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From microscopic to macroscopic scale equations: mean field, hydrodynamic and graph limits

Thierry Paul, Emmanuel Trélat

Abstract

Considering finite particle systems, we elaborate on various ways to pass to the limit as thenumber of agents tends to infinity, either by mean field limit, deriving the Vlasov equation,or by hydrodynamic or graph limit, obtaining the Euler equation. We provide convergenceestimates. We also show how to pass from Liouville to Vlasov or to Euler by taking adequatemoments. Our results encompass and generalize a number of known results of the literature.As a surprising consequence of our analysis, we show that sufficiently regular solutions of anylinear PDE can be approximated by solutions of systems of N particles, to within 1/ log log(N ).

From microscopic to macroscopic scale equations: mean field, hydrodynamic and graph limits

Abstract

Considering finite particle systems, we elaborate on various ways to pass to the limit as thenumber of agents tends to infinity, either by mean field limit, deriving the Vlasov equation,or by hydrodynamic or graph limit, obtaining the Euler equation. We provide convergenceestimates. We also show how to pass from Liouville to Vlasov or to Euler by taking adequatemoments. Our results encompass and generalize a number of known results of the literature.As a surprising consequence of our analysis, we show that sufficiently regular solutions of anylinear PDE can be approximated by solutions of systems of N particles, to within 1/ log log(N ).
Paper Structure (96 sections, 53 theorems, 313 equations, 1 figure)

This paper contains 96 sections, 53 theorems, 313 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting
  3. Structure of the article.
  4. General notations
  5. Hölder and Lipschitz mappings.
  6. Probability Radon measures.
  7. Wasserstein distance.
  8. Disintegration.
  9. Tagged partitions.
  10. From microscopic to mesoscopic scale ("from particle to Vlasov", mean field limit)
  11. Particle system
  12. Lagrangian viewpoint: mean field limit and Vlasov equation
  13. Mean field
  14. Vlasov equation
  15. Particular case: the indistinguishable case.
  16. ...and 81 more sections

Key Result

Lemma 1

[Uniform maximal time] For any compact subset $K$ of $\Omega\times\mathrm{IemR}^d$, there exists a maximal time $T_{\max}(K)\in(0,+\infty]$ such that, for any $N\in\mathrm{IemN}^*$, for any $(X,\Xi(0))\in K^N$,With a slight abuse of notation, $(X,\Xi(0))\in K^N$ means that $(x_i,\xi_i(0))\in K$ for

Figures (1)

  • Figure 1: Relationships between particle (microscopic) system, Liouville (probabilistic) equation, Vlasov (mesoscopic, mean field) equation, Euler (macroscopic, graph limit) equation. We do not write the upperscript $N$ in the various formulas to keep a better readability.

Theorems & Definitions (119)

  • Lemma 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Remark 1
  • Theorem 1
  • Corollary 1
  • proof
  • Proposition 1
  • ...and 109 more