## Helmholtz free energy

For an isolated system S=S(E,V,N), with E,V,N independent variables. For a system in contact with a heat bath at a given temperature, T becomes an independent variable, or control parameter. The energy E and entropy S will then fluctuate about their mean values and . E and S become dependent variables given by equations of state. The change of variables is handled most efficiently by introducing he Helmholtz free energy. In thermodynamics it is defined as

F=E-TS

Imagine a reversible process which takes the system from one equilibrium state to another

We see that the Helmholtz free energy is should be considered to be dependent on the control variables T,V,N. We have

In statistical mechanics we define the Helmholtz free energy as

We wish to show that for a large system

 (19)

Proof: The canonical partition function is

 (20)

We evaluate this integral using the saddle point method. Almost all the contribution to the integral will come from values of E near the value for which E-T(S,E,V,N) = minimum. We let and

Substituting () into () we obtain

 (21)

Using () we find

and from ()

 (22)

The last term in () will be small compared to the first two terms for a large system, and it is possible to choose the tolerance so that it is identically zero. We have therefore shown that () is correct.