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.