Concentration-of-Measure Theory for Structures and Fluctuations of Waves

The emergence of nonequilibrium phenomena in individual complex wave systems has long been of fundamental interest. Its analytic studies remain notoriously difficult. Using the mathematical tool of the concentration of measure, 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 universality, which does not exist in conventional mesoscopic fluctuations associated with the change in disorder realizations (“sample-to-sample” fluctuations), and originates 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 are 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.

Figure 1

In the present setting two high-dimensional geometric bodies appear: $S^{2N-1}$ constituted by distinct incoming current amplitudes $c$ and ${\mathbb{R}}^M$ by distinct disorder realizations $\omega$.

Figure 2

Using Eqs. (5) and (12), we calculate $L_x(c;\omega )$ at $N=800$ for 4 randomly chosen $c$ at fixed $\omega$ and for 4 randomly chosen $\omega$ at fixed $c$, respectively, and calculate $\parallel I_x{\parallel }_{\text{Lip}}$ for 3 large $N$. All profiles collapse into the a single curve. $L=50$.

Figure 3

Simulations show that in a single slab $\omega$ the profiles $I_x(c)$ for distinct $c$ concentrate around $W(x;\omega )$ (right panels), and the data: $[\parallel I_x{\parallel }_{\text{Lip}},\sqrt{{\pi }^{2}N\text{var}(I_x)}]$ (symbols) collapse into a straight line of unit slope for distinct large $N$ while deviate from this line for small $N$ (left panel). $L=50$.

Figure 4

Quasi 1D simulations show that, for both diffusive (a) and localized (b) waves, the distribution of wave-to-wave fluctuations of $I_x(c)$ (black histograms) displays a tail well fit by a Gaussian distribution (pink dashed lines), while that of sample-to-sample fluctuations (blue histograms) is exponential (a) and stretched exponential with a stretching exponent of 0.4 (b), respectively (red dashed lines) [62]. ${\overline{I}}_x=\int I_{x}p(I_x)d{I}_x$.