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Papers by vladimir uchaikin
De Gruyter eBooks, Dec 31, 2006
Успехи физических наук, Nov 1, 2013
Итоги науки и техники. Серия «Современная математика и ее приложения. Тематические обзоры», 2019
В своих недавних работах автор обращал внимание на то, что выделение из замкнутой гамильтоновой с... more В своих недавних работах автор обращал внимание на то, что выделение из замкнутой гамильтоновой системы еe части переводит исходное известное дифференциальное уравнение Лиувилля в интегро-дифференциальное уравнение с запаздывающим временным аргументом, описывающее динамику выделенной подсистемы уже в статусе открытой системы; было показано, что интегральный оператор может быть представлен в форме дробного дифференциального оператора распределeнного порядка. В настоящей работе показано, как преобразуется кинетическая теория системы «атомы$+$фотоны» при рассмотрении подсистемы, образованной возбуждeнными атомами, представлен вывод телеграфного уравнения с запаздыванием, выведено уравнение Бибермана - Холстейна в дробной дифференциальной форме (с оператором Лапласа дробного порядка), рассмотрены граничные эффекты в нелокальной модели переноса. Заключительный раздел посвящeн лазерным технологиям, включающим в себя лазеры на свободных электронах и лазерное охлаждение атомов.
Teoreticheskaya i Matematicheskaya Fizika, 1998
Fractal and Fractional
This paper consists of a general consideration of a seismic system as a subsystem of another, lar... more This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville differential equation of the first order in time, it is described by a fractional differential equation of a distributed equation in the interval (0, 1] order. The key characteristic of its motion is a spectral function, representing the order distribution over the interval. As a specific case of the process, a system with single-point spectrum is investigated. It follows the fractional Poisson process method evolution, obeying via a time-fractional differential equation with a unique order. The article ends with description of statistical estimation of parameters of seismic shocks imitated by Monte Carlo simulated fractional Poisson process.
Universe
The interstellar medium (ISM), serving as a background for the propagation of cosmic rays (CRs) a... more The interstellar medium (ISM), serving as a background for the propagation of cosmic rays (CRs) and other information carriers, has a complex ragged structure. Being chaotically scattered over interstellar space, together with the magnetic field perturbations frozen in them, CRs are connected with each other by a network of magnetic field lines creating long-range correlations of a power-law type, similar to those observed in the spatial distribution of galaxies. These lines solving interstellar transfer problems require the choice of an ISM model, adequately and concisely representing their statistical properties. This article discusses one such model, the Uchaikin–Zolotarev model: a four-parameter approximation of the power spectrum spatial correlations, derived from the generalized Ornstein–Zernike equation. The numerical analysis confirmed that this approximation satisfactorily agrees with the numerical data obtained in the quasi-linear model of plasma turbulence.
Proceedings of The 34th International Cosmic Ray Conference — PoS(ICRC2015), Aug 18, 2016
Commonly used cosmic ray transport equations originate in the standard diffusion model as the mos... more Commonly used cosmic ray transport equations originate in the standard diffusion model as the most known random walk process. However, at least two its features are incompatible with such fixed facts as relativistic boundedness of velocity and multiscale heterogeneity of the interstellar medium. Here is considered the nonlocal relativistic CR-trasport model which is free from both these imperfections, and involving energy dependence of the mean free path length.
Commonly used cosmic ray transport equations originate in the standard diffusion model as the mos... more Commonly used cosmic ray transport equations originate in the standard diffusion model as the most known random walk process. However, at least two its features are incompatible with such fixed facts as relativistic boundedness of velocity and multiscale heterogeneity of turbulent magnetic fields. Here is considered a new transport model called the nonlocal relativistic diffusion (NORD) model which is free from both these imperfections. Numerical comparison with some other models confirmed advantage of the NORD model. KEYWORDS: cosmic ray diffusion, nonlocal operators, fractional derivative, relativistic diffusion I. INTRODUCTION In the frame of large-scale structure of the Universe, cosmic rays (CRs) can be considered as a component of the intergalactic medium, which originate and accelerate inside galaxies, leave them, fill the intergalactic space and have possibility to enter other galaxies and continue acceleration. Multiple repeating, this process may, in principle, produce CRs...
International Journal of Theoretical Physics - INT J THEOR PHYS, 2000
Asymptotic solutions of the m-dimensional Montroll–Weiss'jump problem areobtained. They cove... more Asymptotic solutions of the m-dimensional Montroll–Weiss'jump problem areobtained. They cover both the subdiffusive and the superdiffusive regime, obeyfractional differential equations, and are expressed in terms of stable distributions.Analytical investigation and numerical calculations of anomalous diffusiondistributions are performed and their properties are discussed.
Communications in Nonlinear Science and Numerical Simulation, 2011
Arxiv preprint arXiv:1008.3969, 2010
International Journal of Bifurcation and Chaos, 2008
Fractional generalizations of the Poisson process and branching Furry process are considered. The... more Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Lèvy stable densities are discussed and used for the construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z = N/〈N〉 are found for both processes.
