Experimental investigation of nodal domains in the chaotic microwave rough billiard

TL;DR

This experimental study investigates the properties of nodal domains in chaotic microwave billiards, confirming theoretical predictions about their number and size distributions, and exploring their behavior at high energy levels.

Contribution

The paper provides the first detailed experimental analysis of nodal domains in chaotic microwave billiards, validating theoretical models and percolation theory predictions.

Findings

Number of nodal domains scales linearly with level number N

Distribution of nodal domain areas follows a power law with exponent ~2

Distribution of nodal domain perimeters follows a power law with exponent ~2.1

Abstract

We present the results of experimental study of nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains are regions where a wave function has a definite sign. The wave functions Psi_N of the rough billiard were measured up to the level number N=435. In this way the dependence of the number of nodal domains \aleph_N on the level number $N$ was found. We show that in the limit N->infty a least squares fit of the experimental data reveals the asymptotic number of nodal domains aleph_N/N = 0.058 +- 0.006 that is close to the theoretical prediction aleph_N/N +- 0.062. We also found that the distributions of the areas s of nodal domains and their perimeters l have power behaviors n_s ~ s^{-tau} and n_l ~ l^{-tau'}, where scaling exponents are equal to \tau = 1.99 +- 0.14 and \tau'=2.13 +- 0.23, respectively. These results are in a good agreement with the predictions of percolation theory. Finally, we demonstrate that for higher level numbers N = 220-435 the signed area distribution oscillates around the theoretical limit Sigma_{A} = 0.0386 N^{-1}.