Helmholtz Free Energy - Relation To The Partition Function

Relation To The Partition Function

A system kept at constant volume and temperature is described by the canonical ensemble. The probability to find the system in some energy eigenstate r is given by:

where

Z is called the partition function of the system. The fact that the system does not have a unique energy means that the various thermodynamical quantities must be defined as expectation values. In the thermodynamical limit of infinite system size, the relative fluctuations in these averages will go to zero.

The average internal energy of the system is the expectation value of the energy and can be expressed in terms of Z as follows:

If the system is in state r, then the generalized force corresponding to an external variable x is given by

The thermal average of this can be written as:

Suppose the system has one external variable x. Then changing the system's temperature parameter by and the external variable by dx will lead to a change in :

If we write as:

we get:

This means that the change in the internal energy is given by:

In the thermodynamic limit, the fundamental thermodynamic relation should hold:

This then implies that the entropy of the system is given by:

where c is some constant. The value of c can be determined by considering the limit T → 0. In this limit the entropy becomes where is the ground state degeneracy. The partition function in this limit is where is the ground state energy. Thus, we see that and that: