Viscous attractor for the Galton board

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Abstract
We analyze the Galton Board [or periodic ‘‘Lorentz Gas’’] with a point mass scattered by elastic disks of diameter σ, using a constant driving field g and a constant-viscosity linear drag force −p/τ, where p is the point–mass momentum. This combination leads to a nonequilibrium steady state which depends only upon the dimensionless ratio gτ2/σ. 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 τ] creeping-flow limit describes our results for gτ2/σ 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 gτ2/σ greater than about 2.0.