Relaxation of probability characteristics of the brownian motion of a nonlinear oscillator

We consider an oscillator with nonlinear elasticity and nonlinear damping under the action of a Gaussian delta-correlated random force. The oscillator is treated as a Brownian particle in the corresponding potential profile. We analyze the problem using the analytical-numerical method based on solving the chain of differential equations for the statistical moments, which is broken in a certain manner. For the case of nonlinear elasticity, we find the dependence of the relaxation times of the mean values and variances of both the coordinates and velocities on the system parameters and noise intensity. By analogy, the relaxation of the probability characteristics of the oscillation amplitude is studied for a system with nonlinear damping. In both cases, the evolution of the Gaussian or Rayleigh probability distributions is described on the basis of the moment relaxation. Nizhny Novgorod Architectural and Construction University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 4, pp. 468–478, September, 2000.     

Spectral characteristics of random switchings in bistable systems

Based on the model of the Brownian diffusion in an arbitrary two-well potential profile, the main spectral characteristics of the process of noise-induced switchings of a bistable system from one stable condition into another are considered. It is shown that the maximum value and the band of the spectrum of the random switching process are determined by the times of relaxation of the nonequilibrium probability densities of the Brownian particle coordinate to the equilibrium (Boltzmann) distribution.State University, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, Nos. 1–2, pp. 88–93, January–February, 1995.     

Using bimodal probability distributions in the problems of Brownian diffusion

When analyzing nonlinear stochastic systems, we deal with the chains of differential equations for the moments or cumulants of dynamic variables. To disconnect such chains, the well-known cumulant approach, which is adequate to the quasi-Gaussian expansion of the higher-order moments is used. However, this method is inefficient in the problems of Brownian diffusion in bimodal potential profiles, and the disconnection problem should be solved on the basis of bimodal probability distributions. To this end, we propose to construct bimodal model distributions, in particular, the bi-Gaussian distribution. Cumulants and the expansions of the higher-order moments for symmetric and nonsymmetric bi-Gaussian models. On this basis, we consider relaxation of probability characteristics of one-dimensional Brownian motion in the bimodal potential profile. The dependences of relaxation of the mean value and variance of particle coordinate on the potential barrier “power,” the noise intensity, and the initial distribution of particles are analyzed numerically. In particular, it is shown that relaxation proceeds by stages with different temporal scales in the case of a powerful barrier. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 8, pp. 718–729, August 2006.     

Derivation of the probability distribution of velocities of a detector-recorded brownian particle

The probability distribution of velocities in the given space region (detector) is found for particles of a passive admixture in a stream of external gas. Since direct calculation of the above probability density involves significant difficulties, the solution is based on the classical problem of the probability distribution of coordinates and velocity of a Brownian particle at a fixed time. Analyzing dependence of the solution on the parameters of the initial problem, we obtain conditions under which the assumptions on the character of particle motion hold true. State University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 10, pp. 1301–1313, October 1998.     

Probability characteristics of brownian motion in a stochastic potential profile of a specific type

Some probability characteristics of the Brownian motion in a symmetric potential profile with two equilibrium states subjected to a random force are obtained. Two types of potential fluctuations are considered: the delta-correlated Gaussian noise and the stochastic telegraph process with Poisson statistics of jumps. The stationary probability distributions of the particle coordinate are found, and the dependence on the properties of parametric and additive noise is studied. It is shown that nonzero equilibrium states approach each other and vanish as a result of strong potential fluctuations. Relaxation of intensity and variance of coordinate fluctuations are studied numerically for the case of delta-correlated random forces. The influence of the value of parametric and additive noise, system nonlinearity, and initial conditions on the relaxation process is determined. Architectural and Civil Engineering University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 10, pp. 1290–1300, October 1998.     

Relaxation of the Probability Characteristics of Brownian Motion under the Action of a Dichotomous Random Force

We consider the relaxation of rms characteristics of the coordinates of particles during their Brownian motion in a symmetric potential profile under the action of a dichotomous random force. An analytical-numerical method of analysis based on the numerical solution of a chain of differential equations for coordinate moments and joint correlations is used. The calculation procedure is checked using exact results which can be found in the limiting cases of delta-correlated and quasi-static random action. The dependence of the distribution variance and its relaxation time on the intensity and correlation time of noise is elucidated.     

Relaxation of the Probability Characteristics of Brownian Motion under the Action of a Non-Delta-Correlated Random Force

We consider the relaxation of the moments of the coordinates of one-dimensional Brownian motion of particles in a symmetric potential profile under the action of a Gaussian, exponentially correlated random force. An analytical-numerical method of analysis based on obtaining and numerically solving a chain of differential equations for joint cumulants of some functions of particle coordinates and a random force is used. A priori constraints on the intensity and correlation time of noise are not imposed. Numerical procedure is checked by comparison with analytical results, which can be found in the limiting cases of delta-correlated and quasistatic random force. The dependence of the relaxation of the average value and variance on the intensity and spectrum of a random force and the character of the initial distribution of particles is elucidated. In particular, the presence of a variance minimum during distribution relaxation is established. The evolution of the model probability distribution of particle coordinates is constructed on the basis of the moment relaxation.     

A model of a neuron with oscillatory activity below the excitation threshold

We propose a dynamic model of a neuron with spontaneous periodic oscillations below the excitation threshold. Such neurons, in particular, play an important role in the problem of coordination of motions by the brain specifying the universal rhythm of muscular contractions. The model is constructed on the basis of the known model dynamic systems and is described by a system of fourth-order differential equations. A good qualitative agreement between the model dynamics and experimental data for the actual neurons is obtained. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998). Lobachevsky State University of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1623–1635, December, 1998.     

