Condensation Transition in Polydisperse Hard Rods

TL;DR

This paper investigates a stochastic model of polydisperse hard rods, revealing a condensation transition where a macroscopic aggregate forms beyond a critical density, linking mass transport models with polydisperse fluid behavior.

Contribution

It introduces a factorized steady state for a mass transport model of polydisperse rods, connecting stochastic dynamics with thermodynamic minimization and demonstrating a condensation transition for p>1.

Findings

Condensation transition occurs for p>1.

A macroscopic aggregate coexists with a critical fluid phase.

Steady state distribution aligns with free energy minimization.

Abstract

We study a mass transport model, where spherical particles diffusing on a ring can stochastically exchange volume $v$, with the constraint of a fixed total volume $V=\sum_{i=1}^N v_i$, $N$ being the total number of particles. The particles, referred to as $p$-spheres, have a linear size that behaves as $v_i^{1/p}$ and our model thus represents a gas of polydisperse hard rods with variable diameters $v_i^{1/p}$. We show that our model admits a factorized steady state distribution which provides the size distribution that minimizes the free energy of a polydisperse hard rod system, under the constraints of fixed $N$ and $V$. Complementary approaches (explicit construction of the steady state distribution on the one hand ; density functional theory on the other hand) completely and consistently specify the behaviour of the system. A real space condensation transition is shown to take place for $p>1$: beyond a critical density a macroscopic aggregate is formed and coexists with a critical fluid phase. Our work establishes the bridge between stochastic mass transport approaches and the optimal polydispersity of hard sphere fluids studied in previous articles.