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Fractional Macroscopic Fluctuation Theory for a Superdiffusive Ginzburg-Landau dynamics

Cedric Bernardin, Patricia Gonçalves, João Pedro Mangi

Abstract

We investigate a boundary-driven Ginzburg-Landau dynamics with long-range interactions. In the hydrodynamic limit, the macroscopic evolution is governed by a fractional heat equation with Dirichlet boundary conditions, while the corresponding stationary profile is characterized by a fractional Laplace equation. We establish a dynamical large deviations principle for the empirical measure and derive the associated stationary large deviations principle for the non-equilibrium steady state, which can be computed semi-explicitly. We further show that the stationary rate function coincides with the quasi-potential associated with the dynamical large deviations functional.

Fractional Macroscopic Fluctuation Theory for a Superdiffusive Ginzburg-Landau dynamics

Abstract

We investigate a boundary-driven Ginzburg-Landau dynamics with long-range interactions. In the hydrodynamic limit, the macroscopic evolution is governed by a fractional heat equation with Dirichlet boundary conditions, while the corresponding stationary profile is characterized by a fractional Laplace equation. We establish a dynamical large deviations principle for the empirical measure and derive the associated stationary large deviations principle for the non-equilibrium steady state, which can be computed semi-explicitly. We further show that the stationary rate function coincides with the quasi-potential associated with the dynamical large deviations functional.

Paper Structure

This paper contains 41 sections, 49 theorems, 380 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ginzburg-Landau dynamics and main results
  3. The microscopic model
  4. Fractional Operators
  5. Regional fractional Laplacian on $[0,1]$
  6. Discrete fracional Laplacian on $\tfrac{\Lambda_n}{n}$
  7. Hydrodynamic and Hydrostatic Equations
  8. Hydrodynamic and Hydrostatic Limits
  9. Dynamical Large Deviations
  10. Static Large Deviations
  11. The Quasi-Potential
  12. Hydrodynamic Limit
  13. Dynkin's martingales
  14. Tightness
  15. Characterization of limit points
  16. ...and 26 more sections

Key Result

Lemma 2.1

The NESS $\mu_{ss}^n$ of the boundary driven Ginzburg-Landau dynamics with long range interactions has a density $f_{ss}^n$ with respect to the Lebesgue measure $d\varphi =\prod_{x\in \Lambda_n} d\varphi (x)$ in product form where $\Phi^n_{ss}:\Lambda_n\to[\Phi_\ell,\Phi_r]$ is the unique solution to the equation for all $x\in\Lambda_n$. In particular, $E_{\mu_{ss}^n} [\varphi (x)] = \Phi_{ss}^n

Figures (1)

  • Figure 1: Graph of the stationary profile $\Phi^n_{ss}$ for $\Phi_\ell=1$, $\Phi_r=2$ and $n=200$.

Theorems & Definitions (95)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 2.10: Hydrodynamic limit
  • ...and 85 more