Boltzmann’s Transformation is used (among other things) to convert Fick's second law into an easily solvable ordinary differential equation. It uses the variable $\xi=\frac{x}{2 \sqrt{t}}$.
As far as I can remember, a change of variables has to be a diffeomorphism, and diffeomorphisms are necessarily between manifolds of the same dimension. So there should exist another non trivial variable, function of (x,t), let us call it $ \eta $, such that the transformation $\Phi : (x,t) \rightarrow (\xi,\eta)$ is a diffeomorphism.
I have tried a few possibilities without success and searched the "internets" for a solution in vain, so I thought I would submit it to the community. Would anybody know either the answer to this, of if there exists a general method to find the missing variable?
$\endgroup$1 Answer
$\begingroup$How about $\eta = 2 \sqrt{t}$? Then $x = \eta \xi$, $t = \frac{1}{4} \eta^2$. The manifolds would be $(x,t) \in \mathbb{R} \times (\epsilon,\infty), \epsilon > 0$, $(\xi,\eta) \in \mathbb{R} \times (2\sqrt{\epsilon},\infty)$. You can check that the Jacobian determinant does not vanish.
$\endgroup$
