I am currently going through Pathria's Statistical Mechanics text , and under the Canonical Ensemble description, the author stresses that the partition function of a continuous system is the Laplace transform of the density of states of the said system. While I understand the mathematical description of this relation, I struggle to visualize the concrete physics behind it.
I know that the laplace transform is generally used to turn ODEs into algebraic expressions, but unlike the Fourier transform whose connection to physics (especially through signal processing) is clear, the physics behind the use of the laplace transform in this case of statistical mechanics remains a puzzle for me. Is it simply a mathematical coincidence maybe? Clarification on the subject would be very much appreciated. I provide below the mathematical details
$$Z(V,T) = \int g(E)\exp{-\beta E}dE = \mathcal{L}(g(E))$$
