Condensation transition in polydisperse hard rods

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

Condensation transition in joint large deviations of linear statistics

“…Our next example is a system of hard spheres with variable size, proposed in [9] to sample polydispersity in a hard-sphere fluid. The model consists of N spheres diffusing on an one-dimensional ring of length L and exchanging volume with hardcore interactions.…”

“…This algorithm generates a stationary state of the form (9), with the single site weight f (m) = 1 ‡,…”

“…We use an algorithm introduced in [41] to sample the distribution (9). We consider a chain of L sites with periodic boundary conditions, where each site carries a mass m i ≥ 0, i = 1, .…”

“…In Section 4 we present numerical simulations for the case p = 1/2 that confirm our theoretical predictions. In Section 5, we review several examples of condensation transition: jamming transition in exclusion process, condensation transition in polydisperse rods diffusing on a ring [9] and phase transition in a random pure state of a large bipartite quantum system [36,37]. We also mention a possibly related phenomenon of localised solutions (breathers) of the discrete nonlinear Schrödinger equation [38][39][40][41].…”

“…In this regime, the excess mass forms a condensate which is localised in space, for example at a single lattice site, and coexists with a background fluid in which the remaining mass is evenly distributed over the rest of the system. The condensation phenomenon occurs in many physical contexts such as coalescence in granular systems [3], jamming in traffic [4][5][6], aggregation and absorption on surfaces [7,8], emulsification failure in polydisperse hard-sphere systems [9], etc. In these examples the 'mass' corresponds to some conserved quantity such as the total number of particles, the total length of gaps between the cars or the volume fraction of hard spheres, respectively.…”