Concentration-of-measure theory for structures and fluctuations of waves

TL;DR

This paper develops a concentration-of-measure theory to analyze wave structures and fluctuations in disordered media, revealing universal wave-to-wave fluctuations and criteria for uniform spatial structures in stationary states.

Contribution

It introduces a novel application of concentration-of-measure theory to wave systems, uncovering universal fluctuation behaviors and criteria for spatial uniformity in stationary scattering states.

Findings

Wave-to-wave fluctuations exhibit new universal behaviors.

Stationary scattering states tend to have uniform spatial structures.

Expectations of observables are independent of incoming waves.

Abstract

The emergence of nonequilibrium phenomena in individual complex wave systems has long been of fundamental interests. Its analytic studies remain notoriously difficult. Using the mathematical tool of the concentration of measure (CM), we develop a theory for structures and fluctuations of waves in individual disordered media. We find that, for both diffusive and localized waves, fluctuations associated with the change in incoming waves ("wave-to-wave" fluctuations) exhibit a new kind of universalities, which does not exist in conventional mesoscopic fluctuations associated with the change in disorder realizations ("sample-to-sample" fluctuations), and originate from the coherence between the natural channels of waves -- the transmission eigenchannels. Using the results obtained for wave-to-wave fluctuations, we find the criterion for almost all stationary scattering states to exhibit the same spatial structure such as the diffusive steady state. We further show that the expectations of observables at stationary scattering states are independent of incoming waves and given by their averages with respect to eigenchannels. This suggests the possibility of extending the studies of thermalization of closed systems to open systems, which provides new perspectives for the emergence of nonequilibrium statistical phenomena.