Papers by Alexander Dubkov
Radiophysics and Quantum Electronics, 1981
The bilateral boundary conditions (15) and ( ) which we obtained are the solution of the stated p... more The bilateral boundary conditions (15) and ( ) which we obtained are the solution of the stated problem. Since they were obtained in the Bourret approximation of the theory of multiple scattering, their region of applicability is thereby limited to the allowance for an infinite subsequence of multiple scatterings at random disturbed boundaries 5]. In a number of cases this proves sufficient, which makes them particularly convenient when studying fields formed as a result of multiple scattering on random disturbances, such as for wave propagation in open waveguides with statistically irregular boundaries. In particular, they can prove useful for the investigation of waves and signals in optical fibers having a stepwise variation of the index of refraction over the cross section, in which, as is known, the ohmic losses are now close to the theoretical lower limit, while unavoidable roughness of the boundaries can prove to be an important factor in losses and variation of the dispersion owing to accumulating effects resulting from scattering on roughness over long transmission lines.
L\ue9vy flight in a two competing species dynamics
Population dynamics with L\ue9vy noise source
Stochastic model of memristor based on the length of conductive region
Chaos, Solitons & Fractals, 2021
We propose a stochastic model of a voltage controlled bipolar memristive system, which includes t... more We propose a stochastic model of a voltage controlled bipolar memristive system, which includes the properties of widely used dynamic SPICE models and takes into account the fluctuations inherent in memristors. The proposed model is described by rather simple equations of Brownian diffusion, does not require significant computational resources for numerical modeling, and allows obtaining the exact analytical solutions in some cases. The noise-induced transient bimodality phenomenon, arising under resistive switching, was revealed and investigated theoretically and experimentally in a memristive system, by finding a quite good qualitatively agreement between theory and experiment. Based on the proposed model, the mathematical apparatus of Markov processes for the first passage time of the boundaries can be used to analyse the temporal characteristics of resistive switching
Journal of Statistical Mechanics: Theory and Experiment, 2020
The nonlinear relaxation process in many condensed matter systems proceeds through metastable sta... more The nonlinear relaxation process in many condensed matter systems proceeds through metastable states, giving rise to long-lived states. Stochastic manybody systems, classical and quantum, often display a complex and slow relaxation towards a stationary state. A common phenomenon in the dynamics of out of equilibrium systems is the metastability, and the problem of the lifetime of metastable states involves fundamental aspects of nonequilibrium statistical mechanics. In spite of such ubiquity, the microscopic understanding of metastability and related out of equilibrium dynamics still raise fundamental questions. The aim of this meeting is to bring together scientists interested in the challenging problems connected with dynamics of out of equilibrium classical and quantum physical systems from both theoretical and experimental point of view, within an interdisciplinary context. Specifically, three main areas of outof-equilibrium statistical mechanics will be covered: long range interactions and multistability, anomalous diffusion, and quantum systems. Moreover, the conference will be a discussion forum to promote new ideas in this fertile research field, and in particular new trends such as quantum thermodynamics and novel types of quantum phase transitions occurring in non-equilibrium steady states, and topological phase transitions.
physica status solidi c, 2016
The Spike Noise Based on the Renewal Point Process and Its Possible Applications
We consider a non-Markovian random process in the form of spikes train, where the time intervals ... more We consider a non-Markovian random process in the form of spikes train, where the time intervals between neighboring delta-pulses are mutually independent and identically distributed, i.e. represent the renewal process (1967). This noise can be interpreted as the derivative of well-known continuous time random walk (CTRW) model process with fixed value of jumps. The closed set of equations for the characteristic functional of the noise, useful to split the correlations between stochastic functionals (2008), is obtained. In the particular case of Poisson statistics these equations can be exactly solved and the expression for the characteristic functional coincides with the result for shot noise (2005). Further we analyze the stability of some first-order system with the multiplicative spike noise. We find the momentum stability condition for arbitrary probability distribution of intervals between pulses. The general condition of stability is analyzed for the special probability distribution of intervals between pulses corresponding to so-called dead-time-distorted Poisson process. It means that within some time interval after each delta pulse the occurrence of new one is forbidden (like in neurons). The possible applications of the model to some problems of neural dynamics, epidemiology, ecology, and population dynamics are discussed
Infinitely divisible distributions, generalized Wiener process and Kolmogorov's equation for diffusion induced by nonequilibrium bath
Dynamics of a Lotka-Volterra system in the presence of non-Gaussian noise sources
We consider a Lotka-Volterra system of two competing species subject to multiplicative \\u3b1-sta... more We consider a Lotka-Volterra system of two competing species subject to multiplicative \\u3b1-stable L\\ue9vy noise. The interaction parameter between the species is a random process which obeys a stochastic differential equation with a generalized bistable potential in the presence both of a periodic driving term and an additive alpha-stable L\\ue9vy noise. We study the species dynamics, which is characterized by two different dynamical regimes, exclusion of one species and coexistence of both ones, analyzing the role of the L\\ue9vy noise sources
Izvestiâ vysših učebnyh zavedenij. Prikladnaâ nelinejnaâ dinamika, May 30, 2003
Exact formula for diffusion coefficient оё Brownian particle moving in modulated by white noise p... more Exact formula for diffusion coefficient оё Brownian particle moving in modulated by white noise periodic field is obtained. As it is shown the acceleration of diffusion in comparison with а free diffusion case takes place for ап arbitrary potential profile. Calculations оё effective diffusion constant for different periodic potentials are performed.
