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 φ γ (x)=γφ(γx), γ∈(0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order γ. 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 γ −1 times the velocity itself). Finally we shortly discuss the so called “Vlasov limit,” when time and space are scaled by a factor γ.
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Buttà, P., Caglioti, E. & Marchioro, C. On the Long Time Behavior of Infinitely Extended Systems of Particles Interacting via Kac Potentials. Journal of Statistical Physics 108, 317–339 (2002). https://doi.org/10.1023/A:1015451905014
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DOI: https://doi.org/10.1023/A:1015451905014