02-112 Paolo Butta', Emanuele Caglioti, Carlo Marchioro
On the long time behavior of infinitely extended systems of particles interacting via Kac potentials (273K, PostScript file) Mar 8, 02
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Abstract. We analyze the long time behavior of an infinitely extended system of particles in one dimension, evolving according to the Newton laws and interacting via a non-negative superstable Kac potential $\phi_\ga(x) = \ga\phi(\ga x)$, $\ga\in (0,1]$. We first prove that the velocity of a particle grows at most linearly in time, with rate of order $\ga$. We next study the motion of a fast particle interacting with a background of slow particles, and we prove that its velocity remains almost unchanged for a very long time (at least proportional to $\ga^{-1}$ times the velocity itself). Finally we shortly discuss the so called ``Vlasov limit'', when time and space are scaled by a factor $\ga$.

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