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 t
o 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.
Numerical and analytic results for the exponent theta describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic r
oughness exponent beta, with 0 < beta < 1; for beta = 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent theta_S = 1 - beta. The exponent theta_0 for the flat initial condition appears to be nontrivial. We prove that theta_0 to infty for beta to 0, theta_0 geq theta_S for beta < 1/2 and theta_0 leq theta_S for beta > 1/2, and calculate theta_{0,S} perturbatively to first order in an expansion around the Markovian case beta = 1/2. Using the exact result theta_S = 1 - beta, accurate upper and lower bounds on theta_0 can be derived which show, in particular, that theta_0 geq (1 - beta)^2/beta for small beta.
