Billiards and steep potentials
Vered Rom-Kedar

The behavior of a point particle travelling with 
a constant speed in a region $D$, undergoing elastic 
collisions at the regions's boundary, is known as the
billiard problem. In many applications (molecular dynamics, 
cold atoms optical traps, etc.), the billiard's flow is 
a simplified model which imitates the conservative motion 
of a particle in a smooth steep potential $V_{\varepsilon}$, 
which, in the limit $\varepsilon\rightarrow0$, becomes 
a hard-wall potential. Indeed, one of the underlying 
assumptions of Boltzman hypothesis is that molecules 
behave as hard spheres.

We study rigorously this limit (for arbitrary geometry 
and dimension); on one hand, for regular reflections, 
under some natural assumptions on $V_{\varepsilon}$, 
we provide the asymptotic expansion of the smooth 
solutions in terms of auxiliary billiard approximations, 
with error estimates which are small in the $C^{r}$ norm. 
This seemingly mathematical exercise proves to provide 
a powerful tool for comparing between the smooth and 
the billiard's flow.

On the other hand, in two dimensions, we proved that 
tangent periodic orbits and corner polygons produce 
stability islands even in dispersing geometries for 
which the billiards are mixing. Recently, we 
demonstrated numerically that some smooth three degrees 
of freedom Hamiltonian systems which are arbitrarily
close to three dimensional dispersing billiards have 
islands of effective stability, and hence are non-ergodic.

Joint works with A. Rapoport and D. Turaev.