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Mean field limits of a class of conservative systems with position-dependent transition rates

Xiaofeng Xue

Abstract

In this paper, we are concerned with a class of conservative systems including asymmetric exclusion processes and zero-range processes as examples, where some particles are initially placed on $N$ positions. A particle jumps from a position to another at a rate depending on coordinates of these two positions and numbers of particles on these two positions. We show that the hydrodynamic limit of our model is driven by a nonlinear function-valued ordinary differential equation which is consistent with a mean field analysis. Furthermore, in the case where numbers of particles on all positions are bounded by $\mathcal{K}<+\infty$, we show that the fluctuation of our model is driven by a generalized Ornstein-Uhlenbeck process. A crucial step in proofs of our main results is to show that numbers of particles on different positions are approximately independent by utilizing a graphical method.

Mean field limits of a class of conservative systems with position-dependent transition rates

Abstract

In this paper, we are concerned with a class of conservative systems including asymmetric exclusion processes and zero-range processes as examples, where some particles are initially placed on positions. A particle jumps from a position to another at a rate depending on coordinates of these two positions and numbers of particles on these two positions. We show that the hydrodynamic limit of our model is driven by a nonlinear function-valued ordinary differential equation which is consistent with a mean field analysis. Furthermore, in the case where numbers of particles on all positions are bounded by , we show that the fluctuation of our model is driven by a generalized Ornstein-Uhlenbeck process. A crucial step in proofs of our main results is to show that numbers of particles on different positions are approximately independent by utilizing a graphical method.
Paper Structure (9 sections, 17 theorems, 237 equations)

This paper contains 9 sections, 17 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Approximated independence
  4. The proof of Lemma \ref{['lemma 3.1 approximated independence finite K']}
  5. The proof of Lemma \ref{['lemma 3.2 approximated independence infinite K']}
  6. The proof of Theorem \ref{['theorem 2.1 mean field limit finite']}
  7. The proof of Theorem \ref{['theorem 2.2 mean field limit infinite']}
  8. The proof of Theorem \ref{['theorem 2.3 fluctuation limit']}
  9. Applications

Key Result

Theorem 2.1

If $\mathcal{K}<+\infty$, then the solution $\{\rho^{\mathcal{K}}_{t,k}\}_{0\leq k\leq \mathcal{K}}$ to Equation equ 2.1 mean field ODE finite types exists for $t\in [0, +\infty)$ and is unique. Furthermore, if $\mathcal{K}<+\infty$, then, under Assumption (A), in $L^2$ for any $t\geq 0$, $0\leq k\leq \mathcal{K}$ and $f\in C(\mathbb{T})$.

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 4.1
  • ...and 7 more