On the long time behavior of infinitely extended systems of particles interacting via Kac potentials

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(gamma)(x) = gammaphi(gammax), gammais an element of(0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order gamma. 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 gamma(-1) times the velocity itself). Finally we shortly discuss the so called "Vlasov limit," when time and space are scaled by a factor gamma.