# Condensation transition in polydisperse hard rods

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=Sigma(N)(i=1)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 behavior 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.