Partition function for hard spheres on a lattice

[Yoran91]

Hi everyone,

I'm reading some lecture notes on statistical physics and thermodynamics and I'm stuck at an expression for a partition function which I really don't understand.

The chapter is on mean field theory and the discussion is about hard spheres on a lattice. The interaction of the hard sphere is $\beta U = \infty$ if $r<\sigma$ and $\beta U = 0$ if $r>\sigma$ as usual, where $\sigma$ is the diameter of the spheres.

Now it's said that a single hard sphere is treated exactly and the other spheres are located at their 'ideal' lattice positions. This supposedly leads to $Z_N=\prod_{i=1}^N V_i$ where $V_i$ is the free volume in which the center of mass of particle i can move.

I really don't see this. I'm expecting an $h^3$ or $\Lambda ^3$ to appear somewhere, obtained by integrating over the momenta of such a sphere, but it isn't there. Why is this the partition function?

 

[eljose79]

Thank you for your question regarding the partition function in mean field theory for hard spheres on a lattice. I can understand your confusion as the expression given in the lecture notes may seem counterintuitive. However, let me explain why this is the correct partition function in this case.

Firstly, let's consider the definition of the partition function, which is given by Z = \sum_{\{r_i\}} e^{-\beta U(\{r_i\})}, where \{r_i\} represents the positions of all N particles in the system. In this case, the interaction energy U is given by \beta U = \infty if r<\sigma and \beta U = 0 if r>\sigma, which means that for a single hard sphere, the only allowed positions are those where r>\sigma. This is because the interaction energy becomes infinite if the spheres overlap, and this is not allowed in the system.

Now, when we consider the other N-1 spheres in the system, we assume that they are located at their ideal lattice positions, which means that their positions are fixed and do not contribute to the partition function. Therefore, the only contribution to the partition function comes from the single hard sphere, which can move freely in the available volume. This is why the partition function is given by Z_N=\prod_{i=1}^N V_i, where V_i is the free volume in which the center of mass of particle i can move.

I hope this explanation helps clarify the expression for the partition function in this case. It is important to note that in mean field theory, we make certain approximations and assumptions to simplify the calculations, and in this case, the assumption of fixed lattice positions for the N-1 spheres leads to this form of the partition function. If you still have further questions or concerns, please do not hesitate to ask.