arXiv subjects

Alexander Iomin

Publications and source records attributed to Alexander Iomin.

At least 37 records · Page 2Linked to original sources

arXiv2016

Anomalous diffusion on a fractal mesh

An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets of both backbones and fingers are constructed on the entire (infinite) $y$ and $x$ axises. To this end we suggested a special algorithm of this special construction. The transport properties of the fractal mesh is studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $\langle x^2(t)\rangle\sim t^β$, where the transport exponent $β<1$ is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with $β>1$ has been observed as well when the environment is controlled by means of a memory kernel.

cond-mat.stat-mech↗

arXiv2016

Lévy Transport in Slab Geometry of Inhomogeneous Media

We present a physical example, where a fractional (both in space and time) Schrödinger equation appears only as a formal effective description of diffusive wave transport in complex inhomogeneous media. This description is a result of the parabolic equation approximation that corresponds to the paraxial small angle approximation of the fractional Helmholtz equation. The obtained effective quantum dynamics is fractional in both space and time. As an example, Lévy flights in an infinite potential well are considered numerically. An analytical expression for the effective wave function of the quantum dynamics is obtained as well.

quant-ph↗

arXiv2016

Lévy processes on a generalized fractal comb

Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of the structure can destroy this Markovian transport. These effects can be described by the memory and spatial kernels. In particular, the fractal structure of the fingers, which is controlled by the spatial kernel in both the real and the Fourier spaces, leads to the Lévy processes (Lévy flights) and superdiffusion. This generalization of the fractional diffusion is described by the Riesz space fractional derivative. In the framework of this generalized fractal comb model, Lévy processes are considered, and exact solutions for the probability distribution functions are obtained in terms of the Fox $H$-function for a variety of the memory kernels, and the rate of the superdiffusive spreading is studied by calculating the fractional moments. For a special form of the memory kernels, we also observed a competition between long rests and long jumps. Finally, we considered the fractional structure of the fingers controlled by a Weierstrass function, which leads to the power-law kernel in the Fourier space. It is a special case, when the second moment exists for superdiffusion in this competition between long rests and long jumps.

cond-mat.stat-mech↗

arXiv2016

Comb model with slow and ultraslow diffusion

We consider a generalised diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyse the probability distribution functions and we derive the mean squared displacement in $x$ and $y$ directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both $x$ and $y$ directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.

cond-mat.stat-mech↗

arXiv2015

Mesoscopic description of random walks on combs

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study continuous time random walks on combs and present a generic method to obtain their transport properties. The random walk along the branches may be biased, and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone. We analyze the overall diffusion properties along the backbone and find normal diffusion, anomalous diffusion, and stochastic localization (diffusion failure), respectively, depending on the characteristics of the continuous time random walk along the branches.

cond-mat.stat-mech↗

arXiv2015

Hyperdiffusion of quantum waves in random photonic lattices

A quantum-mechanical analysis of hyper-fast (faster than ballistic) diffusion of a quantum wave packet in random optical lattices is presented. The main motivation of the presented analysis is experimental demonstrations of hyper-diffusive spreading of a wave packet in random photonic lattices [L. Levi \textit{et al.}, Nature Phys. \textbf{8}, 912 (2012)]. A rigorous quantum-mechanical calculation of the mean probability amplitude is suggested, and it is shown that the power law spreading of the mean squared displacement (MSD) is $< x^2(t)>\sim t^α$, where $2<α\leq 3$. The values of the transport exponent $α$ depend on the correlation properties of the random potential $V(x,t)$, which describes random inhomogeneities of the medium. In particular, when the random potential is $δ$ correlated in time, the quantum wave packet spreads according Richardson turbulent diffusion with the MSD $\sim t^3$. Hyper-diffusion with $α=12/5$ is also obtained for arbitrary correlation properties of the random potential.

physics.optics↗

arXiv2015

Fractional diffusion on a fractal grid comb

A grid comb model is a generalization of the well known comb model, and it consists of $N$ backbones. For $N=1$ the system reduces to the comb model where subdiffusion takes place with the transport exponent $1/2$. We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure.

cond-mat.stat-mech↗

arXiv2014

A topological approximation of the nonlinear Anderson model

We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrodinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior, depending on the topology of resonance-overlap in phase space, ranging from a fully developed chaos involving Levy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. We describe this localization-delocalization transition as a percolation transition on a Cayley tree. It is found in vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t\rightarrow+\infty. The second moment grows with time as a powerlaw t^α, with α= 1/3. Also we find for superquadratic nonlinearity that the analog pseudochaotic regime at the edge of chaos is self-controlling in that it has feedback on the topology of the structure on which the transport processes concentrate. Then the system automatically (without tuning of parameters) develops its percolation point. We classify this type of behavior in terms of self-organized criticality dynamics in Hilbert space. For subquadratic nonlinearities, the behavior is shown to be sensitive to details of definition of the nonlinear term. A transport model is proposed based on modified nonlinearity, using the idea of stripes propagating the wave process to large distances. Theoretical investigations, presented here, are the basis for consistency analysis of the different localization-delocalization patterns in systems with many coupled degrees of freedom in association with the asymptotic properties of the transport.

