Fick's three-dimensional law :

\({\vec j_d} = - D\overrightarrow {grad} ({n^*})\)

with \(D\) the coefficient of scattering and \(n^*\) the particle density.

This law shows that scattering can only take place in a medium where the particle density \(n^*\) is not uniform.

Moreover, the gradient being oriented towards the increasing \(n^*\), the - symbol in Fick's law indicates that particles spontaneously scatter from the more concentrated places to the less concentrated ones.

Let's determine the one-dimensional equation of scattering without internal source of creation of particles.

A reasoning similar to the one of thermal transfers (we write the conservation of matter considering a cube, which has a volume \(Sdx\)) :

\(\frac{{\partial {n^*}}}{{\partial t}}Sdxdt = - \frac{{\partial {j_d}}}{{\partial x}}Sdxdt\)

It results in the equation of conservation of matter :

\(\frac{{\partial {n^*}}}{{\partial t}} = - \frac{{\partial {j_d}}}{{\partial x}}\)

Using Fick's law, we obtain the one-dimensional equation of scattering :

\(\frac{{\partial {n^*}}}{{\partial t}} = D\frac{{{\partial ^2}{n^*}}}{{\partial {x^2}}}\)

In three dimensions, we have (see the lesson about "Vector calculus") :

\(\frac{{\partial {n^*}}}{{\partial t}} = - div({{\vec j}_d})\)

And the equation of scattering (without source) :

\(\frac{{\partial {n^*}}}{{\partial t}} = D\Delta {n^*}\)