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Partition function and the canonical distribution

The constant in ([*]) can be determined by normalizing the probability distribution i.e. requiring that

\begin{displaymath}\int p(E)dE=1\end{displaymath}

Let us define the canonical partition function ( $\alpha =microstate)$

\begin{displaymath}Z_c=\sum \limits_\alpha e^{-\beta E(\alpha )}
=\int dE g(E) e^{-\beta E}\end{displaymath}

We find that the probability $p(\alpha )$ that a state is in a given microstate

p(\alpha )={{1}\over{Z_c}}e^{-\beta E}\end{displaymath} (17)

Eq. ([*]) is the canonical distribution. If $x(\alpha
)$ is the value of some physical property in microstate $\alpha
,$ and $E(\alpha )$ the energy of this state then the canonical ensemble average is given by

\langle x\rangle ={{1}\over{Z_c}}\sum \limits_\alpha
)e^{-\beta E(\alpha )}\end{displaymath} (18)

Equation ([*]) is a very useful formula, and we will give many examples of its use.

Birger Bergersen