论文标题
在无限常规晶格上二聚体气体的熵（和病毒系数）的无用膨胀
On the Pernici-Wanless Expansion for the Entropy ( and Virial Coefficients ) of a Dimer Gas on an Infinite Regular Lattice
论文作者
论文摘要
我们使用以下表达式在无限的R-r-groumar晶格lambda（P）= 1/2 [PLN（R）-LN（P）-2（1-P）Ln（1-P）LN（1-P）-P] -P] -P] -P] -P]+SUM_ {K = 2}（p^k）（p^k）的无限型晶格（p）= 1/2 [pln（r）-ln（p）-2（p）-2（1-p）-2（p）Ln（p）（p^k）中，指示sum for d d d d d d d. Pernici计算了k <13的系数D_K。他发现这些D_K是某些有趣的“几何定量”中的多项式，在Wanless的工作中产生。这些数量中的每一个都是某些图的同构映射的数量密度（图）。因此，对于双分晶格D_2 = C_2 D_3 = C_3 D_4 = C_4 + C_4 + C_5 HAT {G} _1 D_5 = C_6 + C_7 HAT {G} _1 _1。 C_I仅取决于R。这里的帽子{g} _1是四个循环图的映射类的密度。 1/v的限制为V量V的晶格的1/V倍，如V到达无限。无限体积限制。有一个简单的线性关系，从d_k的值中得出KTH病毒系数！我们认为这种表达使得迄今为止获得的病毒系数有最深刻的见解。 我们在本文中表明的是，这些几何数量中D_K的多项式关系对于k <28的D_K保留。我们使用与佩里奇相同的计算程序。我们注意到此过程并未严格确定。到目前为止，物理学家的程序，也许不是数学家（他们的损失）。对于数学物理学家来说，提供所需的严谨性是一个值得的挑战。
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.
