Mean field, hydrodynamic and graph limits for deterministic interacting particle systems: a survey with quantitative estimates

TL;DR

This survey unifies various limits of deterministic interacting particle systems with heterogeneous labels, providing quantitative estimates and clarifying the relationships between different limiting procedures.

Contribution

It introduces a unified framework with quantitative estimates for multiple limiting procedures in particle systems, highlighting their interrelations and conditions for commutation.

Findings

Quantitative convergence estimates for graph and mean field limits.

Clear characterization of when continuum limits derive from hydrodynamic equations.

Analysis of limitations involving singular kernels and stochastic dynamics.

Abstract

We present a unified framework, with quantitative estimates, for deterministic interacting particle systems whose pairwise interactions may depend on heterogeneous labels. Heterogeneity is kept at every level by adding a frozen label variable $x\in\Omega$ to the state. Within this framework we compare several limiting procedures: the direct continuum / graph limit, the mean field limit yielding a Vlasov equation on the extended space of labels and states, the Liouville lift of the particle system together with propagation of chaos through marginals of arbitrary order, and the hydrodynamic moment closures. We give a common language for these limits and identify precisely where the various passages commute and where they do not; in particular, we separate the continuum / graph limit equation from the classical hydrodynamic Euler equations and characterize when the former arises as a moment closure of the latter (linearity in $(\xi,\xi')$ or monokinetic ansatz). Along the way, we prove quantitative convergence estimates for the graph limit and for the passages from particles or Liouville to Vlasov, and we discuss the limitations of the framework, in particular concerning singular kernels and stochastic dynamics. The paper is written as a survey with original contributions, with an emphasis on estimates, examples, and a clear delineation of scope.