Some insights in superdiffusive transport

In a wide range of systems, the relaxation in response to an initial pulse has been experimentally found to follow a nonlinear relationship for the mean squared displacement, of the kind 〈x²(t)〉∝t^α$〈x^2(t)〉\propto t^{\alpha }$, where α$\alpha$ may be greater or smaller than 1. Such phenomena have been described under the generic term of anomalous diffusion. “Lévy flights” stochastic processes lead to superdiffusive behaviour (1<α<2)$(1<\alpha <2)$ and have been recently proposed to model—among the others—the subsurface contaminant spread in highly heterogeneous media under the effects of water flow. In this paper, within the continuous-time random walk (CTRW) approach to anomalous diffusion, we compare the analytical solution of the approximated fractional diffusion equation (FDE) with the Monte Carlo one, obtained by simulating the superdiffusive behaviour of an ensemble of particle in a medium. We show that the two are neatly different as the process approaches the standard diffusive behaviour. We argue that this is due to a truncation in the Fourier space expansion introduced by the FDE approach. We propose a second-order correction to this expansion and numerically solve the CTRW model under this hypothesis: the accuracy of the results thus obtained is validated through Monte Carlo simulation over all the superdiffusive range. The same kind of discrepancy is shown to occur also in the derivation of the fractional moments of the distribution: analogous corrections are proposed and validated through the Monte Carlo approach.     

Photon channelling in foams

Experiments by Gittings, Bandyopadhyay and Durian (Europhys. Lett. 65, 414 (2004)) demonstrate that light possesses a higher probability to propagate in the liquid phase of a foam due to total reflection. The authors term this observation photon channelling which we investigate in this article theoretically. We first derive a central relation in the work of Gitting et al. without any free parameters. It links the photon's path-length fraction f in the liquid phase to the liquid fraction ɛ. We then construct two-dimensional Voronoi foams, replace the cell edges by channels to represent the liquid films and simulate photon paths according to the laws of ray optics using transmission and reflection coefficients from Fresnel's formulas. In an exact honeycomb foam, the photons show superdiffusive behavior. It becomes diffusive as soon as disorder is introduced into the foams. The dependence of the diffusion constant on channel width and refractive index is explained by a one-dimensional random-walk model. It contains a photon channelling state that is crucial for the understanding of the numerical results. At the end, we shortly comment on the observation that photon channelling only occurs in a finite range of ɛ.     

Power-law autocorrelated stochastic processes with long-range cross-correlations

We develop a stochastic process with two coupled variables where the absolute values of each variable exhibit long-range power-law autocorrelations and are also long-range cross-correlated. We investigate how the scaling exponents characterizing power-law autocorrelation and long-range cross-correlation behavior in the absolute values of the generated variables depend on the two parameters in our model. In particular, if the autocorrelation is stronger, the cross-correlation is also stronger. We test the utility of our approach by comparing the autocorrelation and cross-correlation properties of the time series generated by our model with data on daily returns over ten years for two major financial indices, the Dow Jones and the S&P500, and on daily returns of two well-known company stocks, IBM and Microsoft, over five years.     

Self-quenched dynamics

We introduce a model for the slow relaxation of an energy landscape caused by its local interaction with a random walker whose motion is dictated by the landscape itself. By choosing relevant measures of time and potential this self-quenched dynamics can be mapped on to the “True” Self-Avoiding Walk model. This correspondence reveals that the average distance of the walker at time t from its starting point is , where for one dimension and 1/2 for all higher dimensions. Furthermore, the evolution of the landscape is similar to that in growth models with extremal dynamics. Received 8 August 2000     

A Cautionary note on modeling with fractional Lévy flights

Temporal scaling and infinite variance are two stylized features often seen together in times series of complex systems. We find that because of their infinite moments samples from fractional Lévy flights produce bi-linear scaling functions which may be incorrectly attributed as evidence of multifractality. We argue that it is unnecessary to consider truncated fractional Lévy flights which are inherently problematic.     

Random Flights in Higher Spaces

We consider in this paper random flights in ℝ ^d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S ₁ ^d . We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ³ are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2. Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.     

A spectroscopic investigation of turbulence in magnetized plasmas

We present a spectroscopic investigation of turbulence in the Tore-Supra edge plasma, where deuterium spectral lines are found to exhibit a power-law behavior in their wings. Such a feature is not predicted by the equilibrium line broadening theory in the conditions of the edge plasma, where the thermal Stark effect is negligible. Therefore, the possible role of turbulence is investigated along two separate paths. Indeed, both the Stark and the Doppler profiles may differ significantly from the equilibrium profiles.     

Energy and number of collision fluctuations in inelastic gases

The two-dimensional Inelastic Maxwell Model (IMM) is studied by numerical simulations. It is shown how the inelasticity of collisions together with the fluctuations of the number of collisions undergone by a particle lead to energy fluctuations. These fluctuations are associated to a shrinking of the available phase space. We find the asymptotic scaling of these energy fluctuations and show how they affect the tail of the velocity distribution during long time intervals. We stress that these fluctuations relax like power laws on much slower time scales than the usual exponential relaxations taking place in kinetic theory.     

Voltage distribution in growing conducting networks

We investigate by random-walk simulations and a mean-field theory how growth by biased addition of nodes affects flow of the current through the emergent conducting graph, representing a digital circuit. In the interior of a large network the voltage varies with the addition time s < t of the node as V(s) ∼ ln(s)/s ^θ when constant current enters the network at last added node t and leaves at the root of the graph which is grounded. The topological closeness of the conduction path and shortest path through a node suggests that the charged random walk determines these global graph properties by using only local search algorithms. The results agree with mean-field theory on tree structures, while the numerical method is applicable to graphs of any complexity. Received 26 August 2002 Published online 29 November 2002     

Spectral line shapes modeling in turbulent plasmas

In this work we investigate the influence of low frequency turbulence on Doppler spectral line shapes in magnetized plasmas. Low frequency refers here to fluctuations whose typical time scale is much larger than those characterizing the atomic processes, such as radiative decay, collisions and charge exchange. This ordering is in particular relevant for drift wave turbulence, ubiquitous in edge plasmas of fusion devices. Turbulent fluctuations are found to affect line shapes through both the spatial and time averages introduced by the measurement process. The profile is expressed in terms of the fluid fields describing the plasma. Assuming the spectrometer acquisition time to be much larger than the turbulent time scale, an ordering generally fulfilled in experiments, allows to develop a statistical formalism. We proceed by successively investigating the effects of density, fluid velocity and temperature fluctuations on the Doppler profile of a spectral line emitted by a charge exchange population of neutrals. Line shapes, and especially line wings are found to be affected by ion temperature or fluid velocity fluctuations, and can in some cases exhibit a power-law behavior. These effects are shown to be measurable with existing techniques, and their interpretation in each particular case would rely on already existing tools. From a fundamental point of view, this study gives some insights in the appearance of non-Boltzmann statistics, such as Lévy statistics, when dealing with averaged experimental data.