We study a mass transport model, where spherical particles diffusing on a ring can stochasticallyexchange volume v, with the constraint of a fixed total volume V=Σ_(i=1)~Nv_i,Nbeing the total number of particles. The particles, referred to as p-spheres, have a linear size that behaves as v_i~(1/p)and ourmodel 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 andV. 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 workestablishes the bridge between stochastic mass transport approaches and the optimal polydispersity of hard sphere fluids studied in previous articles.
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