A gradient model for the Bernstein polynomial basis

TL;DR

This paper introduces a new symmetric gradient exclusion process that models non-linear diffusivity using Bernstein polynomial basis, filling a gap in the class of kinetically constrained lattice gases and generalizing the Porous Media Model.

Contribution

It presents the first gradient property satisfying model within this class, extending the Porous Media Model and establishing connections via an inversion formula.

Findings

Model exhibits non-linear diffusivity based on local density.

Satisfies the gradient property, filling a theoretical gap.

Extends the Porous Media Model with new auxiliary processes.

Abstract

We introduce a symmetric, gradient exclusion process within the class of non-cooperative kinetically constrained lattice gases, modelling a non-linear diffusivity in which the exchange of occupation values between two neighbouring sites depends on the local density in specific boxes surrounding the pair. The existence of such a model satisfying the gradient property is the main novelty of this work, filling a gap in the literature regarding the types of diffusivities attainable within this class of models. The resulting dynamics exhibits similarities with the Bernstein polynomial basis and generalises the Porous Media Model. We also introduce an auxiliary collection of processes, which extend the Porous Media Model in a different direction and are related to the former process via an inversion formula.