We saw that the and could be related by a
Legendre transformation. The partition functions and
can be related by a Laplace transform. Recall that the
Laplace transform
of a function is given by

Let us compute the Laplace transform of with respect to :

Using the -function to do the integral over :

By identifying , we see that the Laplace transform of the microcanonical partition function gives the canonical partition function .

Mark Tuckerman 2004-02-10