We prove by example that the number of elastic collisions of n balls of equal mass and equal size in d-dimensional space can be greater than n 3 /27 for n ≥ 3 and d ≥ 2. The previously known lower bound was of order n 2 .

We investigate the motion of an inert (massive) particle being impinged from below by a particle performing (reflected) Brownian motion. The velocity of the inert particle increases in proportion to the local time of collisions and decreases according to a constant downward gravitational acceleration. We study fluctuations and strong laws of the motion of the particles. We further show that the joint distribution of the velocity of the inert particle and the gap between the two particles converges in total variation distance to a stationary distribution which has an explicit product form.

We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are n balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than n−n, then the total number of collisions is bounded by nfalse(5/2+εfalse)n, for any fixed ε>0 and large n. We also show that if there is a number of collisions larger than ncn for an appropriate c>0, then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.

Consider hard balls in a bounded rotating drum. If there is no gravitation, then there is no Fermi acceleration, i.e., the energy of the balls remains bounded forever. If there is gravitation, Fermi acceleration may arise. A number of explicit formulas for the system without gravitation are given. Some of these are based on an explicit realization, which we derive, of the well-known microcanonical ensemble measure.

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