An introduction to symplectic integrators

1996 ◽  
pp. 45-58 ◽  
Author(s):  
Paul Channell ◽  
F Neri
Keyword(s):  
Symplectic Integrators

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

Symplectic integrators with adaptive time step applied to runaway electron dynamics

2019 ◽  
Vol 81 (4) ◽  
pp. 1295-1309 ◽  
Author(s):  
Yanyan Shi ◽  
Yajuan Sun ◽  
Yang He ◽  
Hong Qin ◽  
Jian Liu
Keyword(s):  
Runaway Electron ◽  
Symplectic Integrators ◽  
Electron Dynamics ◽  
Time Step ◽  
Adaptive Time Step ◽  
Adaptive Time

scholarly journals Poisson brackets, quasi-states and symplectic integrators

2010 ◽  
Vol 28 (4) ◽  
pp. 1455-1468 ◽  
Author(s):  
Michael Entov ◽  
◽  
Leonid Polterovich ◽  
Daniel Rosen ◽  
Keyword(s):  
Symplectic Integrators ◽  
Poisson Brackets

QUASI SYMPLECTIC INTEGRATORS: AN ALTERNATIVE TO NOSÉ HOOVER DYNAMICS?

2005 ◽  
Vol 05 (02) ◽  
pp. L225-L232
Author(s):  
RICCARDO MANNELLA
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulations ◽  
Numerical Integration ◽  
Canonical Ensemble ◽  
Symplectic Integrators ◽  
Langevin Equations ◽  
Dynamics Simulations

Quasi symplectic algorithms for the numerical integration of Langevin equations describing systems in a canonical ensemble are discussed. It is shown that they could be an alternative to molecular dynamics simulations done with a Nosé Hoover booster.

Cheap implicit symplectic integrators

1997 ◽  
Vol 25 (2-3) ◽  
pp. 297-302 ◽  
Author(s):  
Meiqing Zhang ◽  
Robert D. Skeel
Keyword(s):  
Symplectic Integrators

scholarly journals Accelerated methods for direct computation of fusion alpha particle losses within, stellarator optimization

2020 ◽  
Vol 86 (2) ◽  
Author(s):  
Christopher G. Albert ◽  
Sergei V. Kasilov ◽  
Winfried Kernbichler
Keyword(s):  
Alpha Particle ◽  
Symplectic Integrators ◽  
Particle Loss ◽  
Early Classification ◽  
Statistical Computation ◽  
Computational Speed ◽  
Order Of Magnitude ◽  
Speed Up ◽  
Orbit Types ◽  
Optimization Loop

Accelerated statistical computation of collisionless fusion alpha particle losses in stellarator configurations is presented based on direct guiding-centre orbit tracing. The approach relies on the combination