1549380323-Statistical Mechanics Theory and Molecular Simulation

142 Canonical ensemble

μ=−kT

(

∂lnQ

∂N

)

N,V

P=kT

(

∂lnQ

∂V

)

N,T

S=klnQ+kT

(

∂lnQ

∂T

)

N,V

E=−

(

∂

∂β

lnQ

)

N,V

. (4.3.23)

Noting that

kT

∂lnQ

∂T

=kT

∂lnQ

∂β

∂β

∂T

=−kT

∂lnQ

∂β

1

kT^2

=−

1

T

∂lnQ

∂β

=

E

T

, (4.3.24)

one finds that the entropy is given by

S(N,V,T) =klnQ(N,V,T) +

E(N,V,T)

T

, (4.3.25)

which is equivalent toS= (−A+E)/T. Other thermodynamic relations can be ob-
tained as well. For example, the heat capacityCVat constant volume is defined to be

CV=

(

∂E

∂T

)

N,V

. (4.3.26)

Differentiating the last line of eqn. (4.3.19) using∂/∂T=−(kβ^2 )∂/∂βgives

CV=kβ^2

∂^2

∂β^2

lnQ(N,V,β). (4.3.27)

Interestingly, the heat capacity in eqn. (4.3.27) is an extensive quantity. The corre-
sponding intensivemolar heat capacityis obtained from eqn. (4.3.27) by dividing by
the number of moles in the system.

4.4 Energy fluctuations in the canonical ensemble

Since the HamiltonianH(x) is not conserved in the canonical ensemble, it is natural
to ask how the energy fluctuates. Energy fluctuations can be quantified using the
standard statistical measure of variance. The variance of the Hamiltonian is given by

∆E=

√

〈(H(x)−〈H(x)〉)^2 〉, (4.4.1)