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Boundary driven weakly asymmetric Blume-Capel model: Large deviations for mixed Dirichlet-Neumann boundary conditions

Mustapha Mourragui, Nicolas Prévost

Abstract

We consider the Blume-Capel spin model on a finite cylinder with reservoirs at the boundary. A model with spin variable $σ$ taking values in {-1, 0, 1}, with the superposition of two dynamics: in the bulk, the spins evolve according to a weakly asymmetric dynamics; and the boundary dynamics follows a mechanism of creation, annihilation and spin flip, its action is accelerated differently on the left and on the right in a way to produce mixed boundary conditions. For the dynamics in the bulk, two quantities are conserved, the magnetization which corresponds to the sum of the spin values, and the concentration which corresponds to the sum of the squared spin values. We first establish, in the diffusive scaling, the hydrodynamic limit for this model which states that the couple of empirical measures (magnetization, concentration) converges to the solution of a system of coupled equations with mixed boundary conditions. Then we prove the associated dynamical large deviations principle.

Boundary driven weakly asymmetric Blume-Capel model: Large deviations for mixed Dirichlet-Neumann boundary conditions

Abstract

We consider the Blume-Capel spin model on a finite cylinder with reservoirs at the boundary. A model with spin variable taking values in {-1, 0, 1}, with the superposition of two dynamics: in the bulk, the spins evolve according to a weakly asymmetric dynamics; and the boundary dynamics follows a mechanism of creation, annihilation and spin flip, its action is accelerated differently on the left and on the right in a way to produce mixed boundary conditions. For the dynamics in the bulk, two quantities are conserved, the magnetization which corresponds to the sum of the spin values, and the concentration which corresponds to the sum of the squared spin values. We first establish, in the diffusive scaling, the hydrodynamic limit for this model which states that the couple of empirical measures (magnetization, concentration) converges to the solution of a system of coupled equations with mixed boundary conditions. Then we prove the associated dynamical large deviations principle.
Paper Structure (34 sections, 32 theorems, 272 equations)

This paper contains 34 sections, 32 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Description of the model
  3. The results
  4. Notation
  5. Hydrodynamic limit
  6. Dynamical large deviations
  7. Basic tools
  8. Basic inequalities
  9. Sobolev spaces
  10. Classical Sobolev spaces and Trace operator
  11. Weighted Sobolev spaces
  12. Superexponential estimates (Replacement lemmas)
  13. Dirichlet form estimates
  14. Replacement lemma in the bulk
  15. Replacement lemma at the left-hand side boundary
  16. ...and 19 more sections

Key Result

Proposition 3.1

For any sequence of initial probability measures $(\mu_N)_{N\ge 1}$, the sequence of probability measures $(\mathbb{Q}_{\mu_N}^{N,\widehat{b}})_{N\geq 1}$ is weakly relatively compact and all its converging subsequences converge to some limit $\mathbb{Q}^{\widehat{b},*}$ that is concentrated on the Moreover, if the sequence of initial measures $(\mu_N)_{N\geq 1}$ is associated to some continuous

Theorems & Definitions (58)

  • Proposition 3.1
  • Theorem 3.2
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Corollary 4.3
  • Lemma 4.4
  • proof
  • Remark 4.5
  • Lemma 4.6
  • ...and 48 more