the peaks diminish for larger systems indicating that they are just local phenomena. We can associate this phase separation with a phase transition at the critical parameter value $\alpha'$.

![\[phase\_sep\](Colour online) Density profiles for increasing values of $\alpha$ and fixed $\beta=0.9$. At a critical value $\alpha^*$ a high density region at the most right defect occurs (phase separation). For higher $\alpha$ multiple high density regions appear.](densprof_avar_def.eps){width="0.9\columnwidth"}

![\[finsize\_scale1\](Colour online) Density profiles for identical macroscopic parameters $\Omega_a=0.1,\,\, \Omega_d=0.1,\,\,
  \alpha=0.35,\,\, \beta=0.9$ but different system sizes $L$. The left boundary of the high density region (shock) becomes steeper with increasing system size, indicating a macroscopic regime. ](finsize_ps.eps){width="0.9\columnwidth"}

Increasing $\alpha$ further moves the shock position to the left. The density profile right of the defect where phase separation occurred does not change anymore by varying the entry rate. The same is true for the *output current* at the right boundary $J_\text{out}=J(L)$. At some value of $\alpha$ a second high density region starts to form. Thus in a system with many defects multiple shocks can occur associated with alternating domains of high and low density.

Above a critical value $\alpha^*$, where a high density domain extends to the left boundary, the density profile and the current 