Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the correlations of some microscopic random systems

P. Gonçalves, B. Salvador

Abstract

We investigate the two-points correlation function for several boundary-driven interacting particle systems. Our goal is to show that the time evolution of that correlation function is solution to a partial differential equation that can be written in terms of the generator of a two-dimensional random walk, whose jump rates are model dependent. From this, we deduce an asymptotic independence which is shared by many models.

On the correlations of some microscopic random systems

Abstract

We investigate the two-points correlation function for several boundary-driven interacting particle systems. Our goal is to show that the time evolution of that correlation function is solution to a partial differential equation that can be written in terms of the generator of a two-dimensional random walk, whose jump rates are model dependent. From this, we deduce an asymptotic independence which is shared by many models.

Paper Structure

This paper contains 28 sections, 2 theorems, 143 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our result in a nutshell
  3. Consequence of our results
  4. Possible future work
  5. Outline of the article
  6. The models: interacting particles
  7. The dynamics
  8. Invariant measures
  9. Correlation estimates
  10. The discrete density profile
  11. The correlation function
  12. Stationary correlations
  13. Other models
  14. Interacting diffusions
  15. Ginzburg-Landau dynamics
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\varphi^N_t$ denote the two-points correlation function, defined on $(x,y) \in (\Lambda_N)^2$ with $y \neq x$, by Then, there exists a function $f:S \to \mathbb{R}$, that can be written in terms of the quantities $(\eta(x))^2$,$\eta(x)$ and $\varrho_t^N(x)$, and that allows defining the two-points correlation function at the diagonal points $y=x$, i.e. for every $x \in \Lambda_N$ as in such

Figures (2)

  • Figure 2.1: Illustration of the sets $\partial T_N$ (in orange) and $\mathcal{D}_N$ (in cyan).
  • Figure 2.2: Jump rates of the random walk $(\mathcal{X}_{t})_{t \geq 0}$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.1.1
  • Definition 2.1: Two-points correlation function
  • Remark 1
  • Remark 2