  It is known that the non-equilibrium version of the Lorentz gas (a billiard
with dispersing obstacles, electric field and Gaussian thermostat) is
hyperbolic if the field is small. Differently the hyperbolicity of the
non-equilibrium Ehrenfest gas constitutes an open problem, since its obstacles
are rhombi and the techniques so far developed rely on the dispersing nature of
the obstacles. We have developed analytical and numerical investigations which
support the idea that this model of transport of matter has both chaotic
(positive Lyapunov exponent) and non-chaotic steady states with a quite
peculiar sensitive dependence on the field and on the geometry, not observed
before. The associated transport behaviour is correspondingly highly irregular,
with features whose understanding is of both theoretical and technological
interest.
