Statistical properties of transmission spectra of ergodic wave billiards: Random Hamiltonian approach

We used random matrix theory (RMT) to study the probability distribution function and moments of the wave power transmitted inside systems with ergodic wave motion. The results describe either open multichannel systems or their closed counterparts with local-in-space internal dissipation. As the first step, we concentrated on the regime of broken time-reversal invariance and employed two different analytical approaches: the exact supersymmetry method and a simpler technique that uses RMT eigenstatistics for closed nondissipative systems as an input. The results of the supersymmetric method were confirmed by numerical simulation. The simpler method is found to be adequate for closed systems with uniform dissipation, or in the limit of a large number of weak local dampers. Then, we used RMT to study the first two moments of the wave power transmitted inside time-reversal invariant systems with ergodic wave motion. We applied the supersymmetry method to the calculation of these moments, and compared them with a simpler, perturbative calculation that implies eigenstatistics to be those of a closed nondissipative RMT system. The results of the supersymmetric method were, again, confirmed by numerical simulation.; Finally, we considered the experimental realization of the RMT system. We examined the dissipation of diffuse ultrasonic energy in a reverberant body coupled to a waveguide, i.e. in an open ergodic system. A simple model predicts a Porter-Thomas-like distribution of level widths and corresponding nonexponential dissipation, a behavior largely confirmed by measurements. For the case of fully open channels, however, measurements deviate from this model to a statistically significant degree. A supersymmetric calculation is found to model the observed behaviors accurately. In conclusion we discussed a possibility to find a more realistic ‘naive’ model that takes both the RMT and eigenmode complexity into account, but is less technically involved than the supersymmetry method. Such a modification of the naive approach is likely to be useful in studies of power variance and transient decay in chaotic cavities.