Physical review. E, Jun 1, 2016
For a nonlinear dynamical system described by the first-order differential equation with Poisson ... more For a nonlinear dynamical system described by the first-order differential equation with Poisson white noise having exponentially distributed amplitudes of δ pulses, some exact results for the stationary probability density function are derived from the Kolmogorov-Feller equation using the inverse differential operator. Specifically, we examine the "effect of normalization" of non-Gaussian noise by a linear system and the steady-state probability density function of particle velocity in the medium with Coulomb friction. Next, the general formulas for the probability distribution of the system perturbed by a non-Poisson δ-pulse train are derived using an analysis of system trajectories between stimuli. As an example, overdamped particle motion in the bistable quadratic-cubic potential under the action of the periodic δ-pulse train is analyzed in detail. The probability density function and the mean value of the particle position together with average characteristics of the ...
Journal of Statistical Mechanics: Theory and Experiment, 2009
The non-linear dissipation corresponding to a non-Gaussian thermal bath is introduced together wi... more The non-linear dissipation corresponding to a non-Gaussian thermal bath is introduced together with a multiplicative white noise source in the phenomenological Langevin description for the velocity of a particle moving in some potential landscape. Deriving the closed Kolmogorov's equation for the joint probability distribution of the particle displacement and its velocity by use of functional methods and taking into account the well-known Gibbs form of the thermal equilibrium distribution and the condition of 'detailed balance' symmetry, we obtain the exact master equation: given the white noise statistics, this master equation relates the non-linear friction function to the velocitydependent noise function. In particular, for multiplicative Gaussian white noise this operator equation yields a unique inter-relation between the generally nonlinear friction and the (multiplicative) velocity-dependent noise amplitude. This relation allows us to find, for example, the form of velocity-dependent noise function for the case of non-linear Coulomb friction.
This work is devoted to probabilistic analysis of two models of the ideal memristor with an exter... more This work is devoted to probabilistic analysis of two models of the ideal memristor with an external Gaussian noise. First we study the charge-controlled ideal memristor and analyze how the applied voltage in the form of a stationary Gaussian noise influences the probability density function of the charge and resistance. For comparison we pay attention to the case of applied current in the same form. Further the currentcontrolled ideal memristor with the external fluctuations in the form of a stationary Gaussian noise is under consideration. The exact results reported are based on well-known theorems of the probability theory.
We obtain exact result for the mean residence time of anomalous diffusion of a particle in the fo... more We obtain exact result for the mean residence time of anomalous diffusion of a particle in the form of Lévy flights in inverse parabolic potential. The noise-delayed decay phenomenon in such a system is found.
Journal of Statistical Mechanics: Theory and Experiment, 2020
We report on the results of the experimental investigations of the local resistive switching (RS)... more We report on the results of the experimental investigations of the local resistive switching (RS) in the contact of a conductive atomic force microscope (CAFM) probe to a nanometer-thick yttria stabilized zirconia (YSZ) film on a conductive substrate under a Gaussian noise voltage applied between the probe and the substrate. The virtual memristor was found to switch randomly between the low resistance state and the high resistance state as a random telegraph signal (RTS). The potential profile of the virtual memristor calculated from its response to the Gaussian white noise shows two local minima, which is peculiar of a bistable nonlinear system.
Journal of Statistical Mechanics: Theory and Experiment, 2020
We propose a stochastic model for a memristive system by generalizing known approaches and experi... more We propose a stochastic model for a memristive system by generalizing known approaches and experimental results. We validate our theoretical model by experiments carried out on a memristive device based on Au/Ta/ZrO 2 (Y)/Ta 2 O 5 /TiN/Ti multilayer structure. In the framework of the proposed model we obtain the exact analytic expressions for stationary and nonstationary solutions. We analyze the equilibrium and non-equilibrium steady-state distributions of the internal state variable of the memristive system and study the influence of fluctuations on the resistive switching, including the relaxation time to the steady-state. The relaxation time shows a nonmonotonic dependence, with a minimum, on the intensity of the fluctuations. This paves the way for using the intensity of fluctuations as a control parameter for switching dynamics in memristive devices.
Steady-state probability characteristics of Verhulst and Hongler models with multiplicative white Poisson noise
The European Physical Journal B, 2019
Based on recently obtained from Kolmogorov–Feller equation exact analytical results for the stead... more Based on recently obtained from Kolmogorov–Feller equation exact analytical results for the steady-state probability density function of nonlinear dynamical systems driven by white Poisson noise with exponentially distributed amplitudes of pulses we analyze some models of ecology and genetics. Specifically, we find the steady-state probability distribution of the population density in the framework of well-known Verhulst equation with fluctuating population mortality in the form of Poisson sequence with unipolar pulses, leading to an abrupt decrease in population density at random times. As shown, the most probable value of the population density tends to zero with increasing the mean rate of pulses, that is, to an extinction of biological population in perspective. Further, we consider the stochastic Hongler equation which can serve as an approximate model of genetic selection. In the case of multiplicative white Poisson noise having bipolar exponentially distributed amplitudes of pulses we observe noise-induced transition to bimodality (through the trimodal phase) in the steady-state probability distribution with an increase in the mean frequency of pulses. We also discovered a new phenomenon, namely, a direct transition from unimodality to trimodality with a change in the noise intensity, which could not be detected in the framework of the Gaussian perturbation.Graphical abstract
Journal of Statistical Mechanics: Theory and Experiment, 2016
The steady-state correlation characteristics of superdiusion in the form of Lévy flights in one-... more The steady-state correlation characteristics of superdiusion in the form of Lévy flights in one-dimensional confinement potential profiles are investigated both theoretically and numerically. Specifically, for Cauchy stable noise we calculate the steady-state probability density function for an infinitely deep rectangular potential well and for a symmetric steep potential well of the type () ∝ U x x m 2. For these potential profiles and arbitrary Lévy index α, we obtain the asymptotic expression of the spectral power density.
Physical Review E, 2017
Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but... more Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flights and Lévy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.
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Papers by Alexander Dubkov