46 research outputs found
Fractal structures of normal and anomalous diffusion in nonlinear nonhyperbolic dynamical systems
A paradigmatic nonhyperbolic dynamical system exhibiting deterministic
diffusion is the smooth nonlinear climbing sine map. We find that this map
generates fractal hierarchies of normal and anomalous diffusive regions as
functions of the control parameter. The measure of these self-similar sets is
positive, parameter-dependent, and in case of normal diffusion it shows a
fractal diffusion coefficient. By using a Green-Kubo formula we link these
fractal structures to the nonlinear microscopic dynamics in terms of fractal
Takagi-like functions.Comment: 4 pages (revtex) with 4 figures (postscript
Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos
Weakly chaotic non-linear maps with marginal fixed points have an infinite
invariant measure. Time averages of integrable and non-integrable observables
remain random even in the long time limit. Temporal averages of integrable
observables are described by the Aaronson-Darling-Kac theorem. We find the
distribution of time averages of non-integrable observables, for example the
time average position of the particle. We show how this distribution is related
to the infinite invariant density. We establish four identities between
amplitude ratios controlling the statistics of the problem.Comment: 5 pages, 3 figure
Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals
We show that the generalized diffusion coefficient of a subdiffusive
intermittent map is a fractal function of control parameters. A modified
continuous time random walk theory yields its coarse functional form and
correctly describes a dynamical phase transition from normal to anomalous
diffusion marked by strong suppression of diffusion. Similarly, the probability
density of moving particles is governed by a time-fractional diffusion equation
on coarse scales while exhibiting a specific fine structure. Approximations
beyond stochastic theory are derived from a generalized Taylor-Green-Kubo
formula.Comment: 4 pages, 3 eps figure
Understanding deterministic diffusion by correlated random walks
Low-dimensional periodic arrays of scatterers with a moving point particle
are ideal models for studying deterministic diffusion. For such systems the
diffusion coefficient is typically an irregular function under variation of a
control parameter. Here we propose a systematic scheme of how to approximate
deterministic diffusion coefficients of this kind in terms of correlated random
walks. We apply this approach to two simple examples which are a
one-dimensional map on the line and the periodic Lorentz gas. Starting from
suitable Green-Kubo formulas we evaluate hierarchies of approximations for
their parameter-dependent diffusion coefficients. These approximations converge
exactly yielding a straightforward interpretation of the structure of these
irregular diffusion coeficients in terms of dynamical correlations.Comment: 13 pages (revtex) with 5 figures (postscript
Stochastic Model of Virus–Endosome Fusion and Endosomal Escape of pH-Responsive Nanoparticles
In this paper, we set up a stochastic model for the dynamics of active Rab5 and Rab7 proteins on the surface of endosomes and the acidification process that govern the virus–endosome fusion and endosomal escape of pH-responsive nanoparticles. We employ a well-known cut-off switch model for Rab5 to Rab7 conversion dynamics and consider two random terms: white Gaussian and Poisson noises with zero mean. We derive the governing equations for the joint probability density function for the endosomal pH, Rab5 and Rab7 proteins. We obtain numerically the marginal density describing random fluctuations of endosomal pH. We calculate the probability of having a pH level inside the endosome below a critical threshold and therefore the percentage of viruses and pH-responsive nanoparticles escaping endosomes. Our results are in good qualitative agreement with experimental data on viral escape. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.Funding: S.F. and N.K. were funded by EPSRC Grant No. EP/V008641/1. D.A. and I.S. were funded by the Ministry of Science and Higher Education of the Russian Federation (Project No. 075-15-2021-1002)
Electromagnetic Fields on Fractals
Fractals are measurable metric sets with non-integer Hausdorff dimensions. If
electric and magnetic fields are defined on fractal and do not exist outside of
fractal in Euclidean space, then we can use the fractional generalization of
the integral Maxwell equations. The fractional integrals are considered as
approximations of integrals on fractals. We prove that fractal can be described
as a specific medium.Comment: 15 pages, LaTe
Fractional Derivative as Fractional Power of Derivative
Definitions of fractional derivatives as fractional powers of derivative
operators are suggested. The Taylor series and Fourier series are used to
define fractional power of self-adjoint derivative operator. The Fourier
integrals and Weyl quantization procedure are applied to derive the definition
of fractional derivative operator. Fractional generalization of concept of
stability is considered.Comment: 20 pages, LaTe
Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches
Fractional generalization of an exterior derivative for calculus of
variations is defined. The Hamilton and Lagrange approaches are considered.
Fractional Hamilton and Euler-Lagrange equations are derived. Fractional
equations of motion are obtained by fractional variation of Lagrangian and
Hamiltonian that have only integer derivatives.Comment: 21 pages, LaTe
Nonholonomic Constraints with Fractional Derivatives
We consider the fractional generalization of nonholonomic constraints defined
by equations with fractional derivatives and provide some examples. The
corresponding equations of motion are derived using variational principle.Comment: 18 page