The motion of a point particle travelling with a constant speed inside a region D ⊂ Rd, d ≥ 2, undergoing elastic collisions at the regions’s boundary, is known as the billiard problem. Since the days of Boltzmann, scientists have been using various billiard models to approximate the motion in systems with steep potentials (e.g. for studying classical molecular dynamics, the motion of a charged particle and cold atom’s motion in dark optical traps). In this research we develop methodologies for examining the validity and accuracy of this approximation. We consider families of smooth billiard-like potentials Vɛ, that, in the limit ɛ → 0, become singular hard-wall potentials of multi-dimensional billiards. On one hand, we provide natural conditions on these families that insure that regular reflections of the billiards are approached by their smooth trajectories. This gives a tool to understand propertie
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