Tuesday 9 July 2019 16:00 – 17:00 A. Payatakes Seminar Room
“A generalization of Young measures for the Hydrodynamic limit of condensing Zero Range Processes”
Dr. Marios Stamatakis Institute of Applied and Computational Mathematics (IACM)
Abstract
Zero range processes (ZRPs) are stochastic interacting particle systems with zero range interaction. For particular choices of their parameters they exhibit phase separation with the emergence of a condensate. Such ZRPs are referred to as condensing and their hydrodynamic limit is not known, but is expected to be a degenerate non-linear diffusion equation where the diffusivity vanishes above a critical density ρc. In this talk we propose a generalization of the notion of Young-measures which allows to obtain a closed equation as the hydrodynamic limit of condensing ZRPs. We focus on symmetric ZRPs in the discrete torus and prove that the laws of the empirical density of the ZRP in terms of generalized Young-measures are concentrated on generalized Young measure-valued weak solutions π = (πt)t ≥ 0 to the equation θtπ = ΔΦ(π) where Φ(λ) is the mean jump rate of particles under an equilibrium state of mean density λ ≥ 0. Via the barycentric projection of Young measures and partial progress on the replacement lemma we can pass from the description in terms of generalized Young measures to ordinary measures and thus obtain that in the hydrodynamic limit the fluid phase ρ and the condensed phase ρ┴ οf the ZRP satisfy the equation θtρ – ΔΦ (ρ) = – θt ρ┴. Consequently, the fluid phase evolves according to a non-linear diffusion equation with a source term equal to the negative time derivative of the condensed phase.