1Fundamentals of statistical mechanics

II Statistical Physics

1.4 Helmholtz free energy

In the microcanonical ensemble, we discussed the second law of thermodynamics,

namely the entropy increases with time and the maximum is achieved in an

equilibrium.

But this is no longer true in the case of the canonical ensemble, because

we now want to maximize the total entropy of the system plus the heat bath,

instead of just the system itself. Then is there a proper analogous quantity for

the canonical ensemble?

The answer is given by the Helmholtz free energy.

Definition (Helmholtz free energy). The Helmholtz free energy is

F = hEi − T S.

As before, we will often drop the h·i.

In general, in an isolated system,

S

increases, and

S

is maximized in equilib-

rium. In a system with a reservoir,

F

decreases, and

F

minimizes in equilibrium.

In some sense F captures the competition between entropy and energy.

Now is there anything analogous to the first law

dE = T dS − p dV ?

Using this, we can write

dF = dE − d(T S) = −S dT − p dV.

When we wrote down the original first law, we had d

S

and d

V

on the right,

and thus it is natural to consider energy as a function of entropy and volume

(instead of pressure and temperature). Similarly, It is natural to think of

F

as a

function of

T

and

V

. Mathematically, the relation between

F

and

E

is that

F

is the Legendre transform of E.

From this expression, we can immediately write down

S = −

∂F

∂T

V

,

and the pressure is

p = −

∂F

∂V

T

.

As always, we can express the free energy in terms of the partition function.

Proposition.

F = −kT log Z.

Alternatively,

Z = e

−βF

.

Proof. We use the fact that

d

dβ

= kT

2

d

dT

.

Then we can start from

F = E − T S

= −

∂ log Z

∂β

− T S

= kT

2

∂ log Z

∂T

V

− kT

∂

∂T

(T log Z)

V

= −kT log Z,

and we are done. Good.