8. Experimental evidence of multifractality with ultrasonic waves

Taken by Piotr Warchol

The Inverse Participation Ratio (IPR) defined as int{|psi(r)|^4 d^dr} (integral taken over the full d-dimensional space) can be used to measure localization properties of wavefunctions. If the wavefunction is statistically uniformly distributed in a system of size L (along each of the d dimensions), the IPR should scale like 1/L^d. If it is localized with localization length xi, it should scale like 1/xi^d. It turns out that the IPR of actual eigenstates of disordered systems behave differently. This is related to the fractality, or multifractality, of the wavefunctions.

This paper analyzes the propagation of ultrasonic waves in a strongly disordered medium. From the experimental results, it shows evidence for multifractal properties of the wavefunctions. 

The concepts linked to multifractality are far from obvious, but fruitful and at the heart of current research. You may not understand eveything in the paper, but you will learn useful things. Don't be afraid!