Chaos and Noise in Galactic Potentials

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Abstract
This paper summarizes an investigation of the effects of weak friction and noise in time-independent, nonintegrable two-dimensional potentials that admit both regular and stochastic orbits. The aim is to understand the qualitative effects of internal and external irregularities associated with discreteness effects or couplings to an external environment, which stars in any real galaxy must experience. It is found that these irregularities can be important already on timescales much shorter than the natural relaxation timescale tR associated with two-body relaxation. In particular, for stochastic orbits friction and noise result in an exponential divergence from the unperturbed Hamiltonian trajectory, at a rate set by the value of the local Lyapunov exponent, which persists even for relatively large deviations from the unperturbed trajectory. Friction and noise can also have significant effects on the statistical properties of ensembles of stochastic orbits. Stochastic orbits may be divided into two classes, confined or sticky stochastic orbits which are trapped near islands of regularity, and unconfined or filling stochastic orbits that travel unimpeded throughout a stochastic sea. In the absence of friction and noise, transitions between confined and filling stochastic orbits are very slow. However, even very weak friction and noise can drastically accelerate such transitions, leading to an approach toward a statistical equilibrium on timescales tR. In the two-dimensional models studied in this paper, there are cases for which tR exceeds 106 crossing times where friction and noise can induce transitions for more than half the orbits within 100 crossing times, this corresponding in galaxies to a Hubble time tH. Implications for galactic dynamics are discussed.
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