The occupation statistics for indistinguishable dumbbells on a rectangular lattice space. I

Abstract

The method of Hock and McQuistan used recently to solve the occupation statistics for indistinguishable dumbbells (or dimers) on a 2×2×N lattice is extended further to obtain, for the L×M×N lattice, general expressions for the normalization, expectation, and dispersion of the statistics, and their limit as N becomes very large. In particular, an explicit expression of the partition function in the thermodynamic limit Ξ(x) is obtained for any value of the absolute activity x of dimers. The developed mathematical formalism is then applied to planar lattices, 1×M×N, with M=1, 2, 3, and 4. The known results for M=1 and 2 are recovered, and some new ones are obtained. The recurrence relation for the number A(q,N) of arrangements of q dumbbells on a 1×M×N lattice which has 3 and 5 terms when M=1 and 2, respectively, is found to have 15 and 65 terms for M=3 and 4. Analysis and extrapolation of the results enable one to predict the expectation 〈θ〉1MN on a planar 1×M×N lattice to be 63.4%, in the limit as both M and N become infinite. We also find an upper bound on the quantity MN[〈θ2〉−(〈θ〉)2] in the limit as both M and N become infinite. In the thermodynamic limit (M&N→∞) the partition function Ξ(1), for the absolute activity x=1, is found to be equal to 1.95. By limiting the number M of rows of infinite extent (N→∞) to just 4, we find that the error in determining Ξ(1) for the infinite two-dimensional lattice is just 4.5%. In this paper Ξ(x) is obtained for any value of the absolute activity x for M=1 and 2. A more thorough study of Ξ(x), and its fast convergence with increasing values of M, and applications will be presented in a forthcoming article.