Electric fields on a surface of constant negative curvature

Abstract

A one-parameter family of time reversible Anosov flows is studied; physically, it describes a particle moving on a surface of constant negative curvature under the action of an electric field (corresponding to an automorphic form) and of a ‘thermostatting’ force (given by Gauss's least-constraint principle). We show that the flows are dissipative, in the sense that the average volume contraction rate is positive and the Sinai–Ruelle–Bowen measure is singular with respect to the volume: therefore they verify the assumptions for the validity of the continuous time version of Gallavotti–Cohen's fluctuation theorem for the large fluctuations of the average volume contraction rate. If several independent electric fields are considered, it makes sense to ask for the validity of Onsager's reciprocity: we show, by explicitly computing the relevant transport coefficents, that it is indeed obeyed.