Deterministic model of battery, uphill currents, and nonequilibrium phase transitions

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