https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

Acknowledgements 1. Introduction 2. Electromagnetic wave propagation 3. The absorption of light 4. Specular reflection 5. Single particle scattering: perfect spheres 6. Single particle scattering: irregular particles 7. Propagation in a nonuniform medium: the equation of radiative transfer 8. The bidirectional reflectance of a semi-infinite medium 9. The opposition effect 10. A miscellany of bidirectional reflectances and related quantities 11. Integrated reflectances and planetary photometry 12. Photometric effects of large scale roughness 13. Polarization 14. Reflectance spectroscopy 15. Thermal emission and emittance spectroscopy 16. Simultaneous transport of energy by radiation and conduction Appendix A. A brief review of vector calculus Appendix B. Functions of a complex variable Appendix C. The wave equation in spherical coordinates Appendix D. Fraunhoffer diffraction by a circular hole Appendix E. Table of symbols Bibliography Index.

Theory of Reflectance and Emittance Spectroscopy

Modern cryptography relies on algorithmic one-way functions—numerical functions which are easy to compute but very difficult to invert. This dissertation introduces physical one-way functions and physical one-way hash functions as primitives for physical analogs of cryptosystems. 
Physical one-way functions are defined with respect to a physical probe and physical system in some unknown state. A function is called a physical one-way function if (a) there exists a deterministic physical interaction between the probe and the system which produces an output in constant time; (b) inverting the function using either computational or physical means is difficult; (c) simulating the physical interaction is computationally demanding and (d) the physical system is easy to make but difficult to clone. 
Physical one-way hash functions produce fixed-length output regardless of the size of the input. These hash functions can be obtained by sampling the output of physical one-way functions. For the system described below, it is shown that there is a strong correspondence between the properties of physical one-way hash functions and their algorithmic counterparts. In particular, it is demonstrated that they are collision-resistant and that they exhibit the avalanche effect, i.e., a small change in the physical system causes a large change in the hash value. 
An inexpensive prototype authentication system based on physical one-way hash functions is designed, implemented, and analyzed. The prototype uses a disordered three-dimensional microstructure as the underlying physical system and coherent radiation as the probe. It is shown that the output of the interaction between the physical system and the probe can be used to robustly derive a unique tamper-resistant identifier at a very low cost per bit. The explicit use of three-dimensional structures marks a departure from prior efforts. Two protocols, including a one-time pad protocol, that illustrate the utility of these hash functions are presented and potential attacks on the authentication system are considered. 
Finally, the concept of fabrication complexity is introduced as a way of quantifying the difficulty of materially cloning physical systems with arbitrary internal states. Fabrication complexity is discussed in the context of an idealized machine—a Universal Turing Machine augmented with a fabrication head—which transforms algorithmically minimal descriptions of physical systems into the systems themselves. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

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Physical one-way functions

The transport properties of disordered solids have been the subject of much work since at least the 1950s, but with a new burst of activity during the 1980s which has survived up to the present day. There have been numerous reviews of a more or less specialized nature. The present review aims to fill the niche for a non-specialized review of this very active area of research. The basic concepts behind the theory are introduced with more detailed sections covering experimental results, one-dimensional localization, scaling theory, weak localization, magnetic field effects and fluctuations.

https://iopscience.iop.org/article/10.1088/0034-4885/56/12/001/pdf

Localization: theory and experiment

One of the most interesting phenomena in solid-state physics is Anderson localization, which predicts that an electron may become immobile when placed in a disordered lattice. The origin of localization is interference between multiple scatterings of the electron by random defects in the potential, altering the eigenmodes from being extended (Bloch waves) to exponentially localized. As a result, the material is transformed from a conductor to an insulator. Anderson's work dates back to 1958, yet strong localization has never been observed in atomic crystals, because localization occurs only if the potential (the periodic lattice and the fluctuations superimposed on it) is time-independent. However, in atomic crystals important deviations from the Anderson model always occur, because of thermally excited phonons and electron-electron interactions. Realizing that Anderson localization is a wave phenomenon relying on interference, these concepts were extended to optics. Indeed, both weak and strong localization effects were experimentally demonstrated, traditionally by studying the transmission properties of randomly distributed optical scatterers (typically suspensions or powders of dielectric materials). However, in these studies the potential was fully random, rather than being 'frozen' fluctuations on a periodic potential, as the Anderson model assumes. Here we report the experimental observation of Anderson localization in a perturbed periodic potential: the transverse localization of light caused by random fluctuations on a two-dimensional photonic lattice. We demonstrate how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder. Finally, we study how nonlinearities affect Anderson localization. As Anderson localization is a universal phenomenon, the ideas presented here could also be implemented in other systems (for example, matter waves), thereby making it feasible to explore experimentally long-sought fundamental concepts, and bringing up a variety of intriguing questions related to the interplay between disorder and nonlinearity.

