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Equilibrium fluctuations for a multi-species particle system with long jumps

Giuseppe Cannizzaro, Pedro Cardoso, Lukas Gräfner, Alessandra Occelli

Abstract

In the present paper, we study the equilibrium fluctuations of a particle system in infinite volume with two conserved quantities and long-range dependence. More specifically, the model of interest is the so-called ABC model, in which three types of particles (A, B and C) exchange their locations between $x\in\mathbb{Z}$ and $x+z\in\mathbb{Z}$ at a rate that depends on the type of particles involved and is proportional to $|z|^{-γ-1}$ for $γ>0$. After rigorously identifying the normal modes associated to the conserved quantities (the density of particles of types $A$ and $B$, say), we prove that their fluctuations converge to independent fractional stochastic partial differential equations (SPDEs), which are either Gaussian or the Stochastic Burgers equation, and whose nature is determined by the microscopic range of dependence and the strength of the asymmetry.

Equilibrium fluctuations for a multi-species particle system with long jumps

Abstract

In the present paper, we study the equilibrium fluctuations of a particle system in infinite volume with two conserved quantities and long-range dependence. More specifically, the model of interest is the so-called ABC model, in which three types of particles (A, B and C) exchange their locations between and at a rate that depends on the type of particles involved and is proportional to for . After rigorously identifying the normal modes associated to the conserved quantities (the density of particles of types and , say), we prove that their fluctuations converge to independent fractional stochastic partial differential equations (SPDEs), which are either Gaussian or the Stochastic Burgers equation, and whose nature is determined by the microscopic range of dependence and the strength of the asymmetry.

Paper Structure

This paper contains 44 sections, 53 theorems, 326 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The ABC model
  3. Choice for the rates and the transition probability
  4. Invariant measures
  5. Density fluctuation field
  6. Martingale problems
  7. Main results
  8. Proof of Proposition \ref{['tightseqZpm']}
  9. Convergence of $( \mathcal{Z}_0^n(H) \; )_{n \in \mathbb{N}_+}$
  10. Convergence of $\{ ( \mathcal{M}_t^{n} )_{0 \leq t \leq T}: \; n \in \mathbb{N}_+ \}$
  11. Expansion of the Dynkin martingale
  12. Proof of Lemmas \ref{['varnlinvan']} and \ref{['varnlintight']}
  13. Characterization of the limit points
  14. Identification of the drift operator.
  15. Decoupling of the + and - fields.
  16. ...and 29 more sections

Key Result

Proposition 2.2

Let $\nu$ be a probability measure on $\Omega$ satisfying Then, under condition pairbal, the measure $\nu$ is invariant with respect to the dynamics given by ${\mathcalboondox L}^n$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: The ABC model with long jumps on $\mathbb{Z}$. Each site $x\in\mathbb{Z}$ carries exactly one particle of type $A$ (red), $B$ (blue) or $C$ (green). Two particles at sites $x$ and $y$ with distinct types $\alpha=\eta(x)$, $\beta=\eta(y)$ exchange at rate $\Theta(n)\,p(y-x)\,r^n_{\alpha,\beta}$, where $p(z)\sim c^\pm|z|^{-1-\gamma}$ is the long-range jump kernel \ref{['genABC']} and $r^n_{\alpha,\beta}=1+K_n(E_\alpha-E_\beta)$ are the species-dependent rates \ref{['rateABC']}.

Theorems & Definitions (111)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 101 more