Bethe approximation for a system of hard rigid rods: the random locally tree-like layered lattice

TL;DR

This paper introduces a new lattice model that enables exact Bethe approximation analysis of dense packings of long rigid rods, revealing phase transition behaviors dependent on rod length and lattice coordination.

Contribution

The authors define the random locally tree-like layered lattice, allowing exact Bethe approximation for dense rod packings, which was not possible with traditional Bethe lattices.

Findings

For 4-coordinated lattices, rods with k>=4 undergo a continuous phase transition.

For lattices with q>=6, the phase transition occurs only for k above a minimum length and is first order.

The new lattice model accurately captures dense packing phenomena in rod systems.

Abstract

We study the Bethe approximation for a system of long rigid rods of fixed length k, with only excluded volume interaction. For large enough k, this system undergoes an isotropic-nematic phase transition as a function of density of the rods. The Bethe lattice, which is conventionally used to derive the self-consistent equations in the Bethe approximation, is not suitable for studying the hard-rods system, as it does not allow a dense packing of rods. We define a new lattice, called the random locally tree-like layered lattice, which allows a dense packing of rods, and for which the approximation is exact. We find that for a 4-coordinated lattice, k-mers with k>=4 undergo a continuous phase transition. For even coordination number q>=6, the transition exists only for k >= k_{min}(q), and is first order.