nlin.CD↗

arXiv2013

Comb-like models for transport along spiny dendrites

We suggest a modification of a comb model to describe anomalous transport in spiny dendrites. Geometry of the comb structure consisting of a one-dimensional backbone and lateral branches makes it possible to describe anomalous diffusion, where dynamics inside fingers corresponds to spines, while the backbone describes diffusion along dendrites. The presented analysis establishes that the fractional dynamics in spiny dendrites is controlled by fractal geometry of the comb structure and fractional kinetics inside the spines. Our results show that the transport along spiny dendrites is subdiffusive and depends on the density of spines in agreement with recent experiments.

cond-mat.stat-mech↗

arXiv2012

Superdiffusive comb: Application to experimental observation of anomalous diffusion in one dimension

A possible mechanism of superdiffusion of ultra-cold atoms in a one-dimensional polarization optical lattice, observed experimentally in [Phys. Rev. Lett. \textbf{108}, 093002 (2012)], is suggested. The analysis is based on a consideration of anomalous diffusion in a fractal comb [Phys. Rev. E \textbf{83}, 052106 (2011)]. It is shown that the transport exponent is determined by the fractal geometry of the comb due to recoil distributions resulting in Lévy flights of atoms.

cond-mat.stat-mech↗

arXiv2011

Electrostatics in Fractal Geometry: Fractional Calculus Approach

The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume $V_d\sim r^d$ with the fractal dimension $d$ and a complementary host volume $V_h=V_3-V_d$. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.

cond-mat.stat-mech↗

arXiv2011

Fractional-Time Schrödinger Equation: Fractional Dynamics on a Comb

The physical relevance of the fractional time derivative in quantum mechanics is discussed. It is shown that the introduction of the fractional time Scrödinger equation (FTSE) in quantum mechanics by analogy with the fractional diffusion $\frac{\prt}{\prt t}\rightarrow \frac{\prt^α}{\prt t^α}$ can lead to an essential deficiency in the quantum mechanical description, and needs special care. To shed light on this situation, a quantum comb model is introduced. It is shown that for $α=1/2$, the FTSE is a particular case of the quantum comb model. This \textit{exact} example shows that the FTSE is insufficient to describe a quantum process, and the appearance of the fractional time derivative by a simple change $\frac{\prt}{\prt t}\rightarrow \frac{\prt^α}{\prt t^α}$ in the Schrödinger equation leads to the loss of most of the information about quantum dynamics.

quant-ph↗

arXiv2011

Subdiffusion on a Fractal Comb

Subdiffusion on a fractal comb is considered. A mechanism of subdiffusion with a transport exponent different from 1/2 is suggested. It is shown that the transport exponent is determined by the fractal geometry of the comb.

cond-mat.dis-nn↗

arXiv2009

Subdiffusion in the Nonlinear Schroedinger Equation with Disorder

The nonlinear Schroedinger equation in the presence of disorder is considered. The dynamics of an initially localized wave packet is studied. A subdiffusive spreading of the wave packet is explained in the framework of a continuous time random walk. A probabilistic description of subdiffusion is suggested and a transport exponent of subdiffusion is obtained to be 2/5.

cond-mat.stat-mech↗

arXiv2009

On fractional time quantum dynamics

Application of the fractional calculus to quantum processes is presented. In particular, the quantum dynamics is considered in the framework of the fractional time Schrödinger equation (SE), which differs from the standard SE by the fractional time derivative: $\frac{\prt}{\prt t}\to \frac{\prt^α}{\prt t^α}$. It is shown that for $α=1/2$ the fractional SE is isospectral to a comb model. An analytical expression for the Green functions of the systems are obtained. The semiclassical limit is discussed.

math-ph↗

arXiv2009

Lyapunov exponents in 1d disordered system with long-range memory

The Lyapunov exponents for Anderson localization are studied in a one dimensional disordered system. A random Gaussian potential with the power law decay $\sim 1/|x|^q$ of the correlation function is considered. The exponential growth of the moments of the eigenfunctions and their derivative is obtained. Positive Lyapunov exponents, which determine the asymptotic growth rate are found.

cond-mat.stat-mech↗

arXiv2008

Asymptotic localization of stationary states in the nonlinear Schroedinger equation

The mapping of the Nonlinear Schroedinger Equation with a random potential on the Fokker-Planck equation is used to calculate the localization length of its stationary states. The asymptotic growth rates of the moments of the wave function and its derivative for the linear Schroedinger Equation in a random potential are computed analytically and resummation is used to obtain the corresponding growth rate for the nonlinear Schroedinger equation and the localization length of the stationary states.

cond-mat.stat-mech↗

arXiv2008

Probabilistic approach to a proliferation and migration dichotomy in the tumor cell invasion

The proliferation and migration dichotomy of the tumor cell invasion is examined within a two-component continuous time random walk (CTRW) model. The balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration are derived. The transport of tumor cells is formulated in terms of the CTRW with an arbitrary waiting time distribution law, while proliferation is modelled by a logistic growth. The overall rate of tumor cell invasion for normal diffusion and subdiffusion is determined.

q-bio.CB↗