Abstract:
Wave-particle duality finds a natural application for electrons or light propagating in disordered media where coherent corrections to transport are given by two-wave interference. For scatterers with internal degrees of freedom, these corrections are observed to be much smaller than would be expected for structureless scatterers. By examining the basic example of the scattering of one photon by two spin-1/2 atoms--a case-study for coherent backscattering--we demonstrate that the loss of interference strength is associated with which-path information stored by the scattering atoms.

Abstract:
It is well known that, in the Boltzmann-Grad limit, the distribution of the free path length in the Lorentz gas with disordered scatterer configuration has an exponential density. If, on the other hand, the scatterers are located at the vertices of a Euclidean lattice, the density has a power-law tail proportional to xi^{-3}. In the present paper we construct scatterer configurations whose free path lengths have a distribution with tail xi^{-N-2} for any positive integer N. We also discuss the properties of the random flight process that describes the Lorentz gas in the Boltzmann-Grad limit. The convergence of the distribution of the free path length follows from equidistribution of large spheres in products of certain homogeneous spaces, which in turn is a consequence of Ratner's measure classification theorem.

Abstract:
We present a systematical theory for the interplay of strong localization effects and absorption or gain of classical waves in 3-dimensional, disordered dielectrics. The theory is based on the selfconsistent Cooperon resummation, implementing the effects of energy conservation and its absorptive or emissive corrections by an exact, generalized Ward identity. Substantial renormalizations are found, depending on whether the absorption/gain occurs in the scatterers or in the background medium. We find a finite, gain-induced correlation volume which may be significantly smaller than the scale set by the scattering mean free path, even if there are no truly localized modes. Possible consequences for coherent feedback in random lasers as well as the possibility of oscillatory in time behavior induced by sufficiently strong gain are discussed.

Abstract:
We theoretically and numerically investigate the capability of disordered media to enhance the optical path length in dielectric slabs and augment their light absorption efficiency due to scattering. We first perform a series of Monte Carlo simulations of random walks to determine the path length distribution in weakly to strongly (single to multiple) scattering, non-absorbing dielectric slabs under normally incident light and derive analytical expressions for the path length enhancement in these two limits. Quite interestingly, while multiple scattering is expected to produce long optical paths, we find that media containing a vanishingly small amount of scatterers can still provide high path length enhancements due to the very long trajectories sustained by total internal reflection at the slab interfaces. The path length distributions are then used to calculate the light absorption efficiency of media with varying absorption coefficients. We find that maximum absorption enhancement is obtained at an optimal scattering strength, in-between the single-scattering and the diffusive (strong multiple-scattering) regimes. This study can guide experimentalists towards more efficient and potentially low-cost solutions in photovoltaic technologies.

Abstract:
We study the problem of wave transport in a one-dimensional disordered system, where the scatterers of the chain are $n$ barriers and wells with statistically independent intensities and with a spatial extension $\l_c$ which may contain an arbitrary number $\delta/2\pi$ of wavelengths, where $\delta = k l_c$. We analyze the average Landauer resistance and transmission coefficient of the chain as a function of $n$ and the phase parameter $\delta$. For weak scatterers, we find: i) a regime, to be called I, associated with an exponential behavior of the resistance with $n$, ii) a regime, to be called II, for $\delta$ in the vicinity of $\pi$, where the system is almost transparent and less localized, and iii) right in the middle of regime II, for $\delta$ very close to $\pi$, the formation of a band gap, which becomes ever more conspicuous as $n$ increases. In regime II, both the average Landauer resistance and the transmission coefficient show an oscillatory behavior with $n$ and $\delta$. These characteristics of the system are found analytically, some of them exactly and some others approximately. The agreement between theory and simulations is excellent, which suggests a strong motivation for the experimental study of these systems. We also present a qualitative discussion of the results.

Abstract:
The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter $d$, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann-Grad limit, a Poisson distribution of scatters leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the distribution of free path behaves asymptotically like a Cauchy distribution. This paper considers the case when the scatters are distributed on a quasi crystal, i.e. non periodically, but with a long range order. Simulations of a one-dimensional model are presented, showing that the quasi crystal behaves very much like a periodic crystal, and in particular, the distribution of free path lengths is not exponential.

Abstract:
We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

Abstract:
The dynamics of a point particle in a periodic array of spherical scatterers converges, in the limit of small scatterer size, to a random flight process, whose paths are piecewise linear curves generated by a Markov process with memory two. The corresponding transport equation is distinctly different from the linear Boltzmann equation observed in the case of a random configuration of scatterers. In the present paper we provide asymptotic estimates for the transition probabilities of this Markov process. Our results in particular sharpen previous upper and lower bounds on the distribution of free path lengths obtained by Bourgain, Golse and Wennberg.

Abstract:
With the discrete method of the hexagonal cell and three different velocities of particle population in each cell, a two-dimensional lattice Boltzmann model is developed in this paper.1,2] The collision operator in the Boltzmann equation is expanded to fourth order using the Taylor expansion.3,4] With this model, good results have been obtained from the numerical simulation of the reflection phenomenon of the shock wave on the surface of an obstacle, and the numerical stability is also good. Thus the applicability of the D2Q 19 model is verified.

Abstract:
A systematic derivation of Boltzmann equation is presented in the framework of closed-time-path formalism. Introducing a new type of probe, the expectation value of number operator is calculated as a functional of source. Then solving for the source by inverting the relation, the equation of motion for number is obtained when the source is removed, and it turns out to be the Boltzmann equation. The inversion formula is used in the course of derivation.