On the Long Time Behavior of Infinitely Extended Systems of Particles Interacting via Kac Potentials |
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Authors: | Paolo Buttà Emanuele Caglioti Carlo Marchioro |
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Institution: | (1) Dipartimento di Matematica, Università di Roma La Sapienza , P.le Aldo Moro 2, 00185 Roma, Italy |
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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
(x)=![gamma](/content/27u9tb4186av9p2t/xxlarge947.gif) ( x), ![gamma](/content/27u9tb4186av9p2t/xxlarge947.gif) (0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order . 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
–1 times the velocity itself). Finally we shortly discuss the so called Vlasov limit, when time and space are scaled by a factor . |
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Keywords: | infinite particle system Kac potential Vlasov limit |
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