Exact Results for Interacting Hard Rigid Rotors on a d-Dimensional Lattice

Abstract

We study the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d-dimensional regular lattice of lattice spacing a, but can have arbitrary orientations. When the pivoting point is situated asymmetrically on the object, we show that there is a range of lattice spacings a, where in any orientation, a particle can overlap with at most one of its neighbors. In this range, the entropy of the system of particles can be expressed exactly in terms of the grand partition function of coverings of the base lattice by dimers at a finite negative activity. The exact entropy in this range is fully determined by the second virial coefficient. Calculation of the partition function is also shown to be reducible to that of the same model with discretized orientations. We determine the exact functional form of the probability distribution function of orientations at a site. This depends on the density of dimers for the given activity in the dimer problem, that we determine by summing the corresponding Mayer series numerically. These results are verified by numerical simulations.

References in corpus (3)

• Spin-S Kitaev model: Classical Ground States, Order by Disorder and Exact Correlation Functions
• One-dimensional gas of hard needles
• Sequence of phase transitions in a model of interacting rods

Cited by in corpus (5)

• Sequence of phase transitions in a model of interacting rods
• Ordering properties of anisotropic hard bodies in one-dimensional channels
• Hard-needle elastomer in one spatial dimension
• Phases of the hard-plate lattice gas on a three-dimensional cubic lattice
• Cusp singularities in the distribution of orientations of asymmetrically-pivoted hard discs on a lattice