If the number of particles is conserved, we have the continuity equation $$ \begin{equation*} \frac{\partial j(x,t)}{\partial x} = -\frac{\partial w(x,t)}{\partial t}, \end{equation*} $$ which leads to $$ \begin{equation} \tag{5} \frac{\partial w(x,t)}{\partial t} = D\frac{\partial^2w(x,t)}{\partial x^2}, \end{equation} $$ which is the diffusion equation in one dimension.