Continuously Parametrised Porous Media Model and Scaling Limits of Kinetically Constrained Models
G. S. Nahum
TL;DR
The paper develops a continuous interpolation of Porous Media Models within kinetically constrained, one-dimensional symmetric exclusion, via Bernstein-based gradient dynamics that extend PMMs to non-integer parameters. It establishes a robust hydrodynamic limit for a perturbed exclusion process under broad, simple rate conditions, showing convergence of the empirical density to a PDE of the form $\partial_t\rho = \partial_u^2\Phi(\rho)$ in Regime I (nonlinear diffusion) or $\partial_t\rho = \partial_u^2\rho$ in Regime II (linear diffusion). The approach decouples from a specific basis, leveraging gradient structure and replacement lemmas to prove tightness and limit identification; energy estimates then guarantee uniqueness of the limit. This provides a versatile framework for modeling nonlinear diffusion arising from kinetically constrained systems and clarifies how Bernstein-type constructions yield continuous diffusion interpolations between PMM cases, expanding the toolkit for macroscopic limits of interacting particle systems.
Abstract
We investigate the emergence of non-linear diffusivity in kinetically constrained, one-dimensional symmetric exclusion processes satisfying the gradient condition. Recent developments introduced new gradient dynamics based on the Bernstein polynomial basis, enabling richer diffusive behaviours but requiring adaptations of existing techniques. In this work, we exploit these models to generalise the Porous Media Model to non-integer parameters and establish simple conditions on general kinetic constraints under which the empirical measure of a perturbed version of the process converges. This provides a robust framework for modelling non-linear diffusion from kinetically constrained systems.