Transport and Anderson localization in disordered two-dimensional photonic lattices

Recent experiments have confirmed that coherent effects in the multiple scattering of light affect the angular dependence of the intensity reflected by disordered media. By considering the constructive interferences between time-reversed paths of light in a semi-infinite medium, we analyze the experimental line shape of the albedo within the diffusion approximation and explain the observed effects of polarization.

Coherent backscattering of light by disordered media: Analysis of the peak line shape.

We show that the use of the recently proposed thermostatistics based on the generalized entropic form ${S}_{q}\ensuremath{\equiv}\frac{k(1\ensuremath{-}\ensuremath{\Sigma}{i}^{}{p}_{i}^{q})}{(q\ensuremath{-}1)}$ (where $q\ensuremath{\in}R$, with $q=1$ corresponding to the Boltzmann-Gibbs-Shannon entropy $\ensuremath{-}k\ensuremath{\Sigma}{i}^{}{p}_{i} \mathrm{ln} {p}_{i}$), together with the L\'evy-Gnedenko generalization of the central limit theorem, provide a basic step towards the understanding of why L\'evy distributions are ubiquitous in nature. A consistent experimental verification is proposed.

/pdf/statistical-mechanical-foundation-of-the-ubiquity-of-levy-41tpd8n9ig.pdf

Statistical-mechanical foundation of the ubiquity of Lévy distributions in Nature.

A theoretical study of the coherent backscattering effect of light from disordered semi-inifinite media is presented for various situations including time-dependent effects as well as absorption and amplitude modulation. Particular attention is devoted to the case of anisotropic scattering and to polarization in order to explain quantitatively experimental results. A microscopic derivation of the coherent albedo is given which strongly supports the heuristic formula previously established. In addition the coherent albedo of a fractal system is predicted. The validity of the different approximations used are discussed and some further theoretical developments are presented On inclut les effets dependant du temps, les milieux absorbants et les effets lies a la modulation d'amplitude de la lumiere

Theoretical study of the coherent backscattering of light by disordered media

A detailed experimental study of coherent backscattering of light from aqueous suspensions of polystyrene microspheres is presented. Emphasis is on the effects of particle size, of absorption due to added dye and of light polarization on the shape and height of the backscattering cone. For parallel polarization of incident and scattered beams, the scalar diffusion theory, parametrized by the transport mean free path l*, agrees well with our data up to surprizingly large scattering angles (ql * ~ 1) and quantitatively accounts for the rounding of the cones due to absorption. No deviations from the usual Gaussian statistics of scattered fields is observed up to λ/l* ∼ 0.1.

https://hal.archives-ouvertes.fr/jpa-00210675/document

Optical coherent backscattering by random media : an experimental study

A new method of visualizing objects with distinct internal dynamics of the constituent scattering particles embedded in a liquid multiple-scattering medium is presented. We report dynamic multiple-light-scattering experiments and a theoretical model, based on diffusing photon-density waves for concentrated colloidal suspensions in Brownian motion, as a background medium into which is inserted a capillary containing (i) the same suspension under flow, or (ii) suspensions of different particle sizes in Brownian motion. These model objects, with purely dynamic but no static scattering contrast, can be visualized by space-resolved measurements of the time autocorrelation function g2(t) of the scattered light intensity at the sample surface. Maximum contrast occurs at a parameter-dependent finite correlation time t. The physical origin of this effect is outlined. Our data are in excellent quantitative agreement with the model, with no adjustable parameter. © 1997 Optical Society of America. [S0740-3232(97)00701-1]

/pdf/imaging-of-dynamic-heterogeneities-in-multiple-scattering-4mu5l1y8yn.pdf

Roger Maynard

Papers

Coherent backscattering of light by disordered media: Analysis of the peak line shape.

Statistical-mechanical foundation of the ubiquity of Lévy distributions in Nature.

Theoretical study of the coherent backscattering of light by disordered media

Optical coherent backscattering by random media : an experimental study

Imaging of dynamic heterogeneities in multiple-scattering media