4.3 Entropy, Helmholtz Free Energy and the Partition Function

Here is the crucial equation which links the Helmholtz free energy and the partition function:

Since , from the fundamental thermodynamic relation we obtain . Thus

(We first met these in the derivation of Maxwell's relations.) For a magnetic system, we have instead of the equation for P.

Remember, and hence depend on (or ) through the energies of the microstates. For instance the energy levels of a particle in a box of side are proportional to .

These relations are reminiscent of those we met in the case of an isolated system, but there the entropy was the key; here it is the Helmholtz free energy. We can make the following comparison:

It should not surprise us to find that the Helmholtz free energy is the key to a system at fixed temperature (in contrast to the entropy for an isolated system) as that is what we found classically (see here.)

**References**

**Mandl**2.5- (
**Bowley and Sánchez**5.3-6) **Kittel and Kroemer**3