Generalized Young measures and the hydrodynamic limit of condensing zero range processes

Abstract

Condensing zero range processes (ZRPs) are stochastic interacting particle systems that exhibit phase separation with the emergence of a condensate. Standard approaches for deriving a hydrodynamic limit of the density fail in these models, and an effective macroscopic description has not been rigorously established, yet. In this article we prove that the limiting triple $(π,W,σ)$ of the empirical density, the empirical current, and the empirical jump rate of the ZRP satisfies the continuity equation $\partial_tπ=-{\rm{div}}W$ in the sense of distributions. Here $(π_t)_{t\geq 0}$ is a $w^*$-continuous curve of finite non-negative measures on the torus $\mathbb{T}^d$, $σ_t\in H^1(\mathbb{T}^d)$ and $W_t=-\nablaσ_t$ is a vector-valued measure that is absolutely continuous with respect to the Lebesgue measure, for all almost all $t\geq 0$. In order to obtain a closed equation we propose a generalization of Young measures and we prove that for symmetric ZRPs on the torus, the hydrodynamic limit of the density is a generalized Young-measure-valued weak solution $\boldsymbolπ=(\boldsymbolπ_t)_{t\geq 0}$ to a saturated filtration equation $\partial_t\boldsymbolπ=ΔΦ(\boldsymbolπ)$. Furthermore we prove a one-sided two-blocks estimate and we give an equivalent criterion for its validity. Assuming the validity of the two-blocks estimate one obtains the equation $\partial_tπ=ΔΦ(π^{ac})$ for the empirical density, where $π=π^{ac}+π^\perp$ is the Radon-Nikodym decomposition.