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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

\[ \tilde {f} (\lambda) = \int _{0}^{\infty} dx e^{- \lambda x} f (x) \]

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

\[ \tilde {\Omega} (N, V, \lambda ) = C_N \int _{0}^{\infty} dE e^{- \lambda E} \int dx \delta ( H (x) - E ) \]

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

\[\tilde {\Omega} (N, V, \lambda ) = C_N \int dx e^{- \lambda H (x) } \]

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