The exact value of relaxation time for a dynamic system with noise described by an arbitrary symmetric potential profile

We show that the well-known Pontryagin's formula for the average time of first attainment by a Brownian particle of the absorbing boundary can also be used to obtain the exact value of the average relaxation time for the nonequilibrium state of a nonlinear dynamic system with noise and an arbitrary symmetric potential profile.Lobachevskii State University, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, Nos. 3–4, pp. 256–261, March–April, 1995.     

Experimental Remote Sounding of the Atmospheric Turbulence

In this paper, we discuss the problems of studying the atmospheric turbulence by the method of acoustic sounding and present the results of experiments on acoustic sounding of the atmosphere at an altitude of about 500 m, which were carried out in Zimenki (Nizhny Novgorod Region). It is shown that the experimentally determined velocity distribution function in the scattering volume is satisfactorily described by a one-parameter Poisson distribution. The results of numerical simulations are in good agreement with the results of the field experiment.     

On Analysis of Spectro-Correlation Characteristics of One-Dimensional Brownian Motion

We propose an analytical–numerical approach to finding the correlation function of one-dimensional Brownian motion in a one-mode potential profile described by a low-order polynomial. The approach is based on solving chains of differential equations for the statistical moment functions of particle coordinate fluctuations, which are broken in a certain manner. Two methods of such breaking are considered. One method is based upon quasi-linear expansions of the moment functions, and another one, on cumulantless expansions. Spectro-correlation characteristics of Brownian motion in biquadratic potential profiles of two types are studied.     

Quantum–Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion

Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.     

New mechanism of Brownian transport

We propose a new model of transport of Brownian particles by deterministic modulation of the ratchet potential. Such a mechanism results in a directed uniform drift of Brownian particles rather than in their diffusion. Lobachevsky State University of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 1, pp. 87–91, January, 1999.     

Polynomial estimation of probability densities

We propose a new quality criterion and an iterative algorithm for polynomial estimation of probability densities of a random quantity. The properties of the optimal estimate are studied and the conditions and rate of its convergence are found. Antiaircraft Command High School of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 11, pp. 1416–1432, November, 1997.     

Bifurcations of oscillations of the membrane potential and the use of mappings for modeling electrically coupled neurons

In this paper, we present the results of a qualitative analysis of a generalized system of three differential equations that represent the neuron model. The main nontrivial bifurcation sets leading to the appearance of complex motions, i.e., bursts, are given. A two-dimensional mapping that models the flows generated by this system, which is considered to be the simplest model of a neuron, is proposed. The chaotic dynamics of diffusely coupled neurons is studied using the coupled mappings. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998). Lobachevsky State Univesity of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1572–1580, December, 1998.     

Correlation time and structure of the correlation function of nonlinear equilibrium brownian motion in arbitrary-shaped potential wells

We obtain the exact quadrature formula for the correlation time of a stationary Brownian motion in arbitrary-shaped potential wells. We analyze the dependence of the correlation time of the intensity of driving noise for power-law potential profiles and on the height of a potential barrier for bistable profiels. We also study some spectral characteristics of the steady-state motion and find the coefficients of the Taylor series expansion of the correlation function. N. I. Lobachevskii State University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 4, pp. 369–382, April, 2000.     

Dynamics of complex-shape traveling pulses in a fitz hugh-nagumo system with modulated parameters

The dynamics of traveling-pulse solutions is studied for a Fitz Hugh-Nagumo model with modulated parameters. The boundaries of the modulation depth of the system parameters where these solutions still exist are found by numerical simulation. The compensation effect of the parameter modulation acting on the stability of such solutions is obtained. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, (998). Lobachevsky State University of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1586–1592, December, 1998.     

Periodic evolution of space chaos in the one-dimensional complex Ginzburg-Landau equation

Periodic evolution of the space chaos in a one-dimensional distributed system represented by the complex Ginzburg-Landau equation is studied. There exists a region of parameters where spatially chaotic distribution of the field varies periodically with time, and the boundaries of this region are determined. The regime of periodic space chaos was found to exist only for certain initial conditions. A system of ordinary differential equations that describes the space chaos is derived.Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, Nos. 1–2, pp. 37–43, January–February, 1995.     

Cyclotron scattering with mode switching in plasma on degenerate stars I. Radiative relaxation in a homogeneous rarefied plasma

Population balance equations for the Landau levels in a non-relativistic rarefied plasma are written for arbitrary polarization of cyclotron modes. Self-consistent evolution of the transverse distribution of electrons interacting, by virtue of the cyclotron processes, with an isotropic radiation at frequencies near the first harmonic of the electron gyrofrequency is studied. The spectrum of the relaxation times of the system is found for a fixed radiation intensity. It is shown that the time of cyclotron relaxation under the action of the first-harmonic radiation with broad angular and frequency spectra is entirely determined by the rate of spontaneous processes and does not depend on the radiation intensity. Cyclotron radiation transfer coefficients which account for the process of mode switching at the first harmonic are obtained. Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 11, pp. 1035–1053, November, 1999.     

Linear relaxation processes governed by fractional symmetric kinetic equations

The fractional symmetric Fokker-Planck and Einstein-Smoluchowski kinetic equations that describe the evolution of systems influenced by stochastic forces distributed with stable probability laws are derived. These equations generalize the known kinetic equations of the Brownian motion theory and involve symmetric fractional derivatives with respect to velocity and space variables. With the help of these equations, the linear relaxation processes in the force-free case and for the linear oscillator is analytically studied. For a weakly damped oscillator, a kinetic equation for the distribution in slow variables is obtained. Linear relaxation processes are also studied numerically by solving the corresponding Langevin equations with the source given by a discrete-time approximation to white Levy noise. Numerical and analytical results agree quantitatively.