Contemporary Mathematics

Volume 330, 2003

Systems of Particles Involving Permutations and their

Continuous Limits

Y.

Brenier

This paper

is

dedicated to Stan Osher on the occasion of his 60th birthday.

ABSTRACT.

We discuss several elementary dynamical systems of particles in-

volving permutations. We show that they can approximate nontrivial models

of continuum Mechanics and Physics. This includes the simplest model of

adhesion dynamics, linked to one-dimensional scalar conservation laws, the

Euler equations of inviscid incompressible fluids, some models in Electrody-

namics and Geophysics (such as the semigeostrophic equations for atmospheric

fronts) and, finally, isothermal gas dynamics equations through the concept of

harmonic functions "up to rearrangement".

1. Permutations and sticky particles

1.1. Sticky particles.

Let us consider a set of particles, labelled by

a

E

{1, ... , N}, of unit mass, moving along the real axis, with position X(t,

a)

and

velocity X'(t,

a)

at time t

E

[0, T], where T 0 is a fixed horizon time. These

particles are supposed to freely stream until they collide. Then, they stick and

their total momentum is preserved. Typically, as a binary collision occurs at time

t* between particle a and particle

a,

we get

X'(t*

+

0, a)=

X'(t*

+

0, a)=

~(X'(t*-

0, a)+

X'(t*-

0, a)).

Of course, such collisions dissipate energy. So, this model of 'sticky particles',

also known as 'adhesion dynamics'

[3], [15], [13] ... ,

is probably the simplest model

of particle interaction, combining free streaming, conservation of momentum and

maximal dissipation of energy.

1.2. Velocity exchanges.

As well known among specialists of rarefied gas

dynamics and Boltzmann equation, purely elastic collisions between particles in

only one space dimension mean no effective collision at all! Indeed, in one space

dimension, an elastic collision between two particles exactly means an exchange of

velocity. So, nothing has effectively changed after collision, since these two particles,

up to an irrelevant relabelling, continue their way with unchanged velocity. So, we

are very far from sticky particles and adhesion dynamics. However, if we introduce

a slight delay in the exchange process, we can recover the adhesion dynamics in the

©

2003 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/330/05880