thermodynamics - Why is it natural to look for solutions involving dimensionless quantities?

While studying the Heat Equation, I got stuck in a statement in my book. It says:

We have seen that the combination of variables $\displaystyle \frac{x}{\sqrt{Dt}}$ is not only invariant with respect to parabolic dilations but also dimensionless. It is then natural to check if there are solutions of the Heat Equation involving such dimensionless group. (...) it makes sense to look for solutions of the form:

$\displaystyle u^{*}(x,t)=\frac{q}{\sqrt{Dt}}U\left( \frac{x}{\sqrt{Dt}} \right)$

where $U$ is a (dimensionless) function of a single variable."

Why is the bold text so? I don't see why it is intuitive to search for such solutions.

Answer

Let's say your goal is to describe the shape of some object, such as a box.

You could create a completely arbitrary ruler and measure the three axes of the box, coming out for example with lengths of 11.72, 23.44, and 35.16 of your arbitrary ruler units.

Or you might look at your results more closely and think hmm, something is going on here, since the second two lengths are exact multiples of the first one.

So instead of using arbitrary units, you make the box into its own ruler by choosing pairs of sides and looking at the ratios of their lengths. By the rules of division, such ratios are of course dimensionless, since for example your original arbitrary units would cancel out in the ratio.

But look at the result! Instead of (11.72, 23.44, 35.16), you have stripped out the arbitrary noise created when you defined and arbitrary ruler. As a direct result, your "theory of box" instead becomes a much more interesting sequence such as (1, 2, 3) or (1/3, 2/3, 1), depending on which side you pick as your divisor (or "1") unit of natural length. Such simpler ratios immediately suggest that the box is composed of smaller, more fundamental units.

Physics, and in particular the Standard Model, is of course far more complicated that a simple box. But it too is rich in same-to-same ratios (about two dozen total) that measure different magnitudes of identical quantities. For those pairs, you can again use the ratio trick to strip out any "arbitrary ruler" noise and come up with more fundamental, and hopefully insightful, values for fundamental "size and shape" constants that define the exact "shape" of the Standard Model.

For example, the absolute invariance that special relativity imparts to the speed of light makes c into a brain-dead obvious choice for the natural unit of anything involving velocity. Thus c=1, and any other velocity simply uses that unique velocity as its divisor. The charge of an electron is another good divisor choice, though there you can see that there is some wiggle room in the choices, since you could also argue that the -1/3 charge of a down quark might in some way be more "fundamental."