Phase mixing in time - independent Hamiltonian systemsHenry

This paper describes the evolution of localised ensembles of initial conditions in two-and three-dimensional time-independent potentials which admit a coexistence of regular and chaotic orbits. The coarse-grained approach towards an invariant, or near-invariant, distribution was probed by tracking (1) moments hx i y j z k p l x p m y p n z i for i + j + k + l + m + n 4 and (2) binned representations of reduced distributions f(Z a ; Z b) for a 6 = b = x; y; z; p x ; p y ; p z computed at xed intervals t. For ensembles of \unconnned" chaotic orbits in two-dimensional systems not stuck near islands by cantori, the moments evolve exponentially. Quantities like the dispersion px , which start small and eventually asymptote towards a larger value, initially grow exponentially in time at a rate comparable to , the mean value of the largest short time Lyapunov exponent for orbits in the ensemble. Quantities like jhp x ij, that can start large but eventually asymptote towards zero, decrease exponentially. With respect to a discrete L p norm, reduced distributions f(Z a ; Z b) generated from successive snapshots exhibit an overall exponential decay towards a near-invariant f niv (Z a ; Z b), although a plot of Df(t) jjf(t) ? f niv jj can exhibit considerable structure. Implementing an additional coarse-graining by averaging over several successive snapshots can reduce the amount of structure and increase the rate ab at which ensembles asymptote towards f niv. Regular ensembles behave very diierently, both moments and Df evolving in a fashion better represented by a power law time dependence. \Connned" chaotic orbits, initially stuck near regular islands because of cantori, exhibit an intermediate behaviour. The behaviour of ensembles evolved in three-dimensional potentials is qualitatively similar, except that, in this case, it is relatively likely to nd one direction in connguration space which is \less chaotic" than the other two, so that quantities like ab depend more sensitively on which phase space variables one tracks.