Collisional relaxation in a nonintegrable potential

Details Collisional relaxation in a nonintegrable potential Collisional relaxation in a nonintegrable potential

Mar 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994a%26a...283...59k&link_type=abstract

Astronomy and Astrophysics (ISSN 0004-6361), vol. 283, no. 1, p. 59-66

Astronomy and Astrophysics

Astrophysics

6

Boltzmann Transport Equation, Brownian Movements, Galaxies, Harmonic Oscillators, Kinematics, Liapunov Functions, Perturbation Theory, Stellar Orbits, Chaos, Gravitational Fields, Vlasov Equations, White Noise

Scientific paper

In galactic dynamics, a test star is assumed typically to follow a smooth trajectory in some slowly varying mean field potential and, in addition, to be subjected to 'random' close encounters with neighboring field stars which are modeled essentially as a Brownian process. With the noteable exception of Pfenniger (1986), most analyses of the collisonal relaxation induced by these close encounters have assumed that the effects of the mean field may be ignored completely, and that it suffices to consider Brownian particles moving in the absence of any systematic potential. The idealization of zero potential is problematic, as is any integrable potential in which all the mean field orbits are stable. For this reason, the paradigm of Brownian motion is reexamined here, allowing for a time-dependent, non-integrable potential, in which some subset of the mean field trajectories correspond to exponentially unstable orbits with positive Liapounov exponent. A nontrivial deterministic equation of motion is convereted into a stochastic differential equation incorporating dynamical friction and delta-correlated white noise, related via a fluctuation-dissipation theorem, and the effects of the friction and noise are analyzed perturbatively. The principal conclusion is that a coupling to a nonintegrable background potential can decrease by orders of magnitude the time scale on which noise and friction modify positions and velocities along an unperturbed, deterministic trajectory. Specifically, one finds that, even in the weak noise limit, perturbations in position and velocity grow exponentially on a time scale tlambda determined by the Liapounov exponent, although, for a strictly time-dependent potential, perturbations in energy and other collisionless invariants will only grow on a much longer relaxation time tr. Some potential implications of this accelerated collisional relaxation are discussed.

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