Multiple Singularities of the Equilibrium Free Energy in a One-Dimensional Model of Soft Rods

There is a misconception, widely shared among physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at nonzero temperatures, cannot show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counterexample. 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 $\mathcal{F}[(\ell /a),\beta ]$, at inverse temperature $\beta$, has an infinite number of singularities, as a function of $\ell /a$.

Figure 1

A configuration of seven rods on a line. Here, $a$ is the spacing between rods. In the displayed configuration, the number of nearest neighbor overlaps $n_1=3$ and the number of next nearest overlaps $n_2=1$.

Figure 2

First derivative of the free energy ${\mathcal{F}}_1^{\prime }(\kappa )$ for hard-core nearest neighbor interaction between rods ($U_1=\infty$). The inset shows the monotonic increase of ${\mathcal{F}}_1(\kappa )$ as a function of $\kappa$.

Figure 3

Logarithmic divergence of the first derivative of the free energy ${\mathcal{F}}_1^{\prime }(\kappa )$ near $\kappa =1$, for $U_1=\infty$.

Figure 4

Probability distribution of the orientation of the rods generated from the eigenvector ${\psi }_{\kappa }(\theta )$ associated to the largest eigenvalue of the transfer matrix.

Figure 5

The transfer matrix $T_{\kappa }({\theta }^{\prime },\theta )$ on the $\theta \text{−}{\theta }^{\prime }$ plane, for different values of $\kappa$. The shaded regions denote $(\theta ,{\theta }^{\prime })$ values where the rods overlap, and $T_{\kappa }=0$. In the plain regions rods do not overlap and $T_{\kappa }=1$.

Figure 6

The picture shows the matrix $\mathrm{\Delta }T=T_{1-ϵ}-T_1$, for $ϵ=0.02$ on the $\theta \text{−}{\theta }^{\prime }$ plane. In the shaded region $\mathrm{\Delta }T=1$, whereas in the plain region it is 0. The area of the shaded region varies as $ϵ\log (1/ϵ)$, for small $ϵ$.

Figure 7

Variance of the angular distribution of rods generated from Monte Carlo simulations of a system of 100 rods and averaged over $10^6$ sample configurations.