ITLApplied  Computational Mathematics Division
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3D Dimer Constant Computed

March 1999

Since 1938, physicists and mathematicians around the world have been trying to calculate a fundamental quantity known as the dimer constant. This year, the three-dimensional constant was obtained using a unique computational approach by I. Beichl of ITL's Mathematical and Computational Sciences Division in collaboration with F. Sullivan of the IDA Center for Computing Sciences. The dimer constant describes the rate of growth of the number of ways to arrange dominos (in two-dimensions) or bricks (in three dimensions) on a lattice of increasing size. Dimer constants provide key information related to fundamental models of materials and are factors in computing the partition function for the monomer-dimer system. From the partition function, all thermodynamic properties such as specific heat of a material can be computed from first principles. The two-dimensional constant was obtained analytically in 1961, while the three-dimensional problem has defied exact solution. The approximate solution by Beichl and Sullivan is based on the use of importance sampling techniques to estimate the permanent of a related matrix efficiently. The computed value, which comes with rigorous error bars, far exceeds the accuracy of any previously obtained result. A paper describing this work recently appeared in the Journal of Computational Physics (volume 149, number 1, February 1999).

(bullet) Isabel Beichl (NIST/MCSD/OCGG)

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Last updated: 2011-01-12.