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

TL;DR

This paper derives exact entropy formulas for a system of interacting rigid rotors on a lattice, linking it to dimer coverings and providing detailed orientation distributions, verified by simulations.

Contribution

It introduces an exact analytical approach for entropy and orientation distributions of interacting rotors on a lattice, connecting to dimer models and virial coefficients.

Findings

Exact entropy expressed via dimer partition functions.

Probability distribution of orientations depends on dimer density.

Numerical verification confirms theoretical results.

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.