In the beginning we review briefly the evolution of the ideas on the motion of the bodies in our solar system, from Newton's clockwork Universe to the presently accepted ubiquity of chaotic transport in the asteroid belt. Then we discuss the result of chaotic motion, which is transport in phase space, and we introduce the concept of diffusion of an asteroid in action space. We proceed by reviewing recent work on numerical as well as analytical study of asteroids following chaotic trajectories

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... d we summarize the main results. We present several applications of the theoretical modelling of asteroid motion as diffusion in action space, to problems of specific interest. 157 158 Harry Varvoglis model of the elliptic three-body problem. In this way it became widely accepted that chaos is an important phenomenon in the solar system. Therefore Newton's idea of a clockwork universe, which, once started to move, would continue moving in the same way "ad perpetuum", suddenly was proved to be wrong. The bodies of the solar system, especially the minor ones, may follow trajectories that change secularly in time, so that "collisions" (either true or just close encounters) and ejections play an important role as sinks (both) and sources (the former) of bodies, even in the present era. As a result, the question now is not anymore whether chaos affects the dynamics of planetary systems, but on which time-scales it does so and on whether it is the rule or the exception. Recent calculations put the percentage of main belt asteroids on chaotic orbits at 30% (proper elements and LCE computations, Milani & Knežević (2003)). However it was not so easy to understand how to treat mathematically the dynamical evolution of bodies on chaotic trajectories, since this was a novel situation in Celestial Mechanics. Laplace's theory and its continuation, proper elements theory, are not valid in chaotic regions. As numerical experiments clearly show, the values of proper elements in chaotic regions change in a secular, non-quasi-periodic, way. This is how the concepts and methods of Statistical Physics, in particular transport theory, were introduced in Celestial Mechanics. In the rest of this article we will review these methods and show how transport theory may be used in order to obtain useful information in real problems of Celestial Mechanics. † I.e. f (x, t)dt gives the probability that, at time t, the co-ordinate x of a "particle" lies within the interval x and x + dx.