Unusual Brownian motion of photons in open absorbing media

Very recent experiments have discovered that localized light in strongly absorbing media displays intriguing diffusive phenomena. Here we develop a first-principles theory of light propagation in open media with arbitrary absorption strength and sample length. We show analytically that photons in localized open absorbing media exhibit unusual Brownian motion. Specifically, wave transport follows the diffusion equation with the diffusion coefficient exhibiting spatial resolution. Most strikingly, despite that the system is controlled by two parameters -- the ratio of the localization (absorption) length to the sample length -- the spatially resolved diffusion coefficient displays novel single parameter scaling: it depends on the space via the returning probability. Our analytic predictions for this diffusion coefficient are confirmed by numerical simulations. In the strong absorption limit they agree well with the experimental results.

Weak Convergence Methods for Approximation of Path-dependent Functionals

This paper provides convergence analysis for the approximation of a class of path-dependent functionals underlying a continuous stochastic process. In the first part, given a sequence of weak convergent processes, we provide a sufficient condition for the convergence of the path-dependent functional underlying weak convergent processes to the functional of the original process. In the second part, we study the weak convergence of Markov chain approximation to the underlying process when it is given by a solution of stochastic differential equation. Finally, we combine the results of the two parts to provide approximation of option pricing for discretely monitoring barrier option underlying stochastic volatility model. Different from the existing literatures, the weak convergence analysis is obtained by means of metric computations in the Skorohod topology together with the continuous mapping theorem. The advantage of this approach is that the functional under study may be a function of stopping times, projection of the underlying diffusion on a sequence of random times, or maximum/minimum of the underlying diffusion.

Evolution of scaling emergence in large-scale spatial epidemic spreading

Background: Zipfs law and Heaps law are two representatives of the scaling concepts, which play a significant role in the study of complexity science. The coexistence of the Zipfs law and the Heaps law motivates different understandings on the dependence between these two scalings, which is still hardly been clarified. Methodology/Principal Findings: In this article, we observe an evolution process of the scalings: the Zipfs law and the Heaps law are naturally shaped to coexist at the initial time, while the crossover comes with the emergence of their inconsistency at the larger time before reaching a stable state, where the Heaps law still exists with the disappearance of strict Zipfs law. Such findings are illustrated with a scenario of large-scale spatial epidemic spreading, and the empirical results of pandemic disease support a universal analysis of the relation between the two laws regardless of the biological details of disease. Employing the United States(U.S.) domestic air transportation and demographic data to construct a metapopulation model for simulating the pandemic spread at the U.S. country level, we uncover that the broad heterogeneity of the infrastructure plays a key role in the evolution of scaling emergence. Conclusions/Significance: The analyses of large-scale spatial epidemic spreading help understand the temporal evolution of scalings, indicating the coexistence of the Zipfs law and the Heaps law depends on the collective dynamics of epidemic processes, and the heterogeneity of epidemic spread indicates the significance of performing targeted containment strategies at the early time of a pandemic disease.

A complementary media invisibility cloak that can cloak objects at a distance outside the cloaking shell

Based on the concept of complementary media, we propose an invisibility cloak operating at a finite frequency that can cloak an object with a pre-specified shape and size within a certain distance outside the shell. The cloak comprises of a dielectric core, and an anti-object embedded inside a negative index shell. The cloaked object is not blinded by the cloaking shell since it lies outside the cloak. Full-wave simulations in two dimensions have been performed to verify the cloaking effect.