Fick's law and phase transitions in the Lorentz circuit

Nonequilibrium steady states are investigated in a two-dimensional billiard table, called the Lorentz circuit, consisting of two circular urns connected by two rectangular strips: an active channel, where a feedback mechanism operates, and a passive finite-size Lorentz channel, which is a finite network of Sinai billiards, i.e., a specific environment populated with an array of circular scatterers of fixed radius. We show that, in the finite-horizon regime, a variation of any of the four geometrical parameters of the Lorentz channel can trigger a phase transition in the circuit. Our analytic derivations identify the critical parameters governing these transitions and predict whether they are continuous or discontinuous. We find that both transition orders are realized upon changing just one of the geometrical parameters. Our analysis further highlights the validity of Fick's law in the Lorentz channel, which in our model mirrors the validity of Ohm's law in electrical circuits. All analytic results are corroborated by an extensive set of numerical simulations of the particle dynamics.