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 $\langle E\rangle$ and $\langle S\rangle$. 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

$$dE =TdS-PdV+\mu dN=dE(S,V,N)$$

$$dF=dE-TdS-SdT=-SdT-PdV+\mu dN=dF(T,V,N)$$

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

$$S=-{{\partial F}\over{\partial T}}$$

$$P=-{{\partial F}\over{\partial T}}$$

$$\mu ={{\partial F}\over{\partial N}}$$

In statistical mechanics we define the Helmholtz free energy as

$$A =-k_BT \ln Z_c$$

We wish to show that for a large system

$$A=\langle E\rangle -T\langle S\rangle =\langle F\rangle$$ (19)

Proof: The canonical partition function is

$$Z_c=\int dEg(E)e^{-\beta E}$$

$$=\int {{dE}\over{\delta E}}\exp\left\{\beta[E-TS(E,V,N)]\right\}$$ (20)

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

$$E-TS\simeq \langle E\rangle -T\langle S\rangle -{{1} \over{2... ...\langle E\rangle )^2{{\partial ^2S}\over{\partial E^2}}+\cdots$$

Substituting () into () we obtain

$$Z_c\simeq \exp[-\beta (\langle E\rangle -T\langle S\rangle )... ...\exp \left\{{{-(E-\langle E\rangle )^2}\over{2Ck_BT^2}}\right\}$$ (21)

Using () we find

$$Z_c\simeq {{\sqrt{2\pi k_BT^2C}}\over{\delta E}}\exp[- \beta (\langle E\rangle -T\langle S\rangle )]$$

and from ()

$$A=-k_BT\ln Z_c=\langle E\rangle -T\langle S\rangle -k_BT\ln \left[{{\sqrt{2 \pi k_BT^2C}}\over{\delta E}}\right]$$ (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 $\delta E$ so that it is identically zero. We have therefore shown that () is correct.