For theta >= 1, the empirical density of the symmetric zero-range process with slow boundary reservoirs converges in probability to the unique weak solution of the nonlinear heat equation with Robin (theta = 1) or Neumann (theta > 1) boundary conditions.
A consistency-stability approach to scaling limits of zero-range processes
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abstract
We propose a simple quantitative method for studying the hydrodynamic limit of interacting particle systems on lattices. It is applied to the diffusive scaling of the symmetric Zero-Range Process (in dimensions one and two). The rate of convergence is estimated in a Monge-Kantorovich distance asymptotic to the L^1 stability estimate of Kruzkhov, as well as in relative entropy; and it is uniform in time. The method avoids the use of the so-called ``block estimates''. It is based on a modulated Monge-Kantorovich distance estimate and microscopic stability properties.
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Hydrodynamic Limit of the Symmetric Zero-Range Process with Slow Boundary
For theta >= 1, the empirical density of the symmetric zero-range process with slow boundary reservoirs converges in probability to the unique weak solution of the nonlinear heat equation with Robin (theta = 1) or Neumann (theta > 1) boundary conditions.