Transient ensemble dynamics in time-independent galactic potentials

Abstract

This paper summarizes a numerical investigation of the short-time, possibly transient, behaviour of ensembles of stochastic orbits evolving in fixed non-integrable potentials, with the aim of deriving insights into the structure and evolution of galaxies. The simulations involved three different two-dimensional potentials, quite different in appearance. However, despite these differences, ensembles in all three potentials exhibit similar behaviour. This suggests that the conclusions inferred from the simulations are robust, relying only on basic topological properties, e.g., the existence of KAM tori and cantori. Generic ensembles of initial conditions, corresponding to stochastic orbits, exhibit a rapid coarse-grained approach towards a near-invariant distribution on a time-scale ≪ tH, the age of the Universe. This approach is exponential in time, with a rate, ∧, that exhibits a direct correlation with the value of the Liapounov exponent, χ. However, this near-invariant distribution does not correspond to the true invariant measure. If this distribution be evolved for much longer time-scales, one sees systematic evolutionary effects associated with diffusion through cantori, which on short time-scales divide stochastic orbits into two distinct classes, namely confined and unconfined. For the deterministic simulations described herein, the time-scale for this diffusion is ≫tH, although various irregularities associated with external and/or internal irregularities can drastically accelerate this process. A principal tool in the analysis is the notion of a local Liapounov exponent, which provides a statistical characterization of the overall instability of stochastic orbits over finite time intervals. In particular, there is a precise sense in which confined stochastic orbits are less unstable, with small er local Liapounov exponents, than are unconfined stochastic orbits.