19 research outputs found
Random time averaged diffusivities for L\'evy walks
We investigate a L\'evy-Walk alternating between velocities with
opposite sign. The sojourn time probability distribution at large times is a
power law lacking its mean or second moment. The first case corresponds to a
ballistic regime where the ensemble averaged mean squared displacement (MSD) at
large times is , the latter to enhanced diffusion with , . The correlation function and the time averaged
MSD are calculated. In the ballistic case, the deviations of the time averaged
MSD from a purely ballistic behavior are shown to be distributed according to a
Mittag-Leffler density function. In the enhanced diffusion regime, the
fluctuations of the time averages MSD vanish at large times, yet very slowly.
In both cases we quantify the discrepancy between the time averaged and
ensemble averaged MSDs
Asymptotic front behavior in an reaction under subdiffusion
We discuss the front propagation in the reaction under
subdiffusion which is described by continuous time random walks with a
heavy-tailed power law waiting time probability density function. Using a
crossover argument, we discuss the two scaling regimes of the front
propagation: an intermediate asymptotic regime given by the front solution of
the corresponding continuous equation, and the final asymptotics, which is
fluctuation-dominated and therefore lays out of reach of the continuous scheme.
We moreover show that the continuous reaction subdiffusion equation indeed
possesses a front solution that decelerates and becomes narrow in the course of
time. This continuous description breaks down for larger times when the front
gets atomically sharp. We show that the velocity of such fronts decays in time
faster than in the continuous regime
Front propagation in A+B -> 2A reaction under subdiffusion
We consider an irreversible autocatalytic conversion reaction A+B -> 2A under
subdiffusion described by continuous time random walks. The reactants'
transformations take place independently on their motion and are described by
constant rates. The analog of this reaction in the case of normal diffusion is
described by the Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) equation leading
to the existence of a nonzero minimal front propagation velocity which is
really attained by the front in its stable motion. We show that for
subdiffusion this minimal propagation velocity is zero, which suggests
propagation failure
Asymptotic densities of ballistic Lévy walks
We propose an analytical method to determine the shape of density profiles in
the asymptotic long time limit for a broad class of coupled continuous time
random walks which operate in the ballistic regime. In particular, we show that
different scenarios of performing a random walk step, via making an
instantaneous jump penalized by a proper waiting time or via moving with a
constant speed, dramatically effect the corresponding propagators, despite the
fact that the end points of the steps are identical. Furthermore, if the speed
during each step of the random walk is itself a random variable, its
distribution gets clearly reflected in the asymptotic density of random
walkers. These features are in contrast with more standard non-ballistic random
walks
Exactly solvable dynamics of forced polymer loops
Here, we show that a problem of forced polymer loops can be mapped to an asymmetric simple exclusion process with reflecting boundary conditions. The dynamics of the particle system can be solved exactly using the Bethe ansatz. We thus can fully describe the relaxation dynamics of forced polymer loops. In the steady state, the conformation of the loop can be approximated by a combination of Fermi-Dirac and Brownian bridge statistics, while the exact solution is found by using the fermion integer partition theory. With the theoretical framework presented here we establish a link between the physics of polymers and statistics of many-particle systems opening new paths of exploration in both research fields. Our result can be applied to the dynamics of the biopolymers which form closed loops. One such example is the active pulling of chromosomal loops during meiosis in yeast cells which helps to align chromosomes for recombination in the viscous environment of the cell nucleus
Front propagation in the A+B->2A reaction under subdiffusion
We consider an irreversible autocatalytic conversion reaction A+B->2A under subdiffusion described by continuous-time random walks. The reactants transformations take place independently of their motion and are described by constant rates. The analog of this reaction in the case of normal diffusion is described by the Fisher-Kolmogorov-Petrovskii-Piskunov equation leading to the existence of a nonzero minimal front propagation velocity, which is really attained by the front in its stable motion. We show that for subdiffusion, this minimal propagation velocity is zero, which suggests propagation failure
Pulled Polymer Loops as a Model for the Alignment of Meiotic Chromosomes
During recombination, the DNA of parents exchange their genetic information to give rise to a genetically unique offspring. For recombination to occur, homologous chromosomes need to find each other and align with high precision. Fission yeast solves this problem by folding chromosomes in loops and pulling them through the viscous nucleoplasm. We propose a theory of pulled polymer loops to quantify the effect of drag forces on the alignment of chromosomes. We introduce an external force field to the concept of a Brownian bridge and thus solve for the statistics of loop configurations in space