Monodisperse hard rods in external potentials

TL;DR

This paper presents exact analytical methods to study the density and pressure profiles of monodisperse rods in external potentials, revealing microscopic oscillations and their mesoscopic averaging, connecting discrete and continuum models.

Contribution

It introduces two complementary exact analysis methods for monodisperse rods under external potentials, bridging microscopic and mesoscopic descriptions and connecting to continuum limits.

Findings

Identification of microscopic oscillations in density and pressure profiles.

Systematic averaging of microscopic oscillations to obtain mesoscopic profiles.

Validation of methods through comparison with known continuum results.

Abstract

We consider linear arrays of cells of volume $V_\mathrm{c}$ populated by monodisperse rods of size $\sigma V_\mathrm{c}$, $\sigma=1,2,\ldots$, subject to hardcore exclusion interaction. Each rod experiences a position-dependent external potential. In one application we also examine effects of contact forces between rods. We employ two distinct methods of exact analysis with complementary strengths and different limits of spatial resolution to calculate profiles of pressure and density on mesoscopic and microscopic length scales at thermal equilibrium. One method uses density functionals and the other statistically interacting vacancy particles. The applications worked out include gravity, power-law traps, and hard walls. We identify oscillations in the profiles on a microscopic length scale and show how they are systematically averaged out on a well-defined mesoscopic length scale to establish full consistency between the two approaches. The continuum limit, realized as $V_\mathrm{c}\to0$, $\sigma\to\infty$ at nonzero and finite $\sigma V_\mathrm{c}$, connects our highest-resolution results with known exact results for monodisperse rods in a continuum. We also compare the pressure profiles obtained from density functionals with the average microscopic pressure profiles derived from the pair distribution function.