Quantum chaotic scattering with a mixed phase space: The three-disk billiard in a magnetic field

2000 ◽  
Vol 61 (1) ◽  
pp. 382-389 ◽  
Author(s):  
Markus Eichengrün ◽  
Walter Schirmacher ◽  
Wolfgang Bregmann
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Mixed Phase ◽  
Chaotic Scattering

scholarly journals Double layers in the downward current region of the aurora

2003 ◽  
Vol 10 (1/2) ◽  
pp. 45-52 ◽  
Author(s):  
R. E. Ergun ◽  
L. Andersson ◽  
C. W. Carlson ◽  
D. L. Newman ◽  
M. V. Goldman
Keyword(s):  
Magnetic Field ◽  
Electron Beam ◽  
Phase Space ◽  
Electric Field ◽  
Electric Fields ◽  
Double Layers ◽  
Parallel Electric Field ◽  
Electrostatic Waves ◽  
Current Region ◽  
Decisive Evidence

Abstract. Direct observations of magnetic-field-aligned (parallel) electric fields in the downward current region of the aurora provide decisive evidence of naturally occurring double layers. We report measurements of parallel electric fields, electron fluxes and ion fluxes related to double layers that are responsible for particle acceleration. The observations suggest that parallel electric fields organize into a structure of three distinct, narrowly-confined regions along the magnetic field (B). In the "ramp" region, the measured parallel electric field forms a nearly-monotonic potential ramp that is localized to ~ 10 Debye lengths along B. The ramp is moving parallel to B at the ion acoustic speed (vs) and in the same direction as the accelerated electrons. On the high-potential side of the ramp, in the "beam" region, an unstable electron beam is seen for roughly another 10 Debye lengths along B. The electron beam is rapidly stabilized by intense electrostatic waves and nonlinear structures interpreted as electron phase-space holes. The "wave" region is physically separated from the ramp by the beam region. Numerical simulations reproduce a similar ramp structure, beam region, electrostatic turbulence region and plasma characteristics as seen in the observations. These results suggest that large double layers can account for the parallel electric field in the downward current region and that intense electrostatic turbulence rapidly stabilizes the accelerated electron distributions. These results also demonstrate that parallel electric fields are directly associated with the generation of large-amplitude electron phase-space holes and plasma waves.

CHAOTIC SCATTERING IN TIME-DEPENDENT HAMILTONIAN SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 667-679 ◽  
Author(s):  
YING-CHENG LAI ◽  
CELSO GREBOGI
Keyword(s):  
Phase Space ◽  
Measure Zero ◽  
Invariant Set ◽  
Time Dependent ◽  
Invariant Sets ◽  
Physical Origin ◽  
Chaotic Scattering ◽  
Periodic Oscillations ◽  
Surface Of Section ◽  
Oscillating Potential

We consider the classical scattering of particles in a one-degree-of-freedom, time-dependent Hamiltonian system. We demonstrate that chaotic scattering can be induced by periodic oscillations in the position of the potential. We study the invariant sets on a surface of section for different amplitudes of the oscillating potential. It is found that for small amplitudes, the phase space consists of nonescaping KAM islands and an escaping set. The escaping set is made up of a nonhyperbolic set that gives rise to chaotic scattering and remains of KAM islands. For large amplitudes, the phase space contains a Lebesgue measure zero invariant set that gives rise to chaotic scattering. In this regime, we also discuss the physical origin of the Cantor set responsible for the chaotic scattering and calculate its fractal dimension.

scholarly journals The formation of relativistic plasma structures and their potential role in the generation of cosmic ray electrons

2008 ◽  
Vol 15 (6) ◽  
pp. 831-846 ◽  
Author(s):  
M. E. Dieckmann
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Magnetic Energy ◽  
Ion Beam ◽  
Cosmic Ray ◽  
Compact Objects ◽  
Space Structures ◽  
Magnetic Field Generation ◽  
Particle In Cell ◽  
Field Generation

