Effect of random noise on chaotic motion of a particle in a ϕ6 potential

The chaotic behaviors of a particle in a triple well ϕ⁶ potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.     

Diffusion in a lattice with static disorder

Kharkov Physicotechnical Institute, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 89, No. 3, pp. 465–472, December, 1991.     

Diffusion in a locally stationary random environment

This paper deals with homogenization of diffusion processes in a locally stationary random environment. Roughly speaking, such an environment possesses two evolution scales: both a fast microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims at giving a macroscopic approximation that takes into account the microscopic heterogeneities.     

Large deviations from a macroscopic scaling limit for particle systems with Kac interaction and random potential

We consider a lattice gas in a periodic d-dimensional lattice of width γ⁻¹, γ>0, interacting via a Kac's type interaction, with range and strength γ^d, and under the influence of a random one body potential given by independent, bounded, random variables with translational invariant distribution. The system evolves through a conservative dynamics, i.e. particles jump to nearest neighbor empty sites, with rates satisfying detailed balance with respect to the equilibrium measures. In [M. Mourragui, E. Orlandi, E. Saada, Macroscopic evolution of particles systems with random field Kac interactions, Nonlinearity 16 (2003) 2123–2147] it has been shown that rescaling space as γ⁻¹ and time as γ⁻², in the limit γ→0, for dimensions d3, the macroscopic density profile ρ satisfies, a.s. with respect to the random field, a non-linear integral partial differential equation, having the diffusion matrix determined by the statistical properties of the external random field. Here we show an almost sure (with respect to the random field) large deviations principle for the empirical measures of such a process. The rate function, which depends on the statistical properties of the external random field, is lower semicontinuous and has compact level sets.     

Equation of motion for a radiating charged particle in classical electrodynamics

We propose classical equations of motion for charged particles in an electromagnetic field. These are general formulas for the particle acceleration that take the radiation-induced deceleration into account and contain no second derivatives of the particle velocity. In several particular cases considered, the new equations yield results coinciding with those known in the literature and experimentally verified. We show that in the range of ultrahigh energies, classical electrodynamics does not lead to inherent inconsistencies and in principle allows particle motion with energies exceeding the Pomeranchuk limit.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 112–130, April, 2005.     

Tagged particle motion in a clustered random walk system

In this paper a new model is presented of a one-dimensional interacting particle system which we call “a clustered random walk system”, in which a tagged particle has an asymptotically Gaussian distribution with variance βt^1/ (1<2).     

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed.     

Lyapunov control of a quantum particle in a decaying potential

A Lyapunov-based approach for the trajectory generation of an N  -dimensional Schrödinger equation in whole RN${\mathbb{R}}^N$ is proposed. For the case of a quantum particle in an N-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

A new interpretation for the mass of a classical relativistic particle

Based on a recent classification of coadjoint orbits of the full Poincaré group, we give a new group theoretic interpretation for the mass of a classical relativistic particle.     

Long-time asymptotics for a classical particle interacting with a scalar wave field

We consider the Harniltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to a confining external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. We prove that solutions of finite energy converge, in suitable local energy seminorms, to the set of stationary solutions in the long time limit t f oo. The rate of relaxation to a stable stationary solution is determined by spatial decay of initial data. 'Supported partly by French-Russian A.M.Liapunov Center of Moscow State University, by research grants of RFBR (9601-00527) and of Volkswagen-Stiftung.     

Wigner function of a relativistic particle in a time-dependent linear potential

We construct phase-space representations for a relativistic particle in both a constant and a time-dependent linear potential. We obtain explicit expressions for the Wigner distribution functions for these systems and find the correct nonrelativistic limit and free-particle limit for these functions. We derive the relativistic dynamical equation governing the time development of the Wigner distribution function and relativistic equation for the Wigner distribution function of stationary states and also calculate the amplitudes of transitions between energy states.     

Limit velocity for a driven particle in a random medium with mass aggregation

We study a one-dimensional infinite system of particles driven by a constant positive force F which acts only on the leftmost particle which is regarded as the tracer particle (t.p.). All other particles are field neutral, do not interact among themselves, and independently of each other with probability 0<p≤1 are either perfectly inelastic and “stick” to the t.p. after the first collision, or with probability 1−p are perfectly elastic, mechanically identical and have the same mass m. At initial time all particles are at rest, and the initial measure is such that the interparticle distances ξ_i's are i.i.d. r.v.'s. with absolutely continuous density. We show that for any value of the field F>0, the velocity of the t.p. converges to a limit value, which we compute.