Directed transport in a spatially periodic harmonic potential under periodic nonbiased forcing

Transport of a particle in a spatially periodic harmonic potential under the influence of a slowly timedependent unbiased periodic external force is studied. The equations of motion are the same as in the problem of a slowly forced nonlinear pendulum. Using methods of the adiabatic perturbation theory we show that for a periodic external force of a general kind the system demonstrates directed ratchet transport in the chaotic domain on very long time intervals and obtain a formula for the average velocity of this transport. Two cases are studied: The case of the external force of small amplitude, and the case of the external force with amplitude of order one. The obtained formulas can also be used in case of a nonharmonic periodic potential

Charged particle nonlinear resonance with localized electrostatic wave-packets

A resonant wave-particle interaction, in particular a nonlinear resonance characterized by particle phase trapping, is an important process determining charged particle energization in many space and laboratory plasma systems. Although an individual charged particle motion in the nonlinear resonance is well described theoretically, the kinetic equation modeling the long-term evolution of the resonant particle ensemble has been developed only recently. This study is devoted to generalization of this equation for systems with localized wave packets propagating with the wave group velocity different from the wave phase velocity. We limit our consideration to the Landau resonance of electrons and waves propagating in an inhomogeneous magnetic field. Electrons resonate with the wave field-aligned electric fields associated with gradients of wave electrostatic potential or variations of the field-aligned component of the wave vector potential. We demonstrate how wave-packet properties determine the efficiency of resonant particle acceleration and derive the nonlocal integral operator acting on the resonant particle distribution. This operator describes particle distribution variations due to interaction with one wave-packet. We solve kinetic equation with this operator for many wave-packets and show that solutions coincide with the results of the numerical integration of test particle trajectories. To demonstrate the range of possible applications of the proposed approach, we consider the electron evolution induced by the Landau resonances with packets of kinetic Alfven waves, electron acoustic waves, and very oblique whistler waves in the near-Earth space plasma

Kinetic equation for nonlinear wave-particle interaction: solution properties and asymptotic dynamics

We consider a kinetic equation describing evolution of the particle distribution function in a system with nonlinear wave-particle interactions (trappings into resonance and nonlinear scatterings). We study properties of its solutions and show that the only stationary solution is a constant, and that all solutions with smooth initial conditions tend to a constant as time grows. The resulting flattening of the distribution function in the domain of nonlinear interactions is similar to one described by the quasi-linear plasma theory, but the distribution evolves much faster. The results are confirmed numerically for a model problem

Kinetic equation for nonlinear resonant wave-particle interaction

We investigate nonlinear resonant wave-particle interactions including effects of particle (phase) trapping, detrapping, and scattering by high-amplitude coherent waves. After deriving the relation between probability of trapping and velocity of particle drift induced by nonlinear scattering (phase bunching), we substitute this relation and other characteristic equations of wave-particle interaction into a kinetic equation for the particle distribution function. The final equation has the form of a Fokker-Planck equation with peculiar advection and collision terms. This equation fully describes the evolution of particle momentum distribution due to particle diffusion, nonlinear drift, and fast transport in phase-space via trapping. Solutions of the obtained kinetic equation are compared with results of test particle simulations

Nonresonant charged particle acceleration by electrostatic wave propagating across fluctuating magnetic field

In this Letter, we demonstrate the effect of nonresonant charged-particle acceleration by an electrostatic wave propagating across the background magnetic field. We show that in the absence of resonance (i.e., when particle velocities are much smaller than the wave phase velocity) particles can be accelerated by electrostatic waves provided that the adiabaticity of particle motion is destroyed by magnetic field fluctuations. Thus, in a system with stochastic particle dynamics the electrostatic wave should be damped even in the absence of Landau resonance. The proposed mechanism is responsible for the acceleration of particles that cannot be accelerated via resonant wave-particle interactions. Simplicity of this straightforward acceleration scenario indicates a wide range of possible applications

