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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


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

\begin{displaymath}dE =TdS-PdV+\mu dN=dE(S,V,N)\end{displaymath}

\begin{displaymath}dF=dE-TdS-SdT=-SdT-PdV+\mu dN=dF(T,V,N)\end{displaymath}

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

\begin{displaymath}S=-{{\partial F}\over{\partial T}}\end{displaymath}

\begin{displaymath}P=-{{\partial F}\over{\partial T}}\end{displaymath}

\begin{displaymath}\mu ={{\partial F}\over{\partial N}}\end{displaymath}

In statistical mechanics we define the Helmholtz free energy as

\begin{displaymath}A =-k_BT \ln Z_c\end{displaymath}

We wish to show that for a large system

A=\langle E\rangle -T\langle S\rangle =\langle F\rangle\end{displaymath} (19)

Proof: The canonical partition function is

dEg(E)e^{-\beta E}\end{displaymath}

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

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

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\}\end{displaymath} (21)

Using ([*]) we find

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

and from ([*])

A=-k_BT\ln Z_c=\langle E\rangle -T\langle S\rangle -k_BT\ln
\pi k_BT^2C}}\over{\delta E}}\right]\end{displaymath} (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.
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Next: About this document ... Up: Boltzmann statistics Previous: Partition function and the
Birger Bergersen