No abstract

ABBREVIATED ABSTRACT: This paper summarises an investigation of the effects of weak friction and noise in time-independent, nonintegrable potentials which admit both regular and stochastic orbits. The aim is to understand the qualitative effects of internal and external irregularities associated, e.g., with discreteness effects or couplings to an external environment, which stars in any real galaxy must experience. The two principal conclusions are: (1) These irregularities can be important on time scales much shorter than the natural relaxation time scale t_R associated with the friction and noise. For stochastic orbits friction and noise induce an average exponential divergence from the unperturbed Hamiltonian trajectory at a rate set by the value of the local Lyapunov exponent. Even weak noise can make a pointwise interpretation of orbits suspect already on time scales much shorter than t_R. (2) The friction and noise can also have significant effects on the statistical properties of ensembles of stochastic orbits, these also occurring on time scales much shorter than t_R. Potential implications for galactic dynamics are discussed, including the problem of shadowing.Comment: 45 pages, uuencoded PostScript (figures included), LA-UR-94-282

No abstract

This paper summarises a numerical investigation of the short time, possibly transient, behaviour of ensembles of stochastic orbits evolving in xed nonintegrable potentials, with the aim of deriving insights into the structure and evolution of galaxies. The simulations involved three di erent two-dimensional potentials, quite di erent in appearance. However, despite these di erences 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 timescale t H , 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,. However, this near-invariant distribution does not correspond to the true invariant measure: If this distribution be evolved for much longer timescales one sees systematic evolutionary e ects associated with di usion through cantori, which on short timescales divide stochastic orbits into two distinct classes, namely con ned and uncon ned. For the deterministic simulations described herein, the timescale for this di usion is t H , 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 characterisation of the overall instability of stochastic orbits over nite time intervals. In particular, there is a precise sense in which con ned stochastic orbits are less unstable, with smaller local Liapounov exponents, than are uncon ned stochastic orbits.

Results are reported from a numerical investigation of orbits in a truncated Toda potential which is perturbed by weak friction and noise. Two signi cant conclusions are shown to emerge: (1) Despite other nontrivial behaviour, con guration, velocity, and energy space moments associated with these perturbations exhibit a simple scaling in the amplitude of the friction and noise. (2) Even very weak friction and noise can induce an extrinsic di usion through cantori on a time scale much shorter than that associated with intrinsic di usion in the unperturbed system.