Lattice model for fast diffusion equation

Hernández, Freddy; Jara, Milton; Valentim, Fábio

doi:10.1016/j.spa.2019.08.004

Cited by 4 publications

(3 citation statements)

References 18 publications

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“…The Stefan problem has been derived from microscopic dynamics in a few, less degenerate contexts [14,10,5]. We note that the same FDE was derived recently from a non-degenerate zero-range process under a proper high density limit [11].…”

Section: Introductionmentioning

confidence: 65%

Hydrodynamic limit for a facilitated exclusion process

Blondel¹,

Erignoux²,

Sasada³

et al. 2020

Ann. Inst. H. Poincaré Probab. Statist.

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We study the hydrodynamic limit for a periodic 1-dimensional exclusion process with a dynamical constraint, which prevents a particle at site x from jumping to sitex ± 1 unless site x ∓ 1 is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than 1 2 , or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density 1 2 , the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

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Section: Introductionmentioning

confidence: 65%

Hydrodynamic limit for a facilitated exclusion process

Blondel¹,

Erignoux²,

Sasada³

et al. 2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…Related problems are discussed by Caputo et al [3], [4]. Similar idea is used in Hernández et al [23] to derive the fast diffusion equation from zero-range processes. Bertini et al [2] discussed from the viewpoint of large deviation functionals.…”

Section: Our Model and Main Resultsmentioning

confidence: 91%

Motion by Mean Curvature from Glauber–Kawasaki Dynamics

Funaki

Tsunoda

2019

J Stat Phys

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We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen-Cahn equation which is a kind of the reaction-diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a combination of the hydrodynamic and sharp interface limits.

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“…Finally, concerning the fast diffusion case, few results are available in the literature. In [15] the FDE with m = −1 has been derived as the hydrodynamic limit of a zero-range process (the number of particles per site can be any non-negative integer) evolving on the discrete torus, with a jump rate function adjusted to observe frequently a large number of particles, with a specific "weight" associated to each particle. The formalization of the hydrodynamic limit was achieved by using Yau's relative entropy method [22] with some adaptations including spectral gap estimates.…”

Section: Introductionmentioning

confidence: 99%

From Exclusion to Slow and Fast Diffusion

Gonçalves¹,

Nahum²,

Simon³

2023

Preprint

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We construct a nearest-neighbour interacting particle system of exclusion type, which illustrates a transition from slow to fast diffusion. More precisely, the hydrodynamic limit of this microscopic system in the diffusive space-time scaling is the parabolic equation ∂t ρ = ∇(D(ρ)∇ρ), with diffusion coefficient D(ρ) = mρ m−1 where m ∈ (0, 2], including therefore the fast diffusion regime in the range m ∈ (0, 1), and the porous medium equation for m ∈ (1, 2). The construction of the model is based on the generalized binomial theorem, and interpolates continuously in m the already known microscopic porous medium model with parameter m = 2, the symmetric simple exclusion process with m = 1, going down to a fast diffusion model up to any m > 0. The derivation of the hydrodynamic limit for the local density of particles on the one-dimensional torus is achieved via the entropy method -with additional technical difficulties depending on the regime (slow or fast diffusion) and where new properties of the porous medium model need to be derived.

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Lattice model for fast diffusion equation

Cited by 4 publications

References 18 publications

Hydrodynamic limit for a facilitated exclusion process

Hydrodynamic limit for a facilitated exclusion process

Motion by Mean Curvature from Glauber–Kawasaki Dynamics

From Exclusion to Slow and Fast Diffusion

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