Conductivity of the Self-Similar Lorentz Channel

The self-similar Lorentz billiard channel is a spatially extended deterministic dynamical system which consists of an infinite one-dimensional sequence of cells whose sizes increase monotonously according to their indices. This special geometry induces a drift of particles flowing from the small to the large scales. In this article we further explore the dynamical and statistical properties of this billiard. We derive from the ensemble average of the velocity a conductivity formula previously obtained by invoking the equality between phase-space contraction rate and the phenomenological entropy production rate. This formula is valid close to equilibrium. We also review other transport and ergodic properties of this billiard.

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