1549380323-Statistical Mechanics Theory and Molecular Simulation

Ideal gas 273

For other thermodynamic quantities, it is convenient to introduce anew variable

ζ= eβμ (6.4.24)

known as thefugacity. Sinceζandμare directly related, the fugacity can be viewed
as an alternative external control variable for the grand canonical ensemble, and the
partition function can be expressed in terms ofζas

Z(ζ,V,T) =

∑∞

N=0

ζNQ(N,V,T), (6.4.25)

so that
PV
kT

= lnZ(ζ,V,T). (6.4.26)

Since
∂
∂μ

=

∂ζ

∂μ

∂

∂ζ

=βζ

∂

∂ζ

, (6.4.27)

the average particle number can be computed fromZ(ζ,V,T) by

〈N〉=ζ

∂

∂ζ

lnZ(ζ,V,T). (6.4.28)

Thus, the equation of state results when eqn. (6.4.28) is solved forζin terms of〈N〉and
substituted back into eqn. (6.4.26). Other thermodynamic quantities can be obtained
as well. The average energy,E=〈H(x,N)〉, is given by

E=〈H(x,N)〉=

1

Z

∑∞

N=0

ζN

1

N!h^3 N

∫

dxH(x,N)e−βH(x,N)

=−

(

∂

∂β

lnZ(ζ,V,T)

)

ζ,V

. (6.4.29)

In eqn. (6.4.29), it must be emphasized that the average energy is computed as the
derivative with respect toβof lnZat fixedV andζ ratherthan at fixedV andμ.
The entropy is given in terms of the derivative of the free energy with respect toT:

S(μ,V,T) =−

(

∂(−PV)

∂T

)

μ,V

=klnZ(μ,V,T)−kβ

(

∂

∂β

lnZ(μ,V,T)

)

μ,V

. (6.4.30)

For the entropy, the temperature derivative must be taken at fixedμrather than at
fixedζ.