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We consider point particles in a table made of two circular cavitiesconnected by two rectangular channels, forming a closed loop under periodicboundary conditions. In the first channel, a bounce--back mechanism acts whenthe number of particles flowing in one direction exceeds a given threshold $T$.In that case, the particles invert their horizontal velocity, as if collidingwith vertical walls. The second channel is divided in two halves parallel tothe first, but located in the opposite sides of the cavities. In the secondchannel, motion is free. We show that, suitably tuning the sizes of cavities,of the channels and of $T$, non--equilibrium phase transitions take place inthe $N\rightarrow \infty$ limit. This induces a stationary current in thecircuit, thus modeling a kind of battery, although our model is deterministic,conservative, and time reversal invariant.

Deterministic model of battery, uphill currents and non-equilibrium phase transitions