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

TL;DR

This paper analyzes a Blume-Capel spin model with mixed boundary conditions, establishing hydrodynamic limits and large deviations for the coupled magnetization and concentration measures under weakly asymmetric dynamics.

Contribution

It introduces a novel boundary-driven Blume-Capel model with mixed Dirichlet-Neumann conditions and derives hydrodynamic limits and large deviations for the coupled empirical measures.

Findings

Hydrodynamic limit established for the coupled magnetization and concentration.

Large deviations principle proved for the model's dynamics.

Model incorporates asymmetric boundary conditions affecting bulk behavior.

Abstract

We consider the Blume-Capel spin model on a finite cylinder with reservoirs at the boundary. A model with spin variable $\sigma$ 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.