Towards deterministic equations for Levy walks: the fractional material derivative

Levy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Levy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Levy walks in which the fractional equivalent of the material derivative occurs. Our approach will be useful for the dynamical formulation of Levy walks in an external force field or in phase space for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited

Mesoscopic description of reactions under anomalous diffusion: A case study

Reaction-diffusion equations deliver a versatile tool for the description of reactions in inhomogeneous systems under the assumption that the characteristic reaction scales and the scales of the inhomogeneities in the reactant concentrations separate. In the present work we discuss the possibilities of a generalization of reaction-diffusion equations to the case of anomalous diffusion described by continuous-time random walks with decoupled step length and waiting time probability densities, the first being Gaussian or Levy, the second one being an exponential or a power-law lacking the first moment. We consider a special case of an irreversible or reversible A ->B conversion and show that only in the Markovian case of an exponential waiting time distribution the diffusion- and the reaction-term can be decoupled. In all other cases, the properties of the reaction affect the transport operator, so that the form of the corresponding reaction-anomalous diffusion equations does not closely follow the form of the usual reaction-diffusion equations

Tight and loose shapes in flat entangled dense polymers

We investigate the effects of topological constraints (entanglements) on two dimensional polymer loops in the dense phase, and at the collapse transition (Theta point). Previous studies have shown that in the dilute phase the entangled region becomes tight, and is thus localised on a small portion of the polymer. We find that the entropic force favouring tightness is considerably weaker in dense polymers. While the simple figure-eight structure, created by a single crossing in the polymer loop, localises weakly, the trefoil knot and all other prime knots are loosely spread out over the entire chain. In both the dense and Theta conditions, the uncontracted knot configuration is the most likely shape within a scaling analysis. By contrast, a strongly localised figure-eight is the most likely shape for dilute prime knots. Our findings are compared to recent simulations.Comment: 8 pages, 5 figure

Fractional Klein-Kramers equation for superdiffusive transport: normal versus anomalous time evolution in a differential L{\'e}vy walk model

We introduce a fractional Klein-Kramers equation which describes sub-ballistic superdiffusion in phase space in the presence of a space-dependent external force field. This equation defines the differential L{\'e}vy walk model whose solution is shown to be non-negative. In the velocity coordinate, the probability density relaxes in Mittag-Leffler fashion towards the Maxwell distribution whereas in the space coordinate, no stationary solution exists and the temporal evolution of moments exhibits a competition between Brownian and anomalous contributions.Comment: 4 pages, REVTe

Universal Multifractality in Quantum Hall Systems with Long-Range Disorder Potential

We investigate numerically the localization-delocalization transition in quantum Hall systems with long-range disorder potential with respect to multifractal properties. Wavefunctions at the transition energy are obtained within the framework of the generalized Chalker--Coddington network model. We determine the critical exponent $\alpha_0$ characterizing the scaling behavior of the local order parameter for systems with potential correlation length $d$ up to $12$ magnetic lengths $l$. Our results show that $\alpha_0$ does not depend on the ratio $d/l$. With increasing $d/l$, effects due to classical percolation only cause an increase of the microscopic length scale, whereas the critical behavior on larger scales remains unchanged. This proves that systems with long-range disorder belong to the same universality class as those with short-range disorder.Comment: 4 pages, 2 figures, postsript, uuencoded, gz-compresse

Polymer translocation out of confined environments

We consider the dynamics of polymer translocation out of confined environments. Analytic scaling arguments lead to the prediction that the translocation time scales like $\tau\sim N^{\beta+\nu_{2D}}R^{1+(1-\nu_{2D})/\nu}$ for translocation out of a planar confinement between two walls with separation $R$ into a 3D environment, and $\tau \sim N^{\beta+1}R$ for translocation out of two strips with separation $R$ into a 2D environment. Here, $N$ is the chain length, $\nu$ and $\nu_{2D}$ are the Flory exponents in 3D and 2D, and $\beta$ is the scaling exponent of translocation velocity with $N$, whose value for the present choice of parameters is $\beta \approx 0.8$ based on Langevin dynamics simulations. These scaling exponents improve on earlier predictions.Comment: 5 pages, 5 figures. To appear in Phys. Rev.

Blinking statistics of a molecular beacon triggered by end-denaturation of DNA

We use a master equation approach based on the Poland-Scheraga free energy for DNA denaturation to investigate the (un)zipping dynamics of a denaturation wedge in a stretch of DNA, that is clamped at one end. In particular, we quantify the blinking dynamics of a fluorophore-quencher pair mounted within the denaturation wedge. We also study the behavioural changes in the presence of proteins, that selectively bind to single-stranded DNA. We show that such a setup could be well-suited as an easy-to-implement nanodevice for sensing environmental conditions in small volumes.Comment: 14 pages, 5 figures, LaTeX, IOP style. Accepted to J Phys Cond Mat special issue on diffusio

First passage time of N excluded volume particles on a line

Motivated by recent single molecule studies of proteins sliding on a DNA molecule, we explore the targeting dynamics of N particles ("proteins") sliding diffusively along a line ("DNA") in search of their target site (specific target sequence). At lower particle densities, one observes an expected reduction of the mean first passage time proportional to 1/N**2, with corrections at higher concentrations. We explicitly take adsorption and desorption effects, to and from the DNA, into account. For this general case, we also consider finite size effects, when the continuum approximation based on the number density of particles, breaks down. Moreover, we address the first passage time problem of a tagged particle diffusing among other particles.Comment: 9 pages, REVTeX, 6 eps figure