We introduce a Lévy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers {ξi} are independent random variables identically distributed according to the probability density function µ (ξ) ∼ ξ −(1+γ) . We show that under certain conditions the mean square displacement of the particle obeys x 2 (t) ≥ Ct 3−γ for 1 < γ < 2. This behavior is compatible with a renewal Lévy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process.

For anomalous random walkers, whose mean square displacement behaves like ͗x 2 (t)͘ϳt ␦ (␦ 1), the generalized Einstein relation between anomalous diffusion and the linear response of the walkers to an external field F is studied, using a stochastic modeling approach. A departure from the Einstein relation is expected for weak external fields and long times. We investigate such a departure using the Scher-Lax-Montroll model, defined within the context of the continuous time random walk, and which describes electronic transport in a disordered system with an effective exponent ␦Ͻ1. We then consider a collision model which for the force free case may be mapped on a Lévy walk (␦Ͼ1). We investigate the response in such a model to an external driving force and derive the Einstein relation for it both for equilibrium and ordinary renewal processes. We discuss the time scales at which a departure from the Einstein relation is expected. ͓S1063-651X͑98͒07907-0͔

A theoretical study of linear wave scattering by time-periodic spatially localized excitations (discrete breathers (DB)) is presented. We obtain that the wave propagation is strongly influenced by a local coupling between an open and closed channels generated by the DB. A peculiar effect of total reflection occurs due to a Fano resonance when a localized state originating from closed channels resonates with the open channel. For the discrete nonlinear Schrödinger chain we provide with an analytical result for the frequency dependence of the transmission coefficient, including the possibility of resonant reflection. We extend the analysis to chains of weakly coupled anharmonic oscillators and discuss the relevance of the effect for electronic transport spectroscopy of mesoscopic systems. 42.25.Bs, It is a well established fact that various nonlinear spatially discrete systems can support time-periodic spatially localized excitations called discrete breather states (DB) [1]. These states originate from a peculiar interplay between the nonlinearity and discreteness of the lattice rather than from a disorder. While the nonlinearity yields an amplitude-dependent tunability of frequencies of DBs, Ω b , the spatial discreteness of the system leads to finite upper bounds for the frequency spectrum of small amplitude plane waves ω q . This tunability allows one to escape resonances of all multiples of the breather frequency Ω b with the plane wave frequencies ω q , and correspondingly to stabilize the DB state. The frequency dependent localization length of DB's and their stability with respect to small amplitude perturbations have been widely studied [1]. DBs have been observed in experiments covering such diverse fields as interacting Josephson junctions [2], magnetic systems [3] and lattice dynamics of crystals [4].For propagating linear waves a DB acts as a timeperiodic scattering potential, and the transmission coefficient T depends on both the wave vector q of the linear wave and the breather frequency Ω b . The most peculiar effect, observed in many numerical studies of wave scattering by DBs, is the total reflection as T = 0 [5,6]. Note that the presence of a static potential cannot lead to such a total reflection in one-dimensional systems. Similar features are also discussed in other areas, such as electron transport through point contacts, quantum dots and wires [7,8]. The crucial condition allowing a total reflection in these systems is the presence of a few coupled channels connected with the transverse direction of motion. On the other hand, the wave propagation in the presence of a time-periodic scattering potential is characterized by open and closed channels emerging from the Floquet formalism [1,5,6]. The open channel guides the propagating waves, while the eigenfrequencies of closed channels do not match the spectrum of linear waves.In this Letter we show that the total reflection of linear waves in the open channel occurs when a localized state originating from one of the closed channels resonates wi...

Using the analogy between the description of coherent light propagation in a medium with Kerr nonlinearity by means of nonlinear Schrödinger equation and that of a dissipationless liquid, we propose an optical analogue of the Laval nozzle. The optical Laval nozzle will allow one to form a transonic flow, in which one can observe and study very unusual dynamics of classical and quantum fluctuations, including an analogue of the Hawking radiation of real black holes. Theoretical analysis of these dynamics is supported by numerical calculations, and estimates for a possible experimental realization are presented. PACS numbers: 42.65.Sf; 47.40.Hg Black hole radiation is one of the most impressive phenomena at the intersection of general relativity and quantum field theory. Accounting for the quantum nature of the physical vacuum led to the prediction that a black hole, defined classically as an object that even light cannot escape, in fact can be characterized by a temperature, entropy [1] and moreover emits thermal radiation [2]. Since direct experimentation with black holes is hardly possible, it was suggested to consider analogous phenomena in condensed matter physics where the "high-energy" (short-wavelength) physics is known [3]. The suggestion was based on the observation that the derivation of the Hawking radiation uses only the linear wave equation in curved space-time, and not the Einstein equations. The same conditions for wave propagation arise when considering sound propagation in a fluid when the background flow is non-trivial [3]. This similarity was shown to be sufficient for applying the original considerations by Hawking, predicting the existence of thermal radiation of quantum origin from the fluid counterpart of the horizon, which can be called Mach horizon (where the fluid velocity equals the sound velocity, i.e. Mach number M = 1). These ideas were further developed for possible applications in BEC fluids and other systems [4][5][6][7][8]. As for experimental realizations, a white hole horizon was observed in optical fibers [9], where the probe light was back-reflected from a moving soliton, and a black hole horizon was observed in a BEC system [10].We propose an experimental setup capable of creating a Mach horizon in an optical medium with Kerr nonlinearity, which may be called optical Laval nozzle. The propagation of weakly nonlinear coherent optical pulses is described by the non-linear Schrödinger (NLS) equationIt assumes the paraxial approximation, according to which the electric field of the light wave is written as E = A(x, y, t; z)e −iβ0z , where A(x, y, t; z) is weakly z dependent complex amplitude of the light propagating in the z direction. The time coordinate t is converted into the τ coordinate[11] which describes the shape of the light pulse in the moving coordinate system. The radius vector differential is now dr = (dx, dy, dτ ) (anomalous dispersion) or dr = (dx, dy, −idτ ) (normal dispersion). U (x, y) is the equivalent external potential created by spacial variation of the ...

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