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Time evolution for infinitely many hard spheres

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Abstract

We construct the time evolution for infinitely many particles in ℝv interacting by the hard-sphere potential

$$\Phi (x) = \left\{ {\begin{array}{*{20}c} { + \infty } \\ 0 \\ \end{array} } \right. \begin{array}{*{20}c} {|x|< a} \\ {|x| \geqq a} \\ \end{array}$$

. Because there are abundant examples of hard-sphere configurations with more than one solution to the Newtonian equations of motion, we introduce the concept of aregular solution, in which the growth of velocities and crowding of particles at infinity are limited. We prove that (1) regular solutions exist with probability one in every equilibrium state, and (2) any configuration of the infinite system is the initial point of at most one regular solution. Equilibrium states are invariant under the time-evolution.

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Communicated by J. L. Lebowitz

Part of this work forms part of the author's doctoral dissertation written at the University of California, Berkeley, under the direction of O. E. Lanford III

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Alexander, R. Time evolution for infinitely many hard spheres. Commun.Math. Phys. 49, 217–232 (1976). https://doi.org/10.1007/BF01608728

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  • DOI: https://doi.org/10.1007/BF01608728

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