Derivation of the Euler Equations¶from a Caricature of Coulomb Interaction

Abstract

A caricature of collisionless plasma involving 2N particles of opposite charge is introduced. The N first particles are called “ions” and don't move. The N other particles are called “electrons”. At each time, there is a one-to-one matching between electrons and ions and each pair is linked by a “spring” so that each electron oscillates with fixed frequency e−1. The essential point is that the matching between electrons and ions is updated at every discrete time nτ, n≡0,1,2,..., so that the total potential energy of the system stays minimal. This leads to a non trivial interaction which turns out to be a caricature of Coulomb interaction. It is proven that, provided the N ions are equally spaced in a bounded domain D and e, τ and N−1 tend to zero at appropriate rates, the electrons behave as the fluid parcels of an incompressible inviscid liquid moving inside D according to the Euler equations. Our proof relies on a result of P. Lax on the approximation of volume-preserving transformations by permutations.