GENERAL POISSONIAN MODEL OF DIFFUSION IN CHAOTIC COMPONENTS

We extend our recent study of diffusion in strongly chaotic systems (' the random model') to the general systems of mixed-type dynamics, incl uding especially KAM systems, regarding the diffusion in chaotic compo nents. We do this by introducing a Poissonian model as in our previous random model describing the strongly chaotic systems, except that now we allow for different a priori probabilities in different cells of t he discretized phase space (surface of section). Thus the concept of g reyness (of cells), denoted by g, such that 0 less than or equal to g less than or equal to 1, is introduced, as is its distribution w(g). W e derive the relationship between the dynamical property, namely the ( normalized) fraction of chaotic component rho(j) as a function of disc rete time j, and w(g). We predict again the universal scaling law, nam ely that for any w(g), the chaotic fraction rho(j) is a function of j/ N only, and not separately of j and N, where N is the number of cells of equal size 1/N. The random model of exponential 1 - rho(j) = exp(-j /N) is reproduced if all cells have g = 1, i.e. w(g) = delta(1 - g). W e argue that in two-dimensional systems, at any finite N, w(g) is non- trivial due to the fractal dimension of the boundary of the chaotic co mponent, but is such that it goes to delta(1 - g) as N --> infinity, w hilst in systems with three or more degrees of freedom w(g) has a well defined limit with non-zero values also at g < 1. This is due to the existence of the Arnold web. We suggest how-through our formalism-one can calculate the Lebesgue measure of the chaotic component at each fi nite discretization, whose limit exists for N --> infinity. Our findin gs are verified and illustrated for two-and three-dimensional billiard s.