Multiple singularities of the equilibrium free energy in a one-dimensional model of soft rods

TL;DR

This paper presents a one-dimensional model of soft rods demonstrating that the equilibrium free energy can exhibit infinitely many singularities as a function of the ratio of rod length to lattice spacing, countering common misconceptions.

Contribution

It provides a counter-example showing that finite-range interactions in 1D models can lead to multiple singularities in free energy, challenging prevailing beliefs.

Findings

Free energy has infinitely many singularities as a function of rod length to lattice spacing ratio.

Counter-example to the misconception that 1D finite-range models lack free energy singularities.

Demonstrates complex phase behavior in a simple 1D soft-rod system.

Abstract

There is a misconception, widely shared amongst physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at non-zero temperatures, can not show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counter-example. We consider thin rigid linear rods of equal length $2 \ell$ whose centers lie on a one-dimensional lattice, of lattice spacing $a$. The interaction between rods is a soft-core interaction, having a finite energy $U$ per overlap of rods. We show that the equilibrium free energy per rod $\mathcal{F}(\tfrac{\ell}{a}, \beta)$, at inverse temperature $\beta$, has an infinite number of singularities, as a function of $\tfrac{\ell}{a}$.