[Paper Review] Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps

Cédric Bernardin, Patrícia Gonçalves+1 more|arXiv (Cornell University)|Feb 23, 2017

Stochastic processes and statistical mechanics19 references21 citations

TL;DR

This paper studies the symmetric exclusion process with long-range jumps on a finite interval coupled to infinitely extended reservoirs, where reservoir interaction rates scale as $\kappa N^{-\theta}$. For $\gamma > 2$ (finite variance), the hydrodynamic limit yields reaction-diffusion equations with boundary conditions that transition between Dirichlet, Robin, and Neumann types depending on $\theta$, revealing a phase transition in macroscopic behavior as $\theta$ varies.

ABSTRACT

We consider an exclusion process with long jumps in the box $Λ\_N=\{1, \ldots,N-1\}$, for $N \ge 2$, in contact with infinitely extended reservoirs on its left and on its right. The jump rate is described by a transition probability $p(\cdot)$ which is symmetric, with infinite support but with finite variance. The reservoirs add or remove particles with rate proportional to $κN^{-θ}$, where $κ>0$ and $θ\in\mathbb R$. If $θ>0$ (resp. $θ<0$) the reservoirs add and fastly remove (resp. slowly remove) particles in the bulk. According to the value of $θ$ we prove that the time evolution of the spatial density of particles is described by some reaction-diffusion equations with various boundary conditions.

Motivation & Objective

• To understand how reservoir coupling strength, controlled by $\theta$, affects the macroscopic hydrodynamic behavior of an exclusion process with long jumps.
• To extend prior results on nearest-neighbor systems with slow boundaries to long-range jump processes with infinite-variance and symmetric transition rates.
• To characterize the emergence of different boundary conditions (Dirichlet, Robin, Neumann) in the hydrodynamic limit based on the value of $\theta$.
• To establish the hydrodynamic limit for a model with infinitely extended reservoirs and long-range dynamics, under the condition $\gamma > 2$ for finite variance.

Proposed method

• The model uses a symmetric, heavy-tailed jump transition rate $p(z) \sim |z|^{-1-\gamma}$ with $\gamma > 2$, ensuring finite variance.
• Reservoirs at the boundaries inject or remove particles at a rate $\kappa N^{-\theta}$, allowing control over coupling strength via $\theta \in \mathbb{R}$.
• The hydrodynamic limit is derived via a relative entropy method and weak solution techniques, analyzing the time evolution of particle density profiles.
• The analysis identifies five distinct macroscopic phases based on $\theta$, each corresponding to different boundary conditions in the limiting reaction-diffusion equation.
• The proof relies on convergence of empirical measures and weak convergence to solutions of PDEs, with careful estimates using Cauchy-Schwarz and density arguments.
• The generator of the process is decomposed into bulk and boundary components to analyze particle exchange and derive the limiting equations.

Experimental results

Research questions

• RQ1 How does the coupling strength $\theta$ of infinitely extended reservoirs affect the macroscopic hydrodynamic behavior of a symmetric exclusion process with long jumps?
• RQ2 What types of boundary conditions (Dirichlet, Robin, Neumann) emerge in the hydrodynamic limit, and how do they depend on $\theta$?
• RQ3 Does the presence of long-range jumps with finite variance alter the phase structure compared to nearest-neighbor models with slow boundaries?
• RQ4 Can the hydrodynamic limit be rigorously established for a system with infinitely extended reservoirs and long-range dynamics under symmetric, finite-variance jump rates?
• RQ5 What is the role of the parameter $\gamma$ in determining the nature of the limiting PDE and the resulting boundary conditions?

Key findings

• For $\theta \in (2 - \gamma, 1)$, the hydrodynamic limit is the heat equation with Dirichlet boundary conditions, independent of $\theta$ in this interval.
• At $\theta = 1$, the system transitions to a reaction-diffusion equation with Robin-type boundary conditions due to strong reservoir coupling.
• For $\theta > 1$, the boundary conditions become Neumann-type, indicating particle conservation at the boundaries.
• When $\theta < 2 - \gamma$, the system exhibits a different macroscopic behavior, with boundary effects dominating and leading to distinct solution profiles.
• The hydrodynamic limit is characterized as a weak solution of a reaction-diffusion equation, with uniqueness established via energy estimates and density arguments.
• The analysis confirms that the macroscopic behavior undergoes a phase transition as $\theta$ varies, with five distinct phases identified based on the value of $\theta$.