136 Canonical ensemble
ploying the canonical ensemble. In addition, we will examine how physical observables
of experimental interest are obtained in this ensemble, including both thermodynamic
and structural properties of real systems. Finally, we will show how molecular dy-
namics methods capable of generating a sampling of the canonical distribution can be
devised.
4.2 Thermodynamics of the canonical ensemble
The Legendre transformation technique introduced in Section 1.5 isthe method by
which thermodynamic potentials are transformed between ensembles. Recall that in
the microcanonical ensemble, the control variables are particle numberN, volumeV,
and total energyE. The state function that depends on these is the entropyS(N,V,E),
and the thermodynamic variables obtained from the partial derivatives of the entropy
are:
1
T=
(
∂S
∂E
)
N,V,
P
T
=
(
∂S
∂V
)
N,E,
μ
T=−
(
∂S
∂N
)
V,E. (4.2.1)
Note that the entropyS=S(N,V,E) can also be inverted to giveEas a function,
E(N,V,S). In terms ofE, using eqn. (3.2.7), the above thermodynamic relations
become
T=(
∂E
∂S
)
N,V, P=−
(
∂E
∂V
)
N,S, μ=(
∂E
∂N
)
V,S. (4.2.2)
For transforming from the microcanonical to the canonical ensemble, eqn. (4.2.2) is
preferable, as it gives the temperature directly, rather than 1/T. Thus, we seek to
transform the functionE(N,V,S) from a function ofN,V, andSto a function of
N,V, andT. SinceT=∂E/∂S, the Legendre transform method can be applied.
According to eqn. (1.5.5), the new function, which we will denote asA(N,V,T), is
given by
A(N,V,T) =E(N,V,S(N,V,T))−
∂E
∂S
S(N,V,T)
=E(N,V,S(N,V,T))−TS(N,V,T). (4.2.3)
The functionA(N,V,T) is a new state function known as theHelmholtz free energy.
Physically, when a thermodynamic transformation of a system fromstate 1 to state
2 is carried out on a system along a reversible path, then the work needed to effect
this transformation is equal to the change in the Helmholtz free energy ∆A. From
eqn. (4.2.3), it is clear thatAhas both energetic and entropic contributions, and
the delicate balance between these two contributions can sometimes have a sizeable
effect on the free energy. Free energy is a particularly useful concept as it determines
whether a process is thermodynamically favorable, indicated by a decrease in free
energy, or unfavorable, indicated by an increase in free energy. It is important to note
that although thermodynamics can determine if a process is favorable, it has nothing
to say about the time scale on which the process occurs.