Nonlinear diffusion limit for a system with nearest neighbor interactions

Guo, Maozheng; Papanicolaou, George; Varadhan, S. R. S.

doi:10.1007/bf01218476

Cited by 297 publications

(258 citation statements)

References 3 publications

Supporting

Mentioning

256

Contrasting

Order By: Relevance

“…Let Σ N,k denote the hyperplane of configurations with k particles, namely 4) which is invariant under the dynamics. We say that O is an irreducible component of Σ N,k if for every η, ξ ∈ O it is possible to go from η to ξ by a sequence of allowed jumps.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Hydrodynamic limit for a particle system with degenerate rates

Gonçalves¹,

Landim²,

Toninelli³

2009

Ann. Inst. H. Poincaré Probab. Statist.

107

View full text Add to dashboard Cite

Abstract. We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong to the class of kinetically constrained lattice gases (KCLG) which have been introduced and intensively studied in physics literature as simple models for the liquid/glass transition. Due to the degeneracy of rates for KCLG there exists blocked configurations which do not evolve under the dynamics and in general the hyperplanes of configurations with a fixed number of particles can be decomposed into different irreducible sets. As a consequence, both the Entropy and Relative Entropy method cannot be straightforwardly applied to prove the hydrodynamic limit. In particular, some care should be put when proving the One and Two block Lemmas which guarantee local convergence to equilibrium. We show that, for initial profiles smooth enough and bounded away from zero and one, the macroscopic density profile for our KCLG evolves under the diffusive time scaling according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed dynamics obtained by adding jumps of the Symmetric Simple Exclusion. The role of the latter is to remove the degeneracy of rates and at the same time they are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained.

show abstract

Section: Statement Of Resultsmentioning

confidence: 99%

Hydrodynamic limit for a particle system with degenerate rates

Gonçalves¹,

Landim²,

Toninelli³

2009

Ann. Inst. H. Poincaré Probab. Statist.

107

View full text Add to dashboard Cite

show abstract

“…Their proof is an adaptation of the proof in [7] and it requires the reversibility and the translation invariance of the measure ν ρ . In [9], a simpler proof of Proposition 4.1 was obtained, building up in the one-block estimate introduced in [18]. Following the methods in [10], the proof in [8] can be adapted for the mean-zero exclusion process defined in Section 2.2.…”

Section: Proposition 41 ([13 9]) Let F : ω → R Be a Local Functionmentioning

confidence: 99%

“…In [17] an important technical novelty, the so-called second-order Boltzmann-Gibbs principle was introduced. The idea was to extend the one-block and the two-blocks setup introduced in [18] to the fluctuation level.…”

Section: Introductionmentioning

confidence: 99%

Scaling Limits of Additive Functionals of Interacting Particle Systems

Gonçalves

Jara

2013

Comm Pure Appl Math

View full text Add to dashboard Cite

ABSTRACT. Using the renormalization method introduced in [17], we prove what we call the local Boltzmann-Gibbs principle for conservative, stationary interacting particle systems in dimension d = 1. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by-product of these results, we also construct a novel process, related to the stationary solution of the stochastic Burgers equation.

show abstract

“…This is a classical problem in the theory of hydrodynamic limits, and various methods have been devised to tackle it. [28][29][30] In the context of a Ginzburg-Landau model, Guo, Papanicolaou, and Varadhan 28,31 show that a term of the form (23) can be replaced by a function of π N . More precisely, for ψ : T → R smooth and any > 0,…”

Section: The Replacement Lemmamentioning

confidence: 99%

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

Peletier

Redig

Vafayi

2014

Journal of Mathematical Physics

View full text Add to dashboard Cite

We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter m, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP(m) and the KMP, and a nonlinear heat equation for the GBEP(a). We prove the hydrodynamic limit rigorously for the BEP(m), and give a formal derivation for the GBEP(a).We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form − log ρ; they involve dissipation or mobility terms of order ρ 2 for the linear heat equation, and a nonlinear function of ρ for the nonlinear heat equation.

show abstract

Nonlinear diffusion limit for a system with nearest neighbor interactions

Cited by 297 publications

References 3 publications

Hydrodynamic limit for a particle system with degenerate rates

Hydrodynamic limit for a particle system with degenerate rates

Scaling Limits of Additive Functionals of Interacting Particle Systems

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

Contact Info

Product

Resources

About