FRACTAL PROPERTIES OF ESCAPE FROM A 2-DIMENSIONAL POTENTIAL

This paper summarizes a numerical investigation of the escape of particles from the two-dimensional potential V(x, y) = 1/2x2 + 1/2y2 - epsilonx2y2, for variable epsilon but fixed energy h = 0.12. For epsilon < epsilon1 = 1/(4h) almost-equal-to 2.08, escape is impossible, but for epsilon greater-than-or-equal-to epsilon1, particles will escape. For such larger epsilon, the final outcome, as characterized by the time and direction in which particles escape, is a smooth function of initial conditions in certain phase space regions. However, in other regions the evolution evidences an extremely sensitive dependence on initial conditions indicative of a complex microscopic evolution. For sufficiently large values of epsilon > epsilon2 almost-equal-to 4.90 +/- 0.01, these sensitive regions appear to exhibit a fractal structure with simple scaling properties implying that, at a coarse-grained level, the late time behavior is quite regular. Specifically, the evolution is asymptotically Markovian in the sense that, for particles that have not yet escaped, at late times the coarse-grained escape probabilities per unit time tend toward time-independent values p(infinity) > 0 independent of the initial conditions. The asymptotic values p(infinity) and the rate of convergence towards these values were examined as a function of epsilon and the scale of the coarse-graining, and were observed to exhibit simple scaling behaviour. At least for values of epsilon less-than-or-equal-to 5.7, p(infinity) is-proportional-to (epsilon - epsilon2)alpha, with a critical exponent alpha almost-equal-to 0.49 +/- 0.05. For a coarse-graining corresponding to a linear scale r = 0.05, the time T required for convergence towards this value scales as T(epsilon) is-proportional-to (epsilon - epsilon2)-beta, with beta almost-equal-to 0.39(-0.06)+0.14. The difference gamma = alpha - beta = 0.09(-0.02)+0.04. The value p(infinity) appears independent of r, but the convergence time scales as T(r) is-proportional-to r(-delta), with delta = 0.08 +/- 0.03, seemingly independent of epsilon. To within statistical uncertainties, alpha - beta - delta almost-equal-to 0.