Partition function approximation

[homology]

So I've been wrestling with something I was reading in a stat mech text. It's the derivation of the partition function for an ideal gas but I imagine the technique is used again. The author starts with the partition function for a single particle but then approximates the sum as an integral.

I get the idea behind it, the particle is essentially a free particle in a big box and has nearly a continuum of states so the 'Boltzmann-like" function e^(-E/kt) is integrated against density of states.

But I've been trying to make it a bit more rigorous. One idea I wondered about was the use of a discrete measure and maybe showing that the density of states times dE (so D(E)dE) approximates the discrete measure that gives back the sum.

Or perhaps trying to set up some sort of less complicated limiting process where I can show the integral approximation breaks down.

Any ideas? Or, if you're familiar with a more detailed discussion of this feel free to recommend a text/article.

thanks in advance,

Kevin

 

[Valerie Park]

One way to make the derivation more rigorous is to start from the Hamiltonian of the system, which describes the energy of the particles in the system. The partition function is then obtained by taking the trace of the exponential of the Hamiltonian and dividing it by the normalization constant. By expanding the exponential in a Taylor series and using the fact that terms with different energies are orthogonal one can show that the partition function is the sum of an integral and a series of delta-functions. This makes the approximation of the sum as an integral more precise.