Statistical Mechanics - Physics at Oregon State University

2.6. PROPERTIES OF THE HELMHOLTZ FREE ENERGY. 33
function of U and U is an increasing function of T! The ability to switch back
and forth between different sets of independent variables implies constant signs
for certain derivatives, and for response functions. For example, a solid always
expands as a function of temperature. In thermodynamics we related these
properties in a rigorous way to criteria for stability.
2.6 Properties of the Helmholtz free energy.
For a system at constant temperature and volume, the Helmholtz free energy is
the state function determining all physical properties of the system. For example,
consider a system in thermal contact with a large reservoir at temperature
T. The volume of the system is V. Energy can flow back and forth between
the system and the reservoir; the energy of the system is ɛ and the energy of
the combination reservoir plus system is U0. In order to find the value of ɛ
which corresponds to the equilibrium value for the system, i.e. which is equal
to U(T,V), we have to maximize the entropy of the combination system plus
reservoir. Since
Stot(ɛ) = SR(U0 − ɛ) + SS(ɛ) ≈ SR(U0) − 1
T (ɛ − T SS(ɛ)) (2.39)
this means we have to minimize ɛ − T SS(ɛ). In other words, if we specify the
temperature and volume of a system, we can give the energy ɛ an arbitrary
value, but that value will in general not correspond to a system in equilibrium.
The equilibrium value of ɛ is found by minimizing the Helmholtz free energy
ɛ − T S(ɛ). This equilibrium value is called U. Very often, notation is sloppy
and one tells you to minimize U − T S(U). The Helmholtz energies in statistical
mechanics and in thermodynamics therefore have identical properties, and again
the statistical mechanical and thermodynamical descriptions are the same.
The pressure as a function of temperature and volume is related to the
internal energy U and the entropy S by
p(T, V ) = −
∂U
∂V
T
+ T
∂S
∂V T
(2.40)
Both terms can be important, although in general in a solid the first term is
larger and in a gas the second. Beware, however, of the errors introduced by
approximating the pressure by only one of the terms in this expression.
How are they related?
In thermodynamics we have seen that a system as a function of T and V (and
N) is described by the Helmholtz energy F (T, V, N). In statistical mechanics we
needed the partition function Z(T, V, N). These two functions must therefore
be related. It is possible to eliminate S from the definition of the Helmholtz free
energy F, since S = −
∂F
∂T . This gives
V,N