Emergence of particle clusters in a one-dimensional model: connection to condensation processes

Abstract

Abstract. We discuss a simple model of particles hopping in one dimension with attractive interactions. Taking a hydrodynamic limit in which the interaction strength increases with the system size, we observe the formation of multiple clusters of particles, with large gaps between them. These clusters are correlated in space, and the system has a self-similar (fractal) structure. These results are related to condensation phenomena in mass transport models and to a recent mathematical analysis of the hydrodynamic limit in a related model.

Structure of the Condensed Phase in the Inclusion Process

“…The model studied in [34] consists of N particles moving diffusively on a one-dimensional torus of length L, subject to a logarithmic attractive potential and short-range hard-core exclusion. The weak attraction leads to the formation of large gaps between groups of particles, and the distances y = (y 1 , .…”

“…, 1 − β) distribution. Of particular interest in [34] Note that in this model gaps between particles correspond to cluster sizes, and the average cluster size is therefore L/N . A related paper with a hierarchical clustering phenomenon for interacting diffusions on a ring is [35], and to our knowledge these continuous models are the only particle systems where a connection to Poisson-Dirichlet statistics has been recognized so far.…”

“…A related paper with a hierarchical clustering phenomenon for interacting diffusions on a ring is [35], and to our knowledge these continuous models are the only particle systems where a connection to Poisson-Dirichlet statistics has been recognized so far. The Brownian energy process introduced in [12,13] as a dual model to the inclusion process exhibits stationary product measures with chi-squared marginals, and conditioning on the total sum of occupation numbers leads to the same canonical distributions as the model in [34].…”

“…The latter was originally introduced in the context of population genetics [27,28], and has later been identified as the generic stationary distribution of split-merge dynamics [29,30], which is related to its appearance in cycle length distributions of random permutations [31][32][33]. It has further been observed (though not identified) more recently in systems of interacting diffusions [34,35], but to our knowledge is a novelty in the context of condensation in SPS. In general, the condensed phase in SPS with stationary product distributions concentrates on a single lattice site [7,8,10,36].…”