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Partition Function and Free Energy

Put off thy shoes from off thy feet, for the place whereon thou standest is holy ground.

Exodus, 3:5


The normalization constant in the canonical ensemble

Q = \sum_i \exp(-E_i/kT)\end{displaymath}

is called partition function . The quantity

A = -kT\ln Q\end{displaymath}

is called free energy .


Let us calculate $(\partial (A/T)/\partial (1/T))_V$:
\frac{\partial (A/T)}{\partial (1/T)} = -\frac{\partial k\ln Q...
 ...frac{1}{Q} \sum_i E_i \exp(-E_i/kT) = \left\langle E\right\rangle\end{multline*}

\frac{\partial (A/T)}{\partial (1/T)} = E\quad\text{or}
 E = -kT^2 \left(\frac{\partial (A/T)}{\partial T}\right)_V\end{displaymath}

If we know partition function, we can calculate average energy!

Conjugated Fields:
Variable X is a conjugated field to xi, if

\frac{\partial E_i}{\partial X} = -x_i

If one is extensive, another is intensive!
pressure (x) & volume (X), magnetization (x) & field (X),...
Let us calculate derivative:
\frac{\partial A}{\partial X} = -kT \frac{\partial \ln Q}{\par...
 ...{1}{Q} \sum_i x_i\exp(-E_i/kT) = - \left\langle x
 \right\rangle \end{multline*}
If we know partition function, we can calculate averages!

Second derivative:
\frac{\partial^2 A}{\partial X^2} = \\  \frac{1}{Q^2}\frac{\pa...
 ...  \left\langle x \right\rangle^2 - \left\langle x^2 \right\rangle\end{multline*}
This is mean squared fluctuation!

We obtained  
 \frac{\partial A}{\partial X} = - \left\langle x \right\ran...
 ...eft\langle x \right\rangle^2 -
 \left\langle x^2 \right\rangle \end{displaymath} (5)

We can obtain all information about our system from differentiating free energy!

© 1997 Boris Veytsman and Michael Kotelyanskii
Thu Sep 4 21:28:23 EDT 1997