# 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 $$Q (N, V, T )$$.