Particle models with self sustained current

We present some computer simulations run on a stochastic CA (cellular automaton). The CA simulates a gas of particles which are in a channel, the interval $[1,L]$ in $\mathbb Z$, but also in "reservoirs" $\mathcal R_1$ and $\mathcal R_2$. The evolution in the channel simulates a lattice gas with Kawasaki dynamics with attractive Kac interactions, the temperature is chosen smaller than the mean field critical one. There are also exchanges of particles between the channel and the reservoirs and among reservoirs. When the rate of exchanges among reservoirs is in a suitable interval the CA reaches an apparently stationary state with a non zero current, for different choices of the initial condition the current changes sign. We have a quite satisfactory theory of the phenomenon but we miss a full mathematical proof.