scholarly journals Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Nicolas Crouseilles ◽  
Paul-Antoine Hervieux ◽  
Yingzhe Li ◽  
Giovanni Manfredi ◽  
Yajuan Sun
Keyword(s):  
Phase Space ◽  
Maxwell Equations ◽  
Stimulated Raman Scattering ◽  
Extended Phase ◽  
Five Dimensions ◽  
Particle In Cell ◽  
Spin Effects ◽  
Dimensional Phase Space ◽  
Spin Polarized ◽  
Underdense Plasma

We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

scholarly journals Hermite Spline Interpolation on Patches for Parallelly Solving the Vlasov-Poisson Equation

2007 ◽  
Vol 17 (3) ◽  
pp. 335-349 ◽  
Author(s):  
Nicolas Crouseilles ◽  
Guillaume Latu ◽  
Eric Sonnendrücker
Keyword(s):  
Phase Space ◽  
Poisson Equation ◽  
Vlasov Equation ◽  
Parallel Machines ◽  
Spline Interpolation ◽  
Computational Effort ◽  
Type Method ◽  
Particle In Cell ◽  
Dimensional Phase Space ◽  
Hermite Spline

Hermite Spline Interpolation on Patches for Parallelly Solving the Vlasov-Poisson EquationThis work is devoted to the numerical simulation of the Vlasov equation using a phase space grid. In contrast to Particle-In-Cell (PIC) methods, which are known to be noisy, we propose a semi-Lagrangian-type method to discretize the Vlasov equation in the two-dimensional phase space. As this kind of method requires a huge computational effort, one has to carry out the simulations on parallel machines. For this purpose, we present a method using patches decomposing the phase domain, each patch being devoted to a processor. Some Hermite boundary conditions allow for the reconstruction of a good approximation of the global solution. Several numerical results demonstrate the accuracy and the good scalability of the method with up to 64 processors. This work is a part of the CALVI project.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

Explicit symplectic-like integration with corrected map for inseparable Hamiltonian

2020 ◽  
Vol 501 (1) ◽  
pp. 1511-1519
Author(s):  
Junjie Luo ◽  
Weipeng Lin ◽  
Lili Yang
Keyword(s):  
Phase Space ◽  
Space Structure ◽  
Eighth Order ◽  
Time Step ◽  
Extended Phase ◽  
Energy Error ◽  
Phase Space Structure ◽  
Compact Binary ◽  
Three Body

ABSTRACT Symplectic algorithms are widely used for long-term integration of astrophysical problems. However, this technique can only be easily constructed for separable Hamiltonian, as preserving the phase-space structure. Recently, for inseparable Hamiltonian, the fourth-order extended phase-space explicit symplectic-like methods have been developed by using the Yoshida’s triple product with a mid-point map, where the algorithm is more effective, stable and also more accurate, compared with the sequent permutations of momenta and position coordinates, especially for some chaotic case. However, it has been found that, for the cases such as with chaotic orbits of spinning compact binary or circular restricted three-body system, it may cause secular drift in energy error and even more the computation break down. To solve this problem, we have made further improvement on the mid-point map with a momentum-scaling correction, which turns out to behave more stably in long-term evolution and have smaller energy error than previous methods. In particular, it could obtain a comparable phase-space distance as computing from the eighth-order Runge–Kutta method with the same time-step.

TRANSITION STATE THEORY IN EXTENDED PHASE SPACE

1993 ◽  
Vol 07 (19) ◽  
pp. 1263-1268
Author(s):  
H. DEKKER ◽  
A. MAASSEN VAN DEN BRINK
Keyword(s):  
Phase Space ◽  
Transition State Theory ◽  
Unstable Mode ◽  
Planck Equation ◽  
State Theory ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Theoretical Concepts ◽  
Rate Processes ◽  
Mode Energy

Turnover theory (of the escape Γ) à la Grabert will be based solely on Kramers' Fokker–Planck equation for activated rate processes. No recourse to a microscope model or Langevin dynamics will be made. Apart from the unstable mode energy E, the analysis requires new theoretical concepts such as a constrained Gaussian transformation (CGT) and dynamically extended phase space (EPS).

scholarly journals Reality of the Wigner Functions and Quantization

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Sadollah Nasiri ◽  
Samira Bahrami
Keyword(s):  
Phase Space ◽  
Quantum Statistical Mechanics ◽  
Direct Consequence ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Reality Condition ◽  
Energy Quantization ◽  
Extended Action ◽  
Space Formulation ◽  
Quantum Statistical

Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.

A mechanism for spin electron acoustic soliton observed in a spin polarized nanosized electron-hole plasma

2021 ◽  
Author(s):  
Mahmoud Saad Afify ◽  
Zafar Iqbal ◽  
Ghulam Murtaza
Keyword(s):  
Maxwell Equations ◽  
Kdv Equation ◽  
Reductive Perturbation Method ◽  
Electron Hole ◽  
Hole Plasma ◽  
Electron Hole Plasma ◽  
Spin Polarized ◽  
Qhd Model ◽  
Quantum Hydrodynamic ◽  
Electron Acoustic

Abstract The formation and the characteristics of spin electron acoustic (SEA) soliton in a beam interacting spin polarized electron-hole plasma are investigated. These wavepackets are supposed to be the source of heating during the excitation process. We have used the separate spin evolution-quantum hydrodynamic (SSE-QHD) model along with Maxwell equations and derived the Korteweg-de Vries (KdV) equation by using the reductive perturbation method (RPM). We note that the larger values of beam density and spin polarization can change the soliton nature from rarefactive to compressive. Our findings may be important to understand the characteristics of localized spin dependent nonlinear waves in nanosized semiconductor devices.

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