[en] Stochastic processes are widely used in plasma physics and have provided very useful tools to describe the behavior of this disordered medium. The resulting physical effects have received considerable attention and it has motivated the edition of this book, following a workshop on the same subject, held at the 'Institut d'Etudes Scientifiques de Cargese' (Corsica, France) in 1979. The book contains thirty contributed and review papers

[en] Perturbation methods are among the most useful tools in physics. However, in some cases, the convergence of series expansion is doubtful. The aim of this paper is to show on problems on condensed matter that some light can be shed on this question by using the Kolmogorov Arnold Moser theorems for dynamical systems (noted from hereafter K.A.M. theorems) or a closely related theorem of Dinaburg and Sinai. According as convergence does or does not exist, qualitatively different physical properties are obtained concerning the static structure, particularly on the gap existence in the excitation spectrum

[en] A single electrostatic wave in a magnetoplasma causes stochastic ion motion in several physically different situations. Various magnetic fields (uniform, tokamak, and mirror) and various propagation angles with respect to the field have been studied. A brief review of this work shows that all situations can be understood using the concept of overlapping resonances. Analytical calculations of the wave amplitude necessary for stochasticity have been carried out in some cases and compared with computer and laboratory experiments. In the case of an axisymmetric mirror field the calculations predict stochastic motion of ions with energy below a threshold that depends weakly on the wave amplitude and on the scale lengths of the magnetic field. Studies with an azimuthally asymmetric field show that the asymmetry causes substantial changes in the motion of some ions

[en] Nonlinear motion of charged particles in the presence of a uniform magnetic field and of a monochromatic electrostatic wave propagating perpendicularly to the magnetic field is studied. Appearance of islands in phase space is discussed and by taking account of the overlapping of secondary islands, a wave amplitude necessary for the onset of stochasticity is estimated analytically. The results are compared to numerical results. It is shown that high-energy particles are produced most efficiently if the wave frequency is the multiple of the cyclotron frequency. As the wave amplitude increases, numerical calculations show that a steady velocity distribution is insensitive to the wave frequency

[en] A brief summary is made of experimental observations of disruptions in Tokamaks, and of the possible explanations. A discussion of their adequacy is then made, which points towards the importance of small-scale turbulence

[en] Several applications of stochasticity theory to particle motion in quiescent mirror machines are discussed. We briefly review the problem of magnetic moment jumps in a conventional mirror machine, and point out the role of stochasticity in extending the mirror loss cone. We consider magnetic moment jumps in a flat-bottomed magnetic well, such as the solenoid of a tandem mirror machine, and find that, for suitable choices of field parameters, the magnetic moment change per bounce passes through zero at certain values of energy or pitch angle. This system can be modelled by a simple, asymmetric, two-step map; numerical and analytic studies of the map indicate that phase space can contain stochastic layers, separated by nonstochastic layers that serve as barriers for diffusion. We study the motion of guiding centers in a nonaxisymmetric tandem mirror solenoid and show that ions in the Lawrence Livermore Laboratory Tandem Mirror Experiment (TMX) might be stochastic

[en] For nonlinear dynamical problems with two degrees of freedom a transition occurs between regions of the phase space in which stochastic motion is closely bounded by invariant (Kolmogorov, Arnold, Moser or KAM) surfaces, and regions for which the stochastic motion is interconnected over large portions of the space. The mechanism by which this transition takes place is discussed, qualitatively, in terms of the growth of primary and higher order islands. These islands both change the topology of the phase plane and introduce stochastic bands near island separatrices. The relationship between the complete dynamical problem and a two-dimensional mapping is given. Various criteria for the transition are developed. Quantitative values of the strength of the perturbation parameter, required to produce connected stochasticity, are given for the various criteria, as applied to a standard mapping. In particular, the criteria of loss of linear stability of the primary islands, resonance overlap, growth of second order islands, improved overlap criteria, and loss of linear stability at rational iterates of the irrational rotation number connected with a given KAM surface, are considered. The latter two criteria, based on the work of Chirikov and Greene, respectively, are particularly emphasized

[en] A number of problems, including the calculation of magnetic field line trajectories in three-dimensional systems, can be treated as a mapping of a plane onto itself. KAM theory has shown that these trajectories can be either periodic, restricted to a KAM curve, or ergodic and thus filling a finite fraction of the available phase space. The problem addressed here is finding when particular KAM surfaces exist. The method adopted is to study numerically the periodic orbits of a particular mapping. The guiding hypothesis is that the disappearance of a KAM surface is associated with a sudden change from stability to instability of all nearby periodic orbits

[en] The motion of ions in the fields B and E is considered. The equations of motion may be reduced to a set of difference equations. These equations exhibit stochastic behavior when E₀ exceeds a threshold. The diffusion coefficient above the threshold is determined. Far above the threshold, ion Landau damping is recovered. Extension of the method to include parallel propagation is outlined

[en] Toroidally confined plasmas appear to be generally subject to anomalous diffusion. One explanation of the phenomenon is that the diffusion is the result of E x B drifts from a set of waves propagating along rational surfaces. The theory of the drift islands associated with these waves is reviewed, both in the collisionless and highly collisional limits. Assuming that a low frequency resistive-g mode is responsible for the diffusion, the diffusion is calculated, for the Culham Levitron, for the measured values of the fluctuating potential. The results are compared with the measured value of diffusion coefficient, over the range where the theory is thought to apply, giving reasonably good agreement with theory. Other mechanisms for anomalous diffusion in the Levitron are also discussed