On the Pernici-Wanless Expansion for the Entropy ( and Virial Coefficients ) of a Dimer Gas on an Infinite Regular Lattice

Abstract

We work with the following expression for the entropy (density) of a dimer gas on an infinite r-regular lattice lambda(p) = 1/2 [ pln(r)-ln(p)-2(1-p)ln(1-p)-p ]+sum_{k=2}(d_k)(p^k) where the indicated sum converges for density, p, small enough. Pernici has computed the coefficients d_k for k < 13. He found these d_k to be polynomials in certain interesting "geometric quantites" arising in the work of Wanless. Each of these quantities is the number density of isomorphic mappings of some graph into the lattice (graph). So for a bipartite lattice d_2 = c_2 d_3 = c_3 d_4 = c_4 + c_5 hat{G}_1 d_5 = c_6 + c_7 hat{G}_1. The c_i depend only on r. Here hat{G}_1 is the density of mapping classes of the four loop graph into the lattice. The limit of 1/V times the number of such mapping classes into a lattice of volume V as V goes to infinity. The infinite volume limit. There is a simple linear relation that yields the kth virial coefficient from the value of d_k! We feel this expression gives the deepest insight into the virial coefficients so far obtained. What we show in this paper is that such polynomial relations for the d_k in these geometric quantities holds for the d_k for k < 28. Of course we expect it to hold for all k. We use the same computation procedure as Pernici. We note this procedure is not rigorously established. So far a procedure for the physicist, perhaps not the mathematician (their loss). It is a worthy challenge for the mathematical physicist to supply the needed rigor.