The structure of the hard sphere solid

We show that near densest-packing the perturbations of the HCP structure yield higher entropy than perturbations of any other densest packing. The difference between the various structures shows up in the correlations between motions of nearest neighbors. In the HCP structure random motion of each sphere impinges slightly less on the motion of its nearest neighbors than in the other structures.Comment: For related papers see: http://www.ma.utexas.edu/users/radin/papers.htm

Generation of folk song melodies using Bayes transforms

The paper introduces the `Bayes transform', a mathematical procedure for putting data into a hierarchical representation. Applicable to any type of data, the procedure yields interesting results when applied to sequences. In this case, the representation obtained implicitly models the repetition hierarchy of the source. There are then natural applications to music. Derivation of Bayes transforms can be the means of determining the repetition hierarchy of note sequences (melodies) in an empirical and domain-general way. The paper investigates application of this approach to Folk Song, examining the results that can be obtained by treating such transforms as generative models

Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries

Using exact computations we study the classical hard-core monomer-dimer models on m x n plane lattice strips with free boundaries. For an arbitrary number v of monomers (or vacancies), we found a logarithmic correction term in the finite-size correction of the free energy. The coefficient of the logarithmic correction term depends on the number of monomers present (v) and the parity of the width n of the lattice strip: the coefficient equals to v when n is odd, and v/2 when n is even. The results are generalizations of the previous results for a single monomer in an otherwise fully packed lattice of dimers.Comment: 4 pages, 2 figure

Tetromino tilings and the Tutte polynomial

We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial Z\_G(Q,v) (known also to physicists as the partition function of the Q-state Potts model) on an (m-1) x (n-1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.Comment: 8 pages, 6 figure

Phase Transition in a Self-repairing Random Network

We consider a network, bonds of which are being sequentially removed; that is done at random, but conditioned on the system remaining connected (Self-Repairing Bond Percolation SRBP). This model is the simplest representative of a class of random systems for which forming of isolated clusters is forbidden. It qualitatively describes the process of fabrication of artificial porous materials and degradation of strained polymers. We find a phase transition at a finite concentration of bonds $p=p_c$, at which the backbone of the system vanishes; for all $p<p_c$ the network is a dense fractal.Comment: 4 pages, 4 figure

The duality relation between Glauber dynamics and the diffusion-annihilation model as a similarity transformation

In this paper we address the relationship between zero temperature Glauber dynamics and the diffusion-annihilation problem in the free fermion case. We show that the well-known duality transformation between the two problems can be formulated as a similarity transformation if one uses appropriate (toroidal) boundary conditions. This allow us to establish and clarify the precise nature of the relationship between the two models. In this way we obtain a one-to-one correspondence between observables and initial states in the two problems. A random initial state in Glauber dynamics is related to a short range correlated state in the annihilation problem. In particular the long-time behaviour of the density in this state is seen to depend on the initial conditions. Hence, we show that the presence of correlations in the initial state determine the dependence of the long time behaviour of the density on the initial conditions, even if such correlations are short-ranged. We also apply a field-theoretical method to the calculation of multi-time correlation functions in this initial state.Comment: 15 pages, Latex file, no figures. To be published in J. Phys. A. Minor changes were made to the previous version to conform with the referee's Repor

Influence of extended dynamics on phase transitions in a driven lattice gas

Monte Carlo simulations and dynamical mean-field approximations are performed to study the phase transition in a driven lattice gas with nearest-neighbor exclusion on a square lattice. A slight extension of the microscopic dynamics with allowing the next-nearest-neighbor hops results in dramatic changes. Instead of the phase separation into high- and low-density regions in the stationary state the system exhibits a continuous transition belonging to the Ising universality class for any driving. The relevant features of phase diagram are reproduced by an improved mean-field analysis.Comment: 3 pages, 3 figure

Integer Partitions and Exclusion Statistics

We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.Comment: 16 pages, 4 .eps figures include

Density functional theory for nearest-neighbor exclusion lattice gasses in two and three dimensions

To speak about fundamental measure theory obliges to mention dimensional crossover. This feature, inherent to the systems themselves, was incorporated in the theory almost from the beginning. Although at first it was thought to be a consistency check for the theory, it rapidly became its fundamental pillar, thus becoming the only density functional theory which possesses such a property. It is straightforward that dimensional crossover connects, for instance, the parallel hard cube system (three-dimensional) with that of squares (two-dimensional) and rods (one-dimensional). We show here that there are many more connections which can be established in this way. Through them we deduce from the functional for parallel hard (hyper)cubes in the simple (hyper)cubic lattice the corresponding functionals for the nearest-neighbor exclusion lattice gases in the square, triangular, simple cubic, face-centered cubic, and body-centered cubic lattices. As an application, the bulk phase diagram for all these systems is obtained.Comment: 13 pages, 13 figures; needs revtex

Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.Comment: 6 pages, 4 figures, revtex