On Hydrodynamic Equations at the Limit of Infinitely Many Molecules

TL;DR

This paper establishes conditions under which the collective behavior of infinitely many interacting molecules converges to hydrodynamic equations, specifically aligning with Maxwell's macroscopic equations, under certain boundedness and interaction assumptions.

Contribution

It proves that weak convergence and moment conditions lead to hydrodynamic limits for interacting molecules, connecting microscopic interactions to macroscopic Maxwell equations.

Findings

Weak convergence implies hydrodynamic equations at the infinite molecule limit.

Conditions are satisfied with bounded initial conditions and weak molecule interactions.

Hydrodynamic equations match Maxwell's macroscopic equations under these conditions.

Abstract

We show that weak convergence of point measures and $(2+\epsilon)$-moment conditions imply hydrodynamic equations at the limit of infinitely many interacting molecules. The conditions are satisfied whenever the solutions of the classical equations for $N$ interacting molecules obey uniform in $N$ bounds. As an example, we show that this holds when the initial conditions are bounded and that the molecule interaction, a certain $N$-rescaling of potentials that include all $r^{-p}$ for $1<p$, is weak enough at the initial time. In this case the hydrodynamic equations coincide with the macroscopic equations of Maxwell.