W-flows on Weyl manifolds and Gaussian thermostats

Abstract

A relation between Weyl connections and Gaussian thermostats is exposed and exploited.

Particle drift in self-similar billiards

“…As we said, Wojtkowski's theorem [19,20] allows us to transform the trajectories of the flow t into those of the Gaussian thermostatted particle in a periodic Lorentz channel with finite horizon under a small external field e = (e, 0) (whose value is determined by r , precisely e = ln r ; see [3]). Even though Wojtkowski's transformation does not necessarily preserve convexity, the images of the curved boundaries of D 0 will remain convex when ε 0 in Theorem 1 is small enough.…”

Entropy Production in Gaussian Thermostats

“…We emphasize that we do not make any assumptions on g or E except that the underlying isokinetic dynamics is Anosov. Conditions under which the Anosov property holds have been given in [22,23].…”

Log-periodic drift oscillations in self-similar billiards

“…One can nevertheless describe the Gaussian iso-kinetic dynamics by a Hamiltonian formalism, as shown in [11]. A general theorem due to Wojtkowski [12] stipulates that Gaussian iso-kinetic trajectories are geodesic lines of the so-called torsion free connection (also known as the Weyl connection). The Gaussian iso-kinetic Lorentz gas can thus be conformally transformed into a distorted billiard table on which trajectories become straight lines; besides, the conformal transformation preserves the specular character of the collision laws [13].…”