Chaos, ergodic convergence, and fractal instability for a thermostated canonical harmonic oscillator.

The authors thermostat a qp harmonic oscillator using the two additional control variables zeta and xi to simulate Gibbs' canonical distribution. In contrast to the motion of purely Hamiltonian systems, the thermostated oscillator motion is completely ergodic, covering the full four-dimensional [q,p,zeta,xi] phase space. The local Lyapunov spectrum (instantaneous growth rates of a comoving corotating phase-space hypersphere) exhibits singularities like those found earlier for Hamiltonian chaos, reinforcing the notion that chaos requires kinetic-as opposed to statistical-study, both at and away from equilibrium. The exponent singularities appear to have a fractal character.

Viscous attractor for the Galton board.

We analyze the Galton Board [or periodic "Lorentz Gas"] with a point mass scattered by elastic disks of diameter sigma, using a constant driving field g and a constant-viscosity linear drag force -p/tau, where p is the point-mass momentum. This combination leads to a nonequilibrium steady state which depends only upon the dimensionless ratio gtau(2)/sigma. The long-time-averaged trajectory leads to multifractal phase-space structures closely resembling those we found earlier using isokinetic equations of motion derived from Gauss' Principle of Least Constraint. A highly damped [small tau] creeping-flow limit describes our results for gtau(2)/sigma less than about 0.2. The lightly damped Green-Kubo linear-response limit for the model provides an accurate description of the dissipative dynamics for gtau(2)/sigma greater than about 2.0.