Probabilistic approach to nonlinear wave-particle resonant interaction

In this paper we provide a theoretical model describing the evolution of the charged particle distribution function in a system with nonlinear wave particle interactions. Considering a system with strong electrostatic waves propagating in an inhomogeneous magnetic field, we demonstrate that individual particle motion can be characterized by the probability of trapping into the resonance with the wave and by the efficiency of scattering at resonance. These characteristics, being derived for a particular plasma system, can be used to construct a kinetic equation (or generalized Fokker-Planck equation) modelling the long-term evolution of the particle distribution. In this equation, effects of charged particle trapping and transport in phase space are simulated with a nonlocal operator. We demonstrate that solutions of the derived kinetic equations agree with results of test particle tracing. The applicability of the proposed approach for the description of space and laboratory plasma systems is also discussed

Long-term evolution of electron distribution function due to nonlinear resonant interaction with whistler mode waves

Accurately modelling and forecasting of the dynamics of the Earth’s radiation belts with the available computer resources represents an important challenge that still requires significant advances in the theoretical plasma physics field of wave–particle resonant interaction. Energetic electron acceleration or scattering into the Earth’s atmosphere are essentially controlled by their resonances with electromagnetic whistler mode waves. The quasi-linear diffusion equation describes well this resonant interaction for low intensity waves. During the last decade, however, spacecraft observations in the radiation belts have revealed a large number of whistler mode waves with sufficiently high intensity to interact with electrons in the nonlinear regime. A kinetic equation including such nonlinear wave–particle interactions and describing the long-term evolution of the electron distribution is the focus of the present paper. Using the Hamiltonian theory of resonant phenomena, we describe individual electron resonance with an intense coherent whistler mode wave. The derived characteristics of such a resonance are incorporated into a generalized kinetic equation which includes non-local transport in energy space. This transport is produced by resonant electron trapping and nonlinear acceleration. We describe the methods allowing the construction of nonlinear resonant terms in the kinetic equation and discuss possible applications of this equation

Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems

We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedom corresponding to fast motion and the other one corresponding to slow motion. The Hamiltonian function is the sum of potential and kinetic energies, the kinetic energy being a weighted sum of squared momenta. The ratio of time derivatives of slow and fast variables is of order \epsilon « 1. At frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition we prove the existence of many, of order 1/\epsilon, stable periodic trajectories in the domain of the separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order \epsilon. Thus, the total measure of the stability islands is estimated from below by a value independent of \epsilon. We find the location of stable periodic trajectories and an asymptotic formula for the number of these trajectories. As an example, we consider the problem of motion of a charged particle in the parabolic model of magnetic field in the Earth magnetotail

Trapping (capture) into resonance and scattering on resonance: summary of results for space plasma systems

In the present review we survey space plasma systems where the nonlinear resonant interaction between charged particles and electromagnetic waves plays an important role. We focus on particle acceleration by strong electromagnetic waves. We start with presenting a general description of nonlinear resonant interaction based on the theory of slowfast Hamiltonian systems with resonances. Then we turn to several manifestations of the resonance effects in various space plasma systems. We describe a universal approach for evaluating main characteristics of the resonant particle dynamics: probability of trapping into resonance, energy change due to scattering and trapping. Then we demonstrate how effects of nonlinear resonant trapping and scattering can be combined in a generalized kinetic equation. We also discuss the stability of trapped motion and evolution of particle ensemble in systems with trapping. The main objective of this review is to provide a general approach for characterizing plasma systems with nonlinear resonant interactions

Probability of relativistic electron trapping by parallel and oblique whistler-mode waves in Earth's radiation belts

We investigate electron trapping by high-amplitude whistler-mode waves propagating at small as well as large angles relative to geomagnetic field lines. The inhomogeneity of the background magnetic field can result in an effective acceleration of trapped particles. Here, we derive useful analytical expressions for the probability of electron trapping by both parallel and oblique waves, paving the way for a full analytical description of trapping effects on the particle distribution. Numerical integrations of particle trajectories allow to demonstrate the accuracy of the derived analytical estimates. For realistic wave amplitudes, the levels of probabilities of trapping are generally comparable for oblique and parallel waves, but they turn out to be most efficient over complementary energy ranges. Trapping acceleration of 100 keV electrons