On the infinite particle limit in Lagrangian dynamics and convergence of optimal transportation meshfree methods

TL;DR

This paper proves that as the number of particles increases, the discrete Lagrangian systems converge to a continuum model, providing a theoretical foundation for meshfree methods in particle flow simulations.

Contribution

It establishes Gamma-convergence of discrete action functionals to a continuum functional and analyzes the convergence of stationary points and minimizers in particle systems.

Findings

Gamma-convergence of discrete to continuum action functionals

Convergence of stationary points to Euler-Lagrange flow orbits

Validation of meshfree methods for particle flow approximation

Abstract

We consider Lagrangian systems in the limit of infinitely many particles. It is shown that the corresponding discrete action functionals Gamma-converge to a continuum action functional acting on probability measures of particle trajectories. Also the convergence of stationary points of the action is established. Minimizers of the limiting functional and, more generally, limiting distributions of stationary points are investigated and shown to be concentrated on orbits of the Euler-Lagrange flow. We also consider time discretized systems. These results in particular provide a convergence analysis for optimal transportation meshfree methods for the approximation of particle flows by finite discrete Lagrangian dynamics.