arXiv subjects

Alexander Iomin

Publications and source records attributed to Alexander Iomin.

At least 19 recordsLinked to original sources

arXiv2026

Anomalous diffusion in porous fractal media

We suggest a model of a diffusive process inside a fractal sponge structure, which is a generalization of the diffusion processes on a comb and fractal mesh structure. The sponge model is considered as the direct product of Cantor sets. It is shown that the corresponding one-dimensional diffusion process is governed by a generalized Fokker-Planck equation with a power-law memory kernel and a position-dependent diffusion coefficient. That is, the fractal structure of the medium induces memory effects and heterogeneity in the transport system. The considered model may be of interest to describe anomalous heat transport in porous fractal media.

cond-mat.stat-mech↗

arXiv2025

Turbulence spreading and anomalous diffusion on combs

This paper presents a simple model for such processes as chaos spreading or turbulence spillover into stable regions. In this simple model the essential transport occurs via inelastic resonant interactions of waves on a lattice. The process is shown to result universally in a subdiffusive spreading of the wave field. The dispersion of this spreading process is found to depend exclusively on the type of the interaction process (three- or four-wave), but not on a particular instability behind. The asymptotic transport equations for field spreading are derived with the aid of a specific geometric construction in the form of a comb. The results can be summarized by stating that the asymptotic spreading pursues as a continuous-time random walk (CTRW) and corresponds to a kinetic description in terms of fractional-derivative equations. The fractional indexes pertaining to these equations are obtained exactly using the comb model. A special case of the above theory is a situation when two waves with oppositely directed wave vectors couple together to form a bound state with zero momentum. This situation is considered separately and associated with the self-organization of wave-like turbulence into banded flows or staircases. Overall, we find that turbulence spreading and staircasing could be described based on the same mathematical formalism, using the Hamiltonian of inelastic wave-wave interactions and a mapping procedure into the comb space. Theoretically, the comb approach is regarded as a substitute for a more common description based on quasilinear theory. Some implications of the present theory for the fusion plasma studies are discussed and a comparison with the available observational and numerical evidence is given.

nlin.CD↗

arXiv2025

Screening and localization in the nonlinear Anderson problem

We study the spreading dynamics of an initially localized wave packet in 1D nonlinear Schr\"{o}dinger lattices with random potential. It is shown that adding small dielectric coupling to surrounding random medium results in asymptotic localization of the nonlinear field. The nonlinear localization length depends on dielectric loss of the medium at low temperatures and the value of nonlinearity parameter. The model predicts a possibility of self-induced localization when the ``medium" to which the wave field is dielectrically coupled is the nonlinear wave itself.

cond-mat.dis-nn↗

arXiv2023

On quantum free-electron laser: Superradience

A quantum model of a free-electron laser is considered for the many electron system. An exact expression for the evolution of the laser amplitude is obtained in the framework of the coherent state consideration. Reliable conditions for the superradiance of the high-gained laser is discussed for the short time limit of the exact solution.

physics.optics↗

arXiv2023

Fractional Floquet theory

A fractional generalization of the Floquet theorem is suggested for fractional Schrödinger equations (FTSE)s with the time-dependent periodic Hamiltonians. The obtained result, called the fractional Floquet theorem (fFT), is formulated in the form of the Mittag-Leffler function, which is considered as the eigenfunction of the Caputo fractional derivative. The suggested formula makes it possible to reduce the FTSE to the standard quantum mechanics with the time-dependent Hamiltonian, where the standard Floquet theorem is valid. Two examples related to quantum resonances are considered as well to support the obtained result.

quant-ph↗

arXiv2023

Ornstein-Uhlenbeck process and generalizations: particle's dynamics under comb constraints and stochastic resetting

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean-reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equation and the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here, the non-equilibrium stationary state is the main question in task, where the two divergent forces, namely the resetting and the drift towards the mean, lead to compelling results both in the case of the Ornstein-Uhlenbeck process with resetting and its generalization on the two dimensional comb structure.

cond-mat.stat-mech↗

arXiv2023

Dynamical chaos in nonlinear Schrödinger models with subquadratic power nonlinearity

