What does a unit like $C^{1/5}$ or $kg^{1/2}$ physically mean?

Question

I'm more of a math guy than a physics guy so bear with me....

In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal dimension of around $1.2$. You can also define a concept called fractal extent (which is the experimental version of hausdorff measure). This extent is a measure of how large a fractal is. It's even been used to more accurately model the distribution of coastline bird nesting.

Even though fractals work as an interpretation of geometrical units, I don't think they would describe things such as $kg^{1/2}$ very well. In addition, I've heard about white Gaussian noise that has square root time units.

What is the physical meaning of fractional units such as the above? (Since it obviously has little geometric meaning)

Answer 1

Let's take the example of the white Gaussian noise. The total (rms) power of noise is a particular number with units of power. However, it is often more interesting to know how much power there is at a given frequency. And this is where a fractal dimension comes in.

Gaussian white noise has constant power spectral density. In general, when you look at the power density of the noise at a given frequency, this would be given in units of Watt per Hertz (Hertz being the unit of frequency).

Now power, in electrical systems, is proportional to the square of the voltage. The voltage noise will therefore be proportional to the square root of the power noise, which means that it has units of $$V / \sqrt{Hz}$$

So for a system with Gaussian white noise, we can express the voltage noise with a constant figure. For a given bandwidth (center frequency ± range), it is then easy to compute the total noise power. When you are interested in performing signal-to-noise calculations, this is obviously essential. It also shows you right away that a narrower bandwidth system will have better SNR (assuming that the signal of interest is narrow band, of course).

Answer 2

The only context in which I can think of something like this coming up is in distributions near phase transitions. For instance, if you ask about the correlation of two spins in a magnet near the Curie temperature, it will have a power-law dependence on their separation, $$ \langle \vec{s}(x) \cdot \vec{s}(y) \rangle \propto |x - y|^\alpha, $$ where $\alpha$ is not necessarily an integer (it's not necessarily rational either). Obviously, the proportionality constant would have dimensions of $M T^{-1} L^{2-\alpha}$.

This also arises in systems that feature the so-called "self-organized criticality". These systems are not actually undergoing a phase transition, but they naturally tend to states at the edge of stability.

Part of people's interest in these distributions (to my knowledge) with algebraic dependencies comes precisely from the fact that this is practically impossible to interpret. However, there is apparently some relation to fractals, which is described here.