A gradient model for the Bernstein polynomial basis

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.

Figures (3)

• Figure 1: PMM rate ($n=4$) for an exchange in the node $\{x,x+1\}$. The patterned rectangles represent the boxes where there are $n$ aligned particles around $\{x,x+1\}$. The total rate is $2/5$.
• Figure 2: Bernstein model ($n=2,L=4$) rate for an exchange in the node $\{x,x+1\}$. The patterned rectangle represents the box containing $\{x,x+1\}$ with exactly $n+1$ particles. The total rate is $1/5$.
• Figure 3: Reduced PMM ($\ell=2,L=4$) rates for a jump in $\{x,x+1\}$. The patterned rectangles represent the boxes containing $\{x,x+1\}$ with at least two particles. The total rate is $\tfrac{3}{5\binom{4}{2}}$.

Theorems & Definitions (20)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: Bernstein model
• Definition 2.5: Reduced Porous Media Model
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9
• Lemma 3.1: Inversion formula