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Next: Classical Virial Theorem (canonical Up: The canonical ensemble Previous: The partition function

Relation between canonical and microcanonical ensembles

We saw that the $E(N,V,S)$ and $A(N,V,T)$ could be related by a Legendre transformation. The partition functions $\Omega(N,V,E)$ and $Q(N,V,T)$ can be related by a Laplace transform. Recall that the Laplace transform $\tilde{f}(\lambda)$ of a function $f(x)$ is given by

\begin{displaymath}
\tilde{f}(\lambda) = \int_0^{\infty} dx e^{-\lambda x} f(x)
\end{displaymath}

Let us compute the Laplace transform of $\Omega(N,V,E)$ with respect to $E$:

\begin{displaymath}
\tilde{\Omega}(N,V,\lambda) = C_N \int_0^{\infty} dE e^{-\lambda E}
\int d{\rm x}\delta(H({\rm x})-E)
\end{displaymath}

Using the $\delta$-function to do the integral over $E$:

\begin{displaymath}
\tilde{\Omega}(N,V,\lambda) = C_N \int d{\rm x}e^{-\lambda H({\rm x})}
\end{displaymath}

By identifying $\lambda=\beta$, we see that the Laplace transform of the microcanonical partition function gives the canonical partition function $Q(N,V,T)$.



Mark Tuckerman 2004-02-10