Abstract
Ludwig Boltzmann’s kinetic equation for dilute gases involves knowledge of the dynamics of an isolated pair of particles. Attempts to generalize this equation to higher densities necessarily involve knowledge of the collective dynamics of groups of more than two particles. These attempts therefore give rise to the following mathematical question: For the particularly simple case of hard spheres, where only two-particle collisions occur, what is the nature of the sequences of those collisions that can occur in infinite space? In particular, is there a maximum number of collisions among a given number n of hard spheres?
A survey is given of the main results obtained so far:
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1)
The maximum number of collisions among n hard spheres is bounded.
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2)
The maximum number of collisions among three identical hard spheres is four, except in the one-dimensional case. The possible sequences are given and the proof outlined.
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3)
The maximum number of collisions among any three hard spheres constrained to move in one dimension is given in terms of their masses. The collision sequences and the ultimate velocities of the particles are explicitly given in terms of their initial velocities.
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Murphy, T.J., Cohen, E.G.D. (2000). On the Sequences of Collisions Among Hard Spheres in Infinite Space. In: Szász, D. (eds) Hard Ball Systems and the Lorentz Gas. Encyclopaedia of Mathematical Sciences, vol 101. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04062-1_3
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