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Register a new userHamiltonian description of self-consistent wave-particle dynamics in a periodic structure
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Conservation of energy and momentum in the classical theory of radiating electrons has been a challenging problem since its inception. We propose a formulation of classical electrodynamics in Hamiltonian form that satisfies the Maxwell equations and the Lorentz force. The radiated field is represented with eigenfunctions using the Gelfand $beta$-transform. The electron Hamiltonian is the standard one coupling the particles with the propagating fields. The dynamics conserves energy and excludes self-acceleration. A complete Hamiltonian formulation results from adding electrostatic action-at-a-distance coupling between electrons.
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We analyze nonlinear aspects of the self-consistent wave-particle interaction using Hamiltonian dynamics in the single wave model, where the wave is modified due to the particle dynamics. This interaction plays an important role in the emergence of plasma instabilities and turbulence. The simplest case, where one particle (N = 1) is coupled with one wave (M = 1), is completely integrable, and the nonlinear effects reduce to the wave potential pulsating while the particle either remains trapped or circulates forever. On increasing the number of particles (N = 2, M = 1), integrability is lost and chaos develops. Our analyses identify the two standard ways for chaos to appear and grow (the homoclinic tangle born from a separatrix, and the resonance overlap near an elliptic fixed point). Moreover, a strong form of chaos occurs when the energy is high enough for the wave amplitude to vanish occasionally.
To model momentum exchange in nonlinear wave-particle interaction, as in amplification devices like traveling-wave tubes, we use an $N$-body self-consistent hamiltonian description based on Kuznetsovs discrete model, and we provide new formulations for the electromagnetic power and the conserved momentum. This approach leads to fast and accurate numerical simulations in time domain and in one dimensional space.
Single-particle levels of seven magic nuclei are calculated within the Energy Density Functional (EDF) method by Fayans et al. Thr
We consider analytical formulae that describe the chaotic regions around the main periodic orbit $(x=y=0)$ of the H{e}non map. Following our previous paper (Efthymiopoulos, Contopoulos, Katsanikas $2014$) we introduce new variables $(xi, eta)$ in which the product $xieta=c$ (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation $Phi$ to the plane $(x,y)$, giving Moser invariant curves. We find that the series $Phi$ are convergent up to a maximum value of $c=c_{max}$. We give estimates of the errors due to the finite truncation of the series and discuss how these errors affect the applicability of analytical computations. For values of the basic parameter $kappa$ of the H{e}non map smaller than a critical value, there is an island of stability, around a stable periodic orbit $S$, containing KAM invariant curves. The Moser curves for $c leq 0.32$ are completely outside the last KAM curve around $S$, the curves with $0.32<c<0.41$ intersect the last KAM curve and the curves with $0.41leq c< c_{max} simeq 0.49$ are completely inside the last KAM curve. All orbits in the chaotic region around the periodic orbit $(x=y=0)$, although they seem random, belong to Moser invariant curves, which, therefore define a structure of chaos. Orbits starting close and outside the last KAM curve remain close to it for a stickiness time that is estimated analytically using the series $Phi$. We finally calculate the periodic orbits that accumulate close to the homoclinic points, i.e. the points of intersection of the asymptotic curves from $x=y=0$, exploiting a method based on the self-intersections of the invariant Moser curves. We find that all the computed periodic orbits are generated from the stable orbit $S$ for smaller values of the H{e}non parameter $kappa$, i.e. they are all regular periodic orbits.
Levy walk (LW) process has been used as a simple model for describing anomalous diffusion in which the mean squared displacement of the walker grows non-linearly with time in contrast to the diffusive motion described by simple random walks or Brownian motion. In this paper we study a simple extension of the LW model in one dimension by introducing correlation among the velocities of the walker in different (flight) steps. Such correlation is absent in the LW model. The correlations are introduced by making the velocity at a step dependent on the velocity at the previous step in addition to the usual random noise (kick) that the particle gets at random time intervals from the surrounding medium as in the LW model. Consequently the dynamics of the position becomes non-Markovian. We study the statistical properties of velocity and position of the walker at time t, both analytically and numerically. We show how different choices of the distribution of the random time intervals and the degree of correlation, controlled by a parameter r, affect the late time behaviour of these quantities.
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