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=โi=1Nโviโ, N being the total number of particles. The
particles, referred to as p-spheres, have a linear size that behaves as
vi1/pโ and our model thus represents a gas of polydisperse hard rods with
variable diameters vi1/pโ. 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