Boundary conditions

The treatment of finite system requires to define the boundary conditions. In this schematic discussion we consider the case of an homogeneous medium surrounded by vacuum. At the boundary of the medium, there is only an outgoing one sided flux $\overrightarrow{J_{+}}$ while $\overrightarrow{J_{-}} =0.$ This means that the current $\overrightarrow{J} =\overrightarrow{J_{+} } -\overrightarrow{J-} >0$ and thus, that since $grad(\varphi)>0$, $\varphi$ decreases from the inner to the outer region. Extrapolating linearly $\varphi$ in the vacuum region, where the diffusion equation is not valid, the extrapolated value should vanish at some distance d_extra . By comparison with exact calculations one finds that $\varphi$ is a good approximation of the true solution for $d_{extra}\simeq2D.$ This is usually very small compared to the multiplying medium size, so that a simple, but sufficient approximation of $\varphi$, at least for qualitative discussions, is obtained by requiring it to vanish at the boundaries of the medium. For illustration we solve the diffusion equation for a semi infinite homogeneous reactor bounded by two parallel planes[34].