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Structures with heavy-tailed distributions of disorder occur widely in nature. The evolution of such systems, as in foraging for food or the occurrence of earthquakes is generally analyzed in terms of an incoherent series of events. But the study of wave propagation or lasing in such systems requires the consideration of coherent scattering. We consider the distribution of wave energy inside 1D random media in which the spacing between scatterers follow a Levy $alpha$-stable distribution characterized by a power-law decay with exponent $alpha$. We show that the averages of the intensity and logarithmic intensity are given in terms of the average of the logarithm of transmission and the depth into the sample raised to the power $alpha$. Mapping the depth into the sample to the number of scattering elements yields intensity statistics that are identical to those found for Anderson localization in standard random media. This allows for the separation for the impacts of disorder distribution and wave coherence in random media.
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The time that waves spend inside 1D random media with the possibility of performing Levy walks is experimentally and theoretically studied. The dynamics of quantum and classical wave diffusion has been investigated in canonical disordered systems via the delay time. We show that a wide class of disorder--Levy disorder--leads to strong random fluctuations of the delay time; nevertheless, some statistical properties such as the tail of the distribution and the average of the delay time are insensitive to Levy walks. Our results reveal a universal character of wave propagation that goes beyond standard Brownian wave-diffusion.
We study the electromagnetic transmission $T$ through one-dimensional (1D) photonic heterostructures whose random layer thicknesses follow a long-tailed distribution --Levy-type distribution. Based on recent predictions made for 1D coherent transport with Levy-type disorder, we show numerically that for a system of length $L$ (i) the average $<-ln T> propto L^alpha$ for $0<alpha<1$, while $<-ln T> propto L$ for $1lealpha<2$, $alpha$ being the exponent of the power-law decay of the layer-thickness probability distribution; and (ii) the transmission distribution $P(T)$ is independent of the angle of incidence and frequency of the electromagnetic wave, but it is fully determined by the values of $alpha$ and $<ln T>$.
We consider heat transport in one-dimensional harmonic chains attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation $d$ between any two successive impurities is randomly distributed according to a power-law distribution $P(d)sim 1/d^{alpha+1}$, being $alpha>0$. In the regime where the first moment of the distribution is well defined ($1<alpha<2$) the thermal conductivity $kappa$ scales with the system size $N$ as $kappasim N^{(alpha-3)/alpha}$ for fixed boundary conditions, whereas for free boundary conditions $kappasim N^{(alpha-1)/alpha}$ if $Ngg1$. When $alpha=2$, the inverse localization length $lambda$ scales with the frequency $omega$ as $lambdasim omega^2 ln omega$ in the low frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a non-closed form. When $alpha>2$, the thermal conductivity scales as in the uncorrelated disorder case. The situation $alpha<1$ is only analyzed numerically, where $lambda(omega)sim omega^{2-alpha}$ which leads to the following asymptotic thermal conductivity: $kappa sim N^{-(alpha+1)/(2-alpha)}$ for fixed boundary conditions and $kappa sim N^{(1-alpha)/(2-alpha)}$ for free boundary conditions.
We study transport properties of graphene with anisotropically distributed on-site impurities (adatoms) that are randomly placed on every third line drawn along carbon bonds. We show that stripe states characterized by strongly suppressed back-scattering are formed in this model in the direction of the lines. The system reveals Levy-flight transport in stripe direction such that the corresponding conductivity increases as the square root of the system length. Thus, adding this type of disorder to clean graphene near the Dirac point strongly enhances the conductivity, which is in stark contrast with a fully random distribution of on-site impurities which leads to Anderson localization. The effect is demonstrated both by numerical simulations using the Kwant code and by an analytical theory based on the self-consistent $T$-matrix approximation.
We predict the existence of exchange broadening of optical lineshapes in disordered molecular aggregates and a nonuniversal disorder scaling of the localization characteristics of the collective electronic excitations (excitons). These phenomena occur for heavy-tailed Levy disorder distributions with divergent second moments - distributions that play a role in many branches of physics. Our results sharply contrast with aggregate models commonly analyzed, where the second moment is finite. They bear a relevance for other types of collective excitations as well.
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