the bulk, thus the high density regime always fills the whole system left of the current limiting defect.

3.  In systems with only few defects the relation between the average density and the current at a given site is the same as in the homogeneous system. Thus current profiles are almost the same (as long as the maximum current is not exceeded). In disordered systems with a finite fraction of defects, however, the current-density relation is not the same as in the homogeneous system and depends on $q$ and the distribution of defects, since the large number of density peaks have influence on the source term $s(\rho)$ in (\[continuity\_equ\]) on a macroscopic scale. Therefore the current profiles differ from the homogeneous case.

In order to capture these properties, we follow the concept of [@pff_1def] by focusing on the current profiles $J(x)$.

The influence of defects: additional initial conditions
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Locally the current profiles are determined by the continuity equation (\[continuity\_equ\]). Introducing the continuous variable $x:=\frac{i-1}{L-1}$, which is the relative position in the system, one can write $J_{i-1}=J_i-\frac{1}{L}\frac{dJ}{dx}+{\cal O}(1/L^2)$. In the stationary state the continuity equation (\[continuity\_equ\]) becomes $$\label{dJdx}
\frac{dJ}{dx} = S(\rho)+{\cal O}(1/L)$$ where the global source term $S(\rho)=L\, s(\rho)$ was 