Search on a fractal lattice using a quantum random walk

The spatial search problem on regular lattice structures in integer number of dimensions d >= 2 has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

Memory-induced anomalous dynamics in a minimal random walk model

A discrete-time dynamics of a non-Markovian random walker is analyzed using a minimal model where memory of the past drives the present dynamics. In recent work N. Kumar et al., Phys. Rev. E 82, 021101 (2010)] we proposed a model that exhibits asymptotic superdiffusion, normal diffusion, and subdiffusion with the sweep of a single parameter. Here we propose an even simpler model, with minimal options for the walker: either move forward or stay at rest. We show that this model can also give rise to diffusive, subdiffusive, and superdiffusive dynamics at long times as a single parameter is varied. We show that in order to have subdiffusive dynamics, the memory of the rest states must be perfectly correlated with the present dynamics. We show explicitly that if this condition is not satisfied in a unidirectional walk, the dynamics is only either diffusive or superdiffusive (but not subdiffusive) at long times.

A new approach for objective identification of turns and steps in organism movement data relevant to random walk modelling

1. A first step in the analysis of complex movement data often involves discretisation of the path into a series of step-lengths and turns, for example in the analysis of specialised random walks, such as Lévy flights. However, the identification of turning points, and therefore step-lengths, in a tortuous path is dependent on ad-hoc parameter choices. Consequently, studies testing for movement patterns in these data, such as Lévy flights, have generated debate. However, studies focusing on one-dimensional (1D) data, as in the vertical displacements of marine pelagic predators, where turning points can be identified unambiguously have provided strong support for Lévy flight movement patterns. 2. Here, we investigate how step-length distributions in 3D movement patterns would be interpreted by tags recording in 1D (i.e. depth) and demonstrate the dimensional symmetry previously shown mathematically for Lévy-flight movements. We test the veracity of this symmetry by simulating several measurement errors common in empirical datasets and find Lévy patterns and exponents to be robust to low-quality movement data. 3. We then consider exponential and composite Brownian random walks and show that these also project into 1D with sufficient symmetry to be clearly identifiable as such. 4. By extending the symmetry paradigm, we propose a new methodology for step-length identification in 2D or 3D movement data. The methodology is successfully demonstrated in a re-analysis of wandering albatross Global Positioning System (GPS) location data previously analysed using a complex methodology to determine bird-landing locations as turning points in a Lévy walk. For this high-resolution GPS data, we show that there is strong evidence for albatross foraging patterns approximated by truncated Lévy flights spanning over 3·5 orders of magnitude. 5. Our simple methodology and freely available software can be used with any 2D or 3D movement data at any scale or resolution and are robust to common empirical measurement errors. The method should find wide applicability in the field of movement ecology spanning the study of motile cells to humans.

Random walk of magnetic field lines for different values of the energy range spectral index

An analytical nonlinear description of field-line wandering in partially statistically magnetic systems was proposed recently. In this article the influence of the wave spectrum in the energy range onto field-line random walk is investigated by applying this formulation. It is demonstrated that in all considered cases we clearly obtain a superdiffusive behavior of the field-lines. If the energy range spectral index exceeds unity a free-streaming behavior of the field-lines can be found for all relevant length-scales of turbulence. Since the superdiffusive results obtained for the slab model are exact, it seems that superdiffusion is the normal behavior of field-line wandering.

Random walk simulations of fossil proxy data

A wealth of palaeoecological studies (e.g. pollen, diatoms, chironomids and macrofossils from deposits such as lakes or bogs) have revealed major as well as more subtle ecosystem changes over decadal to multimillennial timescales. Such ecosystem changes are usually assumed to have been forced by specific environmental changes. Here, we test if the observed changes in palaeoecological records may be reproduced by random simulations, and we find that simple procedures generate abrupt events, long-term trends, quasi-cyclic behaviour, extinctions and immigrations. Our results highlight the importance of replicated and multiproxy data for reliable reconstructions of past climate and environmental changes.

Detailed analytical investigation of magnetic field line random walk in turbulent plasmas: I. Two-component slab/two-dimensional turbulence

The random displacement of magnetic field lines in the presence of magnetic turbulence in plasmas is investigated from first principles. A two-component (slab/two-dimensional composite) model for the turbulence spectrum is employes. An analytical investigation of the asymptotic behavior of the field-line mean square displacement (FL-MSD) is carried out. It is shown that the magnetic field lines behave superdifusively for every large values of the position variable z, since the FL-MSD sigma varies as sigma similar to z(4/3). An intermediate diffusive regime may also possible exist for finite values of z under conditions which are explicitly determined in terms of the intrinsic turbulent plasma parameters. The superdiffusie asymptotic result is confirmed numerically via an iterative algorithm. The relevance to previous resuslts is discussed.

Detailed analytical investigation of magnetic field line random walk in turbulent plasmas: II. Isotropic turbulence

The random walk of magnetic field lines in the presence of magnetic turbulence in plasmas is investigated from first principles. An isotropic model is employed for the magnetic turbulence spectrum. An analytical investigation of the asymptotic behavior of the field-line mean-square displacement is carried out. in terms of the position variable z. It is shown that varies as similar to z ln z for large distance z. This result corresponds to a superdiffusive behavior of field line wandering. This investigation complements previous work, which relied on a two-component model for the turbulence spectrum. Contrary to that model, quasilinear theory appears to provide an adequate description of the field line random walk for isotropic turbulence.

Nonlinear Field Line Random Walk and Generalized Compound Diffusion of Charged Particles

A new nonlinear theory for the perpendicular transport of charged particles is presented. This approach is based on an improved nonlinear treatment of field line random walk in combination with a generalized compound diffusion model. The generalized compound diffusion model is much more systematic and reliable, in comparison to previous theories. Furthermore, the new theory shows remarkably good agreement with test-particle simulations and heliospheric observations.