The Boundary Driven Zero-Range Process

Frómeta, Susana; Misturini, Ricardo; Neumann, Adriana

doi:10.1007/978-3-030-69784-6_12

Cited by 6 publications

(2 citation statements)

References 26 publications

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“…Here we considered the case Φ(ρ κ ) = (ρ κ ) m and ρ κ ∈ [0, 1], since it was the natural equation arising from our microscopic system. When dealing with, for example, zero-range dynamics or exclusion dynamics with more than one particle per site, this is the natural candidate equation that one should obtain in the hydrodynamic limit for those models, see [7,10]. We intend to pursue the extension of our results to those cases in the near future, but one of the obstacles to be overcome is obtaining a uniform bound on κ for the L 2 -norm of ρ κ t .…”

Section: Introductionmentioning

confidence: 99%

Energy estimates and convergence of weak solutions of the porous medium equation

2021

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We study the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, with respect to the L 2-norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity). Our approach is to consider an underlying microscopic dynamics whose space–time evolution of the density is ruled by the solution of those equations and from this, we derive sufficiently strong energy estimates which are the keystone to the proof of our convergence result.

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Section: Introductionmentioning

confidence: 99%

Energy estimates and convergence of weak solutions of the porous medium equation

2021

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“…We also mention that 1D systems with 'reservoir' boundaries have been studied; see [9] for a discussion of 'hydrostatics', and related references, with respect to exclusion models. In zero-range processes, 'static' reservoir effects are studied and dynamical conjectures are discussed in [15].…”

Section: Introductionmentioning

confidence: 99%

Condensation, boundary conditions, and effects of slow sites in zero-range systems

Sethuraman¹,

Xue²

2022

Preprint

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We consider the space-time scaling limit of the particle mass in zero-range particle systems on a 1D discrete torus Z/N Z with a finite number of defects. We focus on two classes of increasing jump rates g, when g(n) ∼ n α , for 0 < α ≤ 1, and when g is a bounded function. In such a model, a particle at a regular site k jumps equally likely to a neighbor with rate g(n), depending only on the number of particles n at k. At a defect site k j,N , however, the jump rate is slowed down to λ −1 j N −β j g(n) when g(n) ∼ n α , and to λ −1 j g(n) when g is bounded. Here, N is a scaling parameter where the grid spacing is seen as 1/N and time is speeded up by N 2 .Starting from initial measures with O(N ) relative entropy with respect to an invariant measure, we show the hydrodynamic limit and characterize boundary behaviors at the macroscopic defect sites x j = lim N↑∞ k j,N /N , for all defect strengths. For rates g(n) ∼ n α , at critical or super-critical slow sites (β j = α or β j > α), associated Dirichlet boundary conditions arise as a result of interactions with evolving atom masses or condensation at the defects. Differently, when g is bounded, at any slow site (λ j > 1), we find the hydrodynamic density must be bounded above by a threshold value reflecting the strength of the defect. Moreover, due to interactions with masses of atoms stored at the slow sites, the associated boundary conditions bounce between being periodic and Dirichlet.

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Classification of stationary distributions for the stochastic vertex models

Lin

2023

Electron. J. Probab.

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The Boundary Driven Zero-Range Process

Cited by 6 publications

References 26 publications

Energy estimates and convergence of weak solutions of the porous medium equation

Energy estimates and convergence of weak solutions of the porous medium equation

Condensation, boundary conditions, and effects of slow sites in zero-range systems

Classification of stationary distributions for the stochastic vertex models

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