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
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