2.2 Boltzmann's transformation, Parabolic law

Introducing the $\lambda=x/\sqrt t$ parameter the partial differential equation 2.5 transforms to an ordinary differential equation:

$$-\frac{\lambda}{2}\frac{\mathrm{d} \rho}{\mathrm{d} \lambda}=\fra... ...}{\mathrm{d} \lambda}\left(D \frac{\mathrm{d} \rho}{\mathrm{d} \lambda}\right),$$ (2.8)

i.e. the composition depends only on $\lambda$ . From this it follows that a plane with constant composition shifts proportionally to the square root of the time:

$$\rho\left(\frac{x}{\sqrt t}\right)=\mathrm{const} \Rightarrow \frac{x}{\sqrt t}=\mathrm{const} \Rightarrow x \propto \sqrt t.$$ (2.9)

The $x \propto \sqrt t$ relation is often called as parabolic law, since $x^2 \propto t$ . (see also later Fig. 2.1 in section 2.4)

Figure 2.1: A plane with constant composition shifts proportionally to the square root of the time, which also means that the thickness of the diffusion zone (or its half) is also $\propto \sqrt{t}$ .