# Laplace Transform Density of States & Partition function

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

$$Z(V,T) = \int g(E)\exp{-\beta E}dE = \mathcal{L}(g(E))$$

• So you aren't satisfied that we end up with an integral that just happens to be defined as something else that we call the Laplace transform? I am unsure what type of connection you are looking for, and how this would be different from what you think about the Fourier transform. – Aaron Stevens Oct 8 '18 at 11:50
• I am aware that the Fourier Transform is just a special case of the Laplace transform. Yet, in essence when we apply the Fourier transform to some some signal say, it will spit out the fundamental frequencies in Fourier space. Similarly for position and momenta in QM. What I'm wondering is if there is some deep underlying physics or if it is just – Laudicina Corentin Oct 8 '18 at 11:56
• As you mention, the definition of something else we call the Laplace transform – Laudicina Corentin Oct 8 '18 at 11:57
• This is discussed in some detail at physics.stackexchange.com/questions/140932/… and there is an interesting connection (via saddle-point methods) to the Legendre transform which relates the corresponding thermodynamic variables (in the thermodynamic limit). – user197851 Oct 8 '18 at 11:58
• – user197851 Oct 8 '18 at 12:05