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N classical particles diffuse inside the one-dimensional interval [0, L ]. There are reflecting walls at the edges of this interval. The diffusion constant of each particle is assumed independent of its energy and position. The particles interact through short-range repulsive interaction that prevents them from exchanging positions —they stay ordered in their initial ordering on the line. The probability that at time t all the particles are back at the points they were at when t =0 (up to a distance less than a mean free path) and the configuration space average of this probability are calculated. The interaction dominates these quantities, even for times in which it is not expected to affect most of the dynamics of the system. Both results are exact, given in terms of elementary analytic functions, and have the form of a sum over terms that represent different kinds of many body processes, describing different ways of clustering of the particles in the system. Connections to the Bethe ansatz method, to quantum chaos and mesoscopic systems, and to random walks (as appear, e.g., in polymer and protein physics, and in hard-core-exclusion models) are pointed out. |
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