Application of Lie Algebra in Constructing Volume-Preserving Algorithms for Charged Particles Dynamics

2016 ◽  
Vol 19 (5) ◽  
pp. 1397-1408 ◽  
Author(s):  
Ruili Zhang ◽  
Jian Liu ◽  
Hong Qin ◽  
Yifa Tang ◽  
Yang He ◽  
...  
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Charged Particles ◽  
Second Order ◽  
Algebra Structure ◽  
Phase Space Volume ◽  
Volume Preserving ◽  
Particles Dynamics ◽  
Space Volume

AbstractVolume-preserving algorithms (VPAs) for the charged particles dynamics is preferred because of their long-term accuracy and conservativeness for phase space volume. Lie algebra and the Baker-Campbell-Hausdorff (BCH) formula can be used as a fundamental theoretical tool to construct VPAs. Using the Lie algebra structure of vector fields, we split the volume-preserving vector field for charged particle dynamics into three volume-preserving parts (sub-algebras), and find the corresponding Lie subgroups. Proper combinations of these subgroups generate volume preserving, second order approximations of the original solution group, and thus second order VPAs. The developed VPAs also show their significant effectiveness in conserving phase-space volume exactly and bounding energy error over long-term simulations.

Phase-space volume based control of semibatch reactors

2010 ◽  
Vol 88 (3) ◽  
pp. 320-330 ◽  
Author(s):  
José-Manuel Zaldívar ◽  
Fernanda Strozzi
Keyword(s):  
Phase Space ◽  
Phase Space Volume ◽  
Space Volume

scholarly journals Almost additive entropy

2014 ◽  
Vol 11 (05) ◽  
pp. 1450040 ◽  
Author(s):  
Nikos Kalogeropoulos
Keyword(s):  
Phase Space ◽  
Shannon Entropy ◽  
Power Law ◽  
Physical Significance ◽  
Small Deviations ◽  
Phase Space Volume ◽  
Composition Property ◽  
Flat Manifolds ◽  
Special Case ◽  
Space Volume

We explore consequences of a hyperbolic metric induced by the composition property of the Harvda–Charvat/Daróczy/Cressie–Read/Tsallis entropy. We address the special case of systems described by small deviations of the non-extensive parameter q ≈ 1 from the "ordinary" additive case which is described by the Boltzmann/Gibbs/Shannon entropy. By applying the Gromov/Ruh theorem for almost flat manifolds, we show that such systems have a power-law rate of expansion of their configuration/phase space volume. We explore the possible physical significance of some geometric and topological results of this approach.

scholarly journals Relativistic transformation of phase-space distributions

2011 ◽  
Vol 29 (7) ◽  
pp. 1259-1265 ◽  
Author(s):  
R. A. Treumann ◽  
R. Nakamura ◽  
W. Baumjohann
Keyword(s):  
Distribution Function ◽  
Phase Space ◽  
Volume Element ◽  
Plasma Parameters ◽  
Astrophysical Plasmas ◽  
Relativistic Case ◽  
Phase Space Volume ◽  
Radiation Theory ◽  
High Mach ◽  
Space Volume

Abstract. We investigate the transformation of the distribution function in the relativistic case, a problem of interest in plasma when particles with high (relativistic) velocities come into play as for instance in radiation belt physics, in the electron-cyclotron maser radiation theory, in the vicinity of high-Mach number shocks where particles are accelerated to high speeds, and generally in solar and astrophysical plasmas. We show that the phase-space volume element is a Lorentz constant and construct the general particle distribution function from first principles. Application to thermal equilibrium lets us derive a modified version of the isotropic relativistic thermal distribution, the modified Jüttner distribution corrected for the Lorentz-invariant phase-space volume element. Finally, we discuss the relativistic modification of a number of plasma parameters.

scholarly journals SHARP SPECTRAL ESTIMATES IN DOMAINS OF INFINITE VOLUME

2011 ◽  
Vol 23 (06) ◽  
pp. 615-641 ◽  
Author(s):  
LEANDER GEISINGER ◽  
TIMO WEIDL
Keyword(s):  
Phase Space ◽  
Dirichlet Boundary ◽  
Semiclassical Limit ◽  
Dirichlet Boundary Conditions ◽  
Bounded Domains ◽  
Infinite Volume ◽  
Uniform Bounds ◽  
Phase Space Volume ◽  
Spectral Estimates ◽  
Space Volume

We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume — and therefore on the volume of the domain — must fail. Here we present a method on how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit.We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schrödinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.

scholarly journals A Family of Hierarchical Symplectic Maps for N-body Simulations with Post-Newtonian Corrections

2017 ◽  
Vol 45 ◽  
pp. 1760021
Author(s):  
Guilherme Gonçalves Ferrari
Keyword(s):  
Phase Space ◽  
Body Problem ◽  
Hamiltonian Dynamics ◽  
Symplectic Maps ◽  
Time Step ◽  
Phase Space Volume ◽  
Time Stepping ◽  
Adaptive Time ◽  
The Individual ◽  
Space Volume

Symplectic maps are well known for preserving the phase-space volume in Hamiltonian dynamics and are particularly suited for problems that require long integration times, such as the [Formula: see text]-body problem. However, when combined with a varying time-step scheme, they end up losing its symplecticity and become numerically inefficient. We address this problem by using a recursive Hamiltonian splitting based on the time-symmetric value of the individual time-steps required by the particles in the system. We present a family of 48 quasi-symplectic maps with different orders of convergence (2nd-, 4th- & 6th-order) and three time-stepping schemes: i) 16 using constant time-steps, ii) 16 using shared adaptive time-steps, and iii) 16 using hierarchical (individual) time-steps. All maps include post-Newtonian corrections up to order 3.5PN. We describe the method and present some details of the implementation.

scholarly journals A high energy, small phase-space volume muon beam

1969 ◽  
Vol 69 (1) ◽  
pp. 77-88 ◽  
Author(s):  
J. Cox ◽  
F. Martin ◽  
M.L. Perl ◽  
T.H. Tan ◽  
W.T. Toner ◽  
...  
Keyword(s):  
Phase Space ◽  
High Energy ◽  
Muon Beam ◽  
Phase Space Volume ◽  
Small Phase ◽  
Space Volume
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