We study Monte Carlo dynamics of the monomers and center of mass of a model polymer chain functionalized with azobenzene molecules in the presence of an inhomogeneous linearly polarized laser light. ...The simulations use a generalized Bond Fluctuation Model. The mean squared displacements of the monomers and the center of mass are analyzed in a period of Monte Carlo time typical for a build-up of Surface Relief Grating. Approximate scaling laws for mean squared displacements are found and interpreted in terms of sub- and superdiffusive dynamics for the monomers and center of mass. A counterintuitive effect is observed, where the monomers perform subdiffusive motion but the resulting motion of the center of mass is superdiffusive. This result disparages theoretical approaches based on an assumption that the dynamics of single monomers in a chain can be characterized in terms of independent identically distributed random variables.
•A physically based derivation of the fractional porous medium equation is given.•The pressure form of the equation is preferred as a consequence of the nonlocal conservation law.•A hydrological ...application for an important equation in nonlocal analysis is provided.
In this short note we consider a nonlinear and spatially nonlocal PDE modeling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We provide a deterministic derivation which allows us to naturally incorporate the nonlinear effects. This reasoning shows that in our setting the so-called nonlocal pressure form of the porous medium equation is preferred as a description of the evolution. In that case the governing nonlocal operator is the fractional gradient rather than the fractional Laplacian.
Over-fishing may lead to a decrease in fish abundance and a proliferation of jellyfish. Active movements and prey search might be thought to provide a competitive advantage for fish, but here we use ...data-loggers to show that the frequently occurring coastal jellyfish (Rhizostoma octopus) does not simply passively drift to encounter prey. Jellyfish (327 days of data from 25 jellyfish with depth collected every 1 min) showed very dynamic vertical movements, with their integrated vertical movement averaging 619.2 m d−1, more than 60 times the water depth where they were tagged. The majority of movement patterns were best approximated by exponential models describing normal random walks. However, jellyfish also showed switching behaviour from exponential patterns to patterns best fitted by a truncated Lévy distribution with exponents (mean μ = 1.96, range 1.2–2.9) close to the theoretical optimum for searching for sparse prey (μopt ≈ 2.0). Complex movements in these ‘simple’ animals may help jellyfish to compete effectively with fish for plankton prey, which may enhance their ability to increase in dominance in perturbed ocean systems.
It has long been a challenge to accurately and efficiently simulate exciton-phonon dynamics in mesoscale photosynthetic systems with a fully quantum mechanical treatment due to extensive ...computational resources required. In this work, we tackle this seemingly intractable problem by combining the Dirac-Frenkel time-dependent variational method with Davydov trial states and implementing the algorithm in graphic processing units. The phonons are treated on the same footing as the exciton. Tested with toy models, which are nanoarrays of the B850 pigments from the light harvesting 2 complexes of purple bacteria, the methodology is adopted to describe exciton diffusion in huge systems containing more than 1600 molecules. The superradiance enhancement factor extracted from the simulations indicates an exciton delocalization over two to three pigments, in agreement with measurements of fluorescence quantum yield and lifetime in B850 systems. With fractal analysis of the exciton dynamics, it is found that exciton transfer in B850 nanoarrays exhibits a superdiffusion component for about 500 fs. Treating the B850 ring as an aggregate and modeling the inter-ring exciton transfer as incoherent hopping, we also apply the method of classical master equations to estimate exciton diffusion properties in one-dimensional (1D) and two-dimensional (2D) B850 nanoarrays using derived analytical expressions of time-dependent excitation probabilities. For both coherent and incoherent propagation, faster energy transfer is uncovered in 2D nanoarrays than 1D chains, owing to availability of more numerous propagating channels in the 2D arrangement.
We relate the fractional powers of the discrete Laplacian with a standard time-fractional derivative in the sense of Liouville by encoding the iterative nature of the discrete operator through a ...time-fractional memory term.
Survival of an evasive prey Oshanin, G; Vasilyev, O; Krapivsky, P.L ...
Proceedings of the National Academy of Sciences - PNAS,
08/2009, Letnik:
106, Številka:
33
Journal Article
Recenzirano
Odprti dostop
We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears ...within their sighting range. The prey is caught when a predator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range; otherwise the prey stays still. We show that if the sighting range of such a lazy prey is equal to 1 lattice spacing, at least 3 predators are needed in order to catch the prey on a square lattice. In this situation, we establish a simple asymptotic relation lnP ev(t) ~ (N/V)²lnP imm(t) between the survival probabilities of an evasive and an immobile prey. Hence, when the density ρ = N/V of the predators is low, ρ << 1, the lazy evasion strategy leads to the spectacular increase of the survival probability. We also argue that a short-sighting prey (its sighting range is smaller than the sighting range of the predators) undergoes an effective superdiffusive motion, as a result of its encounters with the predators, whereas a far-sighting prey performs a diffusive-type motion.
The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher ...randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems involve multiple searchers, and the important timescale is the time it takes the fastest searcher to find a target, which is called an extreme FHT. In this paper, we study extreme FHTs for any stochastic process that is a random time change of Brownian motion by a Lévy subordinator. This class of stochastic processes includes superdiffusive Lévy flights in any space dimension, which are processes described by a Fokker–Planck equation with a fractional Laplacian. We find the short-time distribution of a single FHT for any Lévy subordinate Brownian motion and use this to find the full distribution and moments of extreme FHTs as the number of searchers grows. We illustrate these rigorous results in several examples and numerical simulations.
In order to study the behaviour of discrete dynamical systems under adiabatic cyclic variations of their parameters, we consider discrete versions of adiabatically-rotated rotators. Parallelling the ...studies in continuous systems, we generalize the concept of geometric phase to discrete dynamics and investigate its presence in these rotators. For the rotated sine circle map, we demonstrate an analytical relationship between the geometric phase and the rotation number of the system. For the discrete version of the rotated rotator considered by Berry, the rotated standard map, we further explore this connection as well as the role of the geometric phase at the onset of chaos. Further into the chaotic regime, we show that the geometric phase is also related to the diffusive behaviour of the dynamical variables and the Lyapunov exponent.
•We extend the concept of geometric phase to maps.•For the rotated sine circle map, we demonstrate an analytical relationship between the geometric phase and the rotation number.•For the rotated standard map, we explore the role of the geometric phase at the onset of chaos.•We show that the geometric phase is related to the diffusive behaviour of the dynamical variables and the Lyapunov exponent.
Some aspects of nonlocal dynamics on directed and undirected networks for an initial value problem whose Jacobian matrix is a variable-order fractional power of a Laplacian matrix are discussed here. ...This is a new extension to non-stationary behaviour of a class of non-local phenomena on complex networks for which both directed and undirected graphs are considered. Under appropriate assumptions, the existence, uniqueness, and uniform asymptotic stability of the solutions of the underlying initial value problem are proved. Some examples giving a sample of the behaviour of the dynamics are also included.
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