Phase transitions in a system of long hard rods on a lattice

Abstract

A system of hard rigid rods of length $k \gg1$ on hypercubic lattices in dimensions $d \geq2$, is known to undergo two phase transitions when chemical potential is increased: from a low-density phase to an intermediate density nematic phase, and on further increase to a high-density phase with no nematic order. I will present non-rigorous arguments to support the conjecture that for large $k$, the second phase transition is a first-order transition with a discontinuity in density in all dimensions greater than $1$. The chemical potential at the transition is $\approx A k \ln k$ for large $k$, and that the density of uncovered sites drops from a value $\approx B (\ln k)/ k^2$ , to a value of order $\exp(−ck)$, where $c$ is some constant, across the transition. We conjecture that these results are asymptotically exact, and $A = B= 1$, in all dimensions $d ≥ 2$.