Dynamics of condensation in the symmetric inclusion process

Abstract

The inclusion process is a stochastic lattice gas, which is a natural bosonic counterpart of the well-studied exclusion process and has strong connections to models of heat conduction and applications in population genetics. Like the zero-range process, due to attractive interaction between the particles, the inclusion process can exhibit a condensation transition. In this paper we present first rigorous results on the dynamics of the condensate formation for this class of models. We study the symmetric inclusion process on a finite set $S$ with total number of particles $N$ in the regime of strong interaction, i.e. with independent diffusion rate $m=m_N\to 0$. For the case $N{m}_N\to \infty$ we show that on the time scale $1/m_N$ condensates emerge from general homogeneous initial conditions, and we precisely characterize their limiting dynamics. In the simplest case of two sites or a fully connected underlying random walk kernel, there is a single condensate hopping over $S$ as a continuous-time random walk. In the non fully connected case several condensates can coexist and exchange mass via intermediate sites in an interesting coarsening process, which consists of a mixture of a diffusive motion and a jump process, until a single condensate is formed. Our result is based on a general two-scale form of the generator, with a fast-scale neutral Wright-Fisher diffusion and a slow-scale deterministic motion. The motion of the condensates is described in terms of the generator of the deterministic motion and the harmonic projection corresponding to the absorbing states of the Wright-Fisher diffusion.

30 pages, 2 figures; minor corrections, including proof of Lemma 3.5

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• Arrhenius law for interacting diffusive systems
• Condensation of Non-Reversible Zero-Range Processes
• Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions
• Coarsening dynamics in condensing zero-range processes and size-biased birth death chains
• Condensation and Metastable Behavior of Non-reversible Inclusion Processes
• Derivation of mean-field equations for stochastic particle systems
• Current large deviations for partially asymmetric particle systems on a ring
• Explosive condensation in symmetric mass transport models
• Monotonicity and Condensation in Homogeneous Stochastic Particle Systems
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• Weakly Non-Equilibrium Properties of Symmetric Inclusion Process with Open Boundaries
• Emerging universality classes in thermally-assisted activation of interacting diffusive systems: A perturbative hydrodynamic approach
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• Size-biased diffusion limits and the inclusion process
• Second Time Scale of the Metastability of Reversible Inclusion Processes
• Coupling independent walkers and the inclusion process
• Quantitative results for the Fleming-Viot Particle system and quasi-stationary distributions in discrete space
• Spectral gap of the symmetric inclusion process
• Condensation in zero-range processes with a fast rate