Fractional Kinetics in Space
Fractional Kinetics in Space
Fractional Kinetics in Space
De Gruyter eBooks, Dec 31, 2006
Успехи физических наук, Nov 1, 2013
Итоги науки и техники. Серия «Современная математика и ее приложения. Тематические обзоры», 2019
В своих недавних работах автор обращал внимание на то, что выделение из замкнутой гамильтоновой с... more В своих недавних работах автор обращал внимание на то, что выделение из замкнутой гамильтоновой системы еe части переводит исходное известное дифференциальное уравнение Лиувилля в интегро-дифференциальное уравнение с запаздывающим временным аргументом, описывающее динамику выделенной подсистемы уже в статусе открытой системы; было показано, что интегральный оператор может быть представлен в форме дробного дифференциального оператора распределeнного порядка. В настоящей работе показано, как преобразуется кинетическая теория системы «атомы$+$фотоны» при рассмотрении подсистемы, образованной возбуждeнными атомами, представлен вывод телеграфного уравнения с запаздыванием, выведено уравнение Бибермана - Холстейна в дробной дифференциальной форме (с оператором Лапласа дробного порядка), рассмотрены граничные эффекты в нелокальной модели переноса. Заключительный раздел посвящeн лазерным технологиям, включающим в себя лазеры на свободных электронах и лазерное охлаждение атомов.
Teoreticheskaya i Matematicheskaya Fizika, 1998
Fractal and Fractional
This paper consists of a general consideration of a seismic system as a subsystem of another, lar... more This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville differential equation of the first order in time, it is described by a fractional differential equation of a distributed equation in the interval (0, 1] order. The key characteristic of its motion is a spectral function, representing the order distribution over the interval. As a specific case of the process, a system with single-point spectrum is investigated. It follows the fractional Poisson process method evolution, obeying via a time-fractional differential equation with a unique order. The article ends with description of statistical estimation of parameters of seismic shocks imitated by Monte Carlo simulated fractional Poisson process.
Universe
The interstellar medium (ISM), serving as a background for the propagation of cosmic rays (CRs) a... more The interstellar medium (ISM), serving as a background for the propagation of cosmic rays (CRs) and other information carriers, has a complex ragged structure. Being chaotically scattered over interstellar space, together with the magnetic field perturbations frozen in them, CRs are connected with each other by a network of magnetic field lines creating long-range correlations of a power-law type, similar to those observed in the spatial distribution of galaxies. These lines solving interstellar transfer problems require the choice of an ISM model, adequately and concisely representing their statistical properties. This article discusses one such model, the Uchaikin–Zolotarev model: a four-parameter approximation of the power spectrum spatial correlations, derived from the generalized Ornstein–Zernike equation. The numerical analysis confirmed that this approximation satisfactorily agrees with the numerical data obtained in the quasi-linear model of plasma turbulence.
Proceedings of The 34th International Cosmic Ray Conference — PoS(ICRC2015), Aug 18, 2016
Commonly used cosmic ray transport equations originate in the standard diffusion model as the mos... more Commonly used cosmic ray transport equations originate in the standard diffusion model as the most known random walk process. However, at least two its features are incompatible with such fixed facts as relativistic boundedness of velocity and multiscale heterogeneity of the interstellar medium. Here is considered the nonlocal relativistic CR-trasport model which is free from both these imperfections, and involving energy dependence of the mean free path length.
Commonly used cosmic ray transport equations originate in the standard diffusion model as the mos... more Commonly used cosmic ray transport equations originate in the standard diffusion model as the most known random walk process. However, at least two its features are incompatible with such fixed facts as relativistic boundedness of velocity and multiscale heterogeneity of turbulent magnetic fields. Here is considered a new transport model called the nonlocal relativistic diffusion (NORD) model which is free from both these imperfections. Numerical comparison with some other models confirmed advantage of the NORD model. KEYWORDS: cosmic ray diffusion, nonlocal operators, fractional derivative, relativistic diffusion I. INTRODUCTION In the frame of large-scale structure of the Universe, cosmic rays (CRs) can be considered as a component of the intergalactic medium, which originate and accelerate inside galaxies, leave them, fill the intergalactic space and have possibility to enter other galaxies and continue acceleration. Multiple repeating, this process may, in principle, produce CRs...
International Journal of Theoretical Physics - INT J THEOR PHYS, 2000
Asymptotic solutions of the m-dimensional Montroll–Weiss'jump problem areobtained. They cove... more Asymptotic solutions of the m-dimensional Montroll–Weiss'jump problem areobtained. They cover both the subdiffusive and the superdiffusive regime, obeyfractional differential equations, and are expressed in terms of stable distributions.Analytical investigation and numerical calculations of anomalous diffusiondistributions are performed and their properties are discussed.
Communications in Nonlinear Science and Numerical Simulation, 2011
Arxiv preprint arXiv:1008.3969, 2010
International Journal of Bifurcation and Chaos, 2008
Fractional generalizations of the Poisson process and branching Furry process are considered. The... more Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Lèvy stable densities are discussed and used for the construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z = N/〈N〉 are found for both processes.
Fractional Kinetics in Space
Fractional Kinetics in Space
Fractional Kinetics in Space