We devise an analytical method to deal with a class of nonlinear Schrödinger lattices with random potential and subquadratic power nonlinearity. An iteration algorithm is proposed based on multinomial theorem, using Diophantine equations and a mapping procedure onto a Cayley graph. Based on this algorithm, we were able to obtain several hard results pertaining to asymptotic spreading of the nonlinear field beyond a perturbation theory approach. In particular, we show that the spreading process is subdiffusive and has complex microscopic organization involving both long-time trapping phenomena on finite clusters and long-distance jumps along the lattice consistent with Lévy flights. The origin of the flights is associated with the occurrence of degenerate states in the system; the latter are found to be a characteristic of the subquadratic model. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border, above which the field can spread to long distances on a stochastic process and below which it is Anderson localized similarly to a linear field.

quant-ph↗

arXiv2021

Fractional Schrödinger equation in gravitational optics

This paper addresses issues surrounding the concept of fractional quantum mechanics, related to lights propagation in inhomogeneous nonlinear media, specifically restricted to a so called gravitational optics. Besides Schrödinger Newton equation, we have also concerned with linear and nonlinear Airy beam accelerations in flat and curved spaces and fractal photonics, related to nonlinear Schrödinger equation, where impact of the fractional Laplacian is discussed. Another important feature of the gravitational optics' implementation is its geometry with the paraxial approximation, when quantum mechanics, in particular, fractional quantum mechanics, is an effective description of optical effects. In this case, fractional-time differentiation reflexes this geometry effect as well.

quant-ph↗

arXiv2020

Quantum dynamics and relaxation in comb turbulent diffusion

Continuous time quantum walks in the form of quantum counterparts of turbulent diffusion in comb geometry are considered. The interplay between the backbone inhomogeneous advection $δ(y)x\partial_x$ along the $x$ axis, which takes place only at the $y=0$, and normal diffusion inside fingers $\partial_y^2$ along the $y$ axis leads to turbulent diffusion. This geometrical constraint of transport coefficients due to comb geometry and properties of a dilatation operator lead to consideration of two possible scenarios of quantum mechanics. These two variants of continuous time quantum walks are described by non-Hermitian operators of the form $\hat{\cal H}=\hat{A}+i\hat{B}$. Operator $\hat{A}$ is responsible for the unitary transformation, while operator $i\hat{B}$ is responsible for quantum/classical relaxation. At the first quantum scenario, the initial wave packet can move against the classical streaming. This quantum swimming upstream is due to the dilatation operator, which is responsible for the quantum (not unitary) dynamics along the backbone, while the classical relaxation takes place in fingers. In the second scenario, the dilatation operator is responsible for the quantum relaxation in the form of an imaginary optical potential, while the quantum unitary dynamics takes place in fingers. Rigorous analytical analysis is performed for both wave and Green's functions.

cond-mat.stat-mech↗

arXiv2020

Stochastic resetting on comb-like structures

We study a diffusion process on a three-dimensional comb under stochastic resetting. We consider three different types of resetting: global resetting from any point in the comb to the initial position, resetting from a finger to the corresponding backbone and resetting from secondary fingers to the main fingers. The transient dynamics along the backbone in all three cases is different due to the different resetting mechanisms, finding a wide range of dynamics for the mean squared displacement. For the particular geometry studied herein, we compute the stationary solution and the mean square displacement and find that the global resetting breaks the transport in the three directions. Regarding the resetting to the backbone, the transport is broken in two directions but it is enhanced in the main axis. Finally, the resetting to the fingers enhances the transport in the backbone and the main fingers but reaches a steady value for the mean squared displacement in the secondary fingers.

cond-mat.stat-mech↗

arXiv2019

From power law to Anderson localization in nonlinear Schrödinger equation with nonlinear randomness

We study the propagation of coherent waves in a nonlinearly-induced random potential, and find regimes of self-organized criticality and other regimes where the nonlinear equivalent of Anderson localization prevails. The regime of self-organized criticality leads to power-law decay of transport [Phys. Rev. Lett. 121, 233901 (2018)], whereas the second regime exhibits exponential decay.

cond-mat.dis-nn↗

arXiv2019

Richardson diffusion in neurons

The dynamics of an initial wave packed affected by random noise is considered in the framework of a comb model. The model is relevant to a diffusion problem in neurons where the transport of ions can be accelerated by an external random field due to synapse fluctuations. In the present specific case, it acts as boundary conditions, which lead to a reaction transport equation with multiplicative noise. The temporal behavior of the mean squared displacement is estimated analytically, and it is shown that the spreading of the initial wave packet corresponds to Richardson diffusion.

physics.bio-ph↗

arXiv2019

Subdiffusive Lévy flights in quantum nonlinear Schrödinger lattices with algebraic power nonlinearity

We report a new result concerning the dynamics of an initially localized wave packet in quantum nonlinear Schrödinger lattices with a disordered potential. A class of nonlinear lattices with subquadratic power nonlinearity is considered. We show that there exists a parameter range for which an initially localized wave packet can spread along the lattice to unlimited distances, but the phenomenon is purely quantum and is hindered in the corresponding classical lattices. The mechanism for this spreading assumes that the components of the wave field may form coupled states by tunneling under the topological barriers caused by multiple discontinuities in the operator space. Then these coupled states thought of as quasiparticle states can propagate to long distances on Lévy flights with a distribution of waiting times. The overall process is subdiffusive and occurs as a competition between long-distance jumps of the quasiparticle states, on the one hand, and long-time trapping phenomena mediated by clustering of unstable modes in wave number space, on the other hand. The kinetic description of the transport, discussed in this work, is based on fractional-derivative equations allowing for both non-Markovianity of the spreading process as a result of attractive interaction among the unstable modes and the effect of long-range correlations in wave number space tending to introduce fast channels for the transport, the so-called stripes. We argue that the notion of stripes is key to understand the topological constraints behind the quantum spreading, and we involve the idea of stripy ordering to obtain self-consistently the parameters of the associated waiting-time and jump-length distributions. Finally, we predict the asymptotic laws for quantum transport and show that the relevant parameter determining these laws is the exponent of the power-law defining the type of the nonlinearity.

cond-mat.stat-mech↗

arXiv2018

Finite-velocity diffusion on a comb

A Cattaneo equation for a comb structure is considered. We present a rigorous analysis of the obtained fractional diffusion equation, and corresponding solutions for the probability distribution function are obtained in the form of the Fox $H$-function and its infinite series. The mean square displacement along the backbone is obtained as well in terms of the infinite series of the Fox $H$-function. The obtained solutions describe the transition from normal diffusion to subdiffusion, which results from the comb geometry.

cond-mat.stat-mech↗

arXiv2017

Heterogeneous diffusion in comb and fractal grid structures

We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior $\langle x^2(t)\rangle\sim t^{1/(2-α)}$, where $α$ is the power-law exponent of the position dependent diffusion coefficient $D(x)\sim |x|^α$. Depending on the value of $α$ we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., $\langle x^2(t)\rangle\sim t^{(1+ν)/(2-α)}$, where $0<ν<1$ is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the Fox $H$-functions.

cond-mat.stat-mech↗

arXiv2017

Langevin dynamics for ramified structures

We propose a generalized Langevin formalism to describe transport in combs and similar ramified structures. Our approach consists of a Langevin equation without drift for the motion along the backbone. The motion along the secondary branches may be described either by a Langevin equation or by other types of random processes. The mean square displacement (MSD) along the backbone characterizes the transport through the ramified structure. We derive a general analytical expression for this observable in terms of the probability distribution function of the motion along the secondary branches. We apply our result to various types of motion along the secondary branches of finite or infinite length, such as subdiffusion, superdiffusion, and Langevin dynamics with colored Gaussian noise and with non-Gaussian white noise. Monte Carlo simulations show excellent agreement with the analytical results. The MSD for the case of Gaussian noise is shown to be independent of the noise color. We conclude by generalizing our analytical expression for the MSD to the case where each secondary branch is $n$ dimensional.

cond-mat.stat-mech↗

arXiv2017

Destruction of Anderson localization in quantum nonlinear Schrödinger lattices

The four-wave interaction in quantum nonlinear Schrödinger lattices with disorder is shown to destroy the Anderson localization of waves, giving rise to unlimited spreading of the nonlinear field to large distances. Moreover, the process is not thresholded in the quantum domain, contrary to its "classical" counterpart, and leads to an accelerated spreading of the subdiffusive type, with the dispersion $\langle(Δn)^2\rangle \sim t^{1/2}$ for $t\rightarrow+\infty$. The results, presented here, shed new light on the origin of subdiffusion in systems with a broad distribution of relaxation times.

cond-mat.dis-nn↗

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↗