Boltzmann’s transformation converts Fick's second law into an easily solvable ordinary differential equation.
Assuming a diffusion coefficient D that is in general a function of concentration c, Fick's second law is where t is time, and x is distance.
Observing the previous equation, a trivial solution is found for the case dc/dξ = 0, that is when concentration is constant over ξ.
[2] Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys.
The initial conditions are: Also, the alloys on both sides are assumed to stretch to infinity, which means in practice that they are large enough that the concentration at their other ends is unaffected by the transient for the entire duration of the experiment.
It was not necessary to introduce one as Boltzmann's transformation worked fine without a specific reference for x; it is easy to verify that the Boltzmann transformation holds also when using x-XM instead of plain x. XM is often indicated as the Matano interface, and is in general not coincident with x=0: since D is in general variable with concentration c, the concentration profile is not necessarily symmetric.
Since the denominator term dc/dx goes to zero for c → cL (as the concentration profile flattens out), the integral in the numerator must also tend to zero in the same conditions.