Abstract. Recent particle-in-cell (PIC) simulation studies have addressed particle acceleration and magnetic field generation in relativistic astrophysical flows by plasma phase space structures. We discuss the astrophysical environments such as the jets of compact objects, and we give an overview of the global PIC simulations of shocks. These reveal several types of phase space structures, which are relevant for the energy dissipation. These structures are typically coupled in shocks, but we choose to consider them here in an isolated form. Three structures are reviewed. (1) Simulations of interpenetrating or colliding plasma clouds can trigger filamentation instabilities, while simulations of thermally anisotropic plasmas observe the Weibel instability. Both transform a spatially uniform plasma into current filaments. These filament structures cause the growth of the magnetic fields. (2) The development of a modified two-stream instability is discussed. It saturates first by the formation of electron phase space holes. The relativistic electron clouds modulate the ion beam and a secondary, spatially localized electrostatic instability grows, which saturates by forming a relativistic ion phase space hole. It accelerates electrons to ultra-relativistic speeds. (3) A simulation is also revised, in which two clouds of an electron-ion plasma collide at the speed 0.9c. The inequal densities of both clouds and a magnetic field that is oblique to the collision velocity vector result in waves with a mixed electrostatic and electromagnetic polarity. The waves give rise to growing corkscrew distributions in the electrons and ions that establish an equipartition between the electron, the ion and the magnetic energy. The filament-, phase space hole- and corkscrew structures are discussed with respect to electron acceleration and magnetic field generation.

Direct Measurement of Excitation and Growth of Large Amplitude Cyclotron Waves by Reflected Ion Beams in Front of Shock

2020 ◽  
Author(s):  
Jiansen He ◽  
Chuanpeng Hou ◽  
Xingyu Zhu ◽  
Qiaowen Luo ◽  
Daniel Verscharen ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Electromagnetic Field ◽  
Phase Space ◽  
Large Amplitude ◽  
Particle Interaction ◽  
Field Vector ◽  
Ion Beams ◽  
Magnetic Field Vector ◽  
Wave Particle Interaction

<p>Wave-particle interaction plays a critical role in producing the newborn waves/turbulence in the foreshock region in front of supercritical shock, which is prevalent in the heliosphere. It has been a long-lasting goal to catch and witness the excitation and growth of waves/turbulence by identifying the ongoing process of wave-particle interaction. This goal cannot be fulfilled until the arrival of the MMS’s era, during which we can simultaneously measure the electromagnetic fields and particle phase space densities with the unprecedented data quality. By surveying the data of burst mode, we are lucky to find some good examples illustrating the clear signals of wave activities in front of the shock. The active waves are diagnosed to be right-handed cyclotron waves, being highly circularly polarized and rotating right-handed about the background magnetic field vector. The waves are large amplitude with dB being greatly dominant over B0, or in other words, almost the whole magnetic field vector is involved in the circular rotation. Furthermore, we investigate the growth evolution of the large-amplitude cyclotron waves by calculating the spectrum of dJ.dE and its ratio to the electromagnetic energy spectrum. As far as we know, it is the first time to provide the spectrum of growth rate from in-situ measurements. Interestingly, we find that the contribution to the growth rate spectrum mainly comes from dJ<sub>e,perp</sub>·dE<sub>perp</sub> rather than dJ<sub>e,para</sub>·dE<sub>para</sub> or d<strong>J</strong><sub>i</sub>·d<strong>E</strong>. Although the eigen mode to couple the oscillating electromagnetic field is the electron bulk oscillation, the ultimate free energy to make the eigen mode unstable comes from the ion beams, which are reflected from the shock. The dynamics of 3D phase space densities for both ion and electron species are also studied in detail together with the fluctuating electromagnetic field, demonstrating the ongoing energy conversion during the wave-particle process.</p><p> </p>

Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field

1986 ◽  
Vol 403 (1824) ◽  
pp. 135-166 ◽  
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Periodic Function ◽  
Degrees Of Freedom ◽  
Two Dimensions ◽  
Effective Hamiltonian ◽  
Hall Conductance ◽  
Von Neumann ◽  
Wannier Functions ◽  
Bloch Electrons

The problem of Bloch electrons in a magnetic field in two dimensions can be reduced to a one-dimensional problem with a Hamiltonian Ĥ that is a periodic function of x ^ and p ^ . Wannier functions can be defined for the sub-bands of the spectrum of this effective Hamiltonian. When the Chern class (quantized Hall conductance integer) of the sub-band is zero, the Weyl-Wigner formalism can be used to represent these Wannier functions by a von Neumann lattice. It is shown how this von Neumann lattice of Wannier functions can be defined for irrational as well as rational magnetic fields. An important benefit from using the Weyl-Wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective Hamiltonian in phase space. It is shown how the Wannier functions can be defined so that their Wigner functions have the point symmetries of the effective Hamiltonian. An example of how these results can prove useful is given: if we take matrix elements of the Hamiltonian between the Wannier states of a sub-band, we derive a new effective Hamiltonian describing this sub-band, which is again a periodic function of coordinate and momentum operators. Since, by projecting onto a sub-band, we have also reduced the number of degrees of freedom, this operation is a renormalization group transformation. It is shown that the symmetry of the new effective Hamil­tonian in phase space is the same as that of the original one. This preservation of symmetry helps to explain some unusual properties of the spectrum when the Hamiltonian has fourfold symmetry.

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