^1 )

Q(N,V,T)N 1 !h^3 N^1

)

1

(N−N 1 )!h3(N−N^1 )

∫

dx 2 e−βH^2 (x^2 ,N−N^1 )

=

Q 2 (N−N 1 ,V−V 1 ,T)

Q(N,V,T)

1

N 1 !h^3 N^1

e−βH^1 (x^1 ,N^1 ), (6.4.10)

satisfies the normalization condition

∑N

N 1 =0

∫

dx 1 f(x 1 ,N 1 ) = 1. (6.4.11)

Since the total partition function is canonical,Q(N,V,T) = exp[−βA(N,V,T)] where
A(N,V,T) is the Helmholtz free energy, and it follows that

Q 2 (N−N 1 ,V−V 1 ,T)

Q(N,V,T)

= e−β[A(N−N^1 ,V−V^1 ,T)−A(N,V,T)], (6.4.12)

where we have assumed that system 1 and system 2 are described by the same set of
physical interactions, so that the functional form of the free energy is the same for
both systems and for the total system. SinceN≫N 1 andV≫V 1 , we may expand
A(N−N 1 ,V−V 1 ,T) aboutN 1 = 0 andV 1 = 0. To first order, the expansion yields

A(N−N 1 ,V−V 1 ,T)≈A(N,V,T)−

∂A

∂N

N 1 −

∂A

∂V

V 1

=A(N,V,T)−μN 1 +PV 1. (6.4.13)

Thus, the phase space distribution of system 1 becomes

f(x 1 ,N 1 ) =

1

N 1 !h^3 N^1

eβμN^1 e−βPV^1 e−βH^1 (x^1 ,N^1 )

=

1

N 1 !h^3 N^1

eβμN^1

1

eβPV^1

e−βH^1 (x^1 ,N^1 ). (6.4.14)

Since system 2 quantities no longer appear in eqn. (6.4.14), we may drop the “1”
subscript and write the phase space distribution for the grand canonical ensemble as

f(x,N) =

1

N!h^3 N

eβμN

1

eβPV

e−βH(x,N). (6.4.15)

Moreover, taking the thermodynamic limit, the summation overNis now unrestricted
(N∈[0,∞)), so the normalization condition becomes

∑∞

N=0

∫

dxf(x,N) = 1, (6.4.16)

which implies that