Helmholtz free energy

Since we have just shown that the microcanonical and the canonical ensembles are equivalent we expect that the thermodynamics of a system can be derived also from the canonical ensemble, but how? What we now want to show is that the Helmholtz free energy of a system (see Thermodynamic potentials) finds a natural place in the canonical ensemble. In particular we want to show that , where is the free energy of the system. This way, once the canonical partition function has been computed we can obtain the Helmholtz free energy as , and with appropriate derivatives we can obtain all the thermodynamics of the system (seeThermodynamic potentials).

The first thing we can note is that the canonical partition function can be written in terms of the microcanonical phase space volume :

(where we have only inserted a ). This means that the canonical ensemble can be thought of as an ensemble of microcanonical systems each weighted with . Furthermore, if the Hamiltonian is bounded below then we can always shift it by a constant amount so that , and thus:
which is a Laplace transform! In other words is the Laplace transform of , and is the variable conjugated to . From the microcanonical definition of entropy we have , so that can be rewritten as:
Now, both and are extensive so we can use the saddle point approximation (see the appendix The saddle point approximation) to compute the integral. In particular, in order to do so we need to determine the value of the energy that maximizes , namely minimizes . Therefore:
Thus, if (which is always the case since it means that giving heat to a system its temperature will increase) then is indeed a minimum. If we now expand the integrand of around :
and the values of that contribute significantly to are those such that , namely : for this reason in the thermodynamic limit we can integrate on the whole real axis (even if does not span ), since in this way we introduce a perfectly negligible error. Therefore:
The first term of is extensive, while the second is proportional to ; in the thermodynamic limit this last contribution vanishes and so:
This is indeed the free energy of the system if we have . However:
because by definition is the value of that extremizes . Therefore , and the expression in is indeed the free energy of the system:
We also see that although the partition function formally allows the system to have any value of energy, it is largely "dominated" by the configurations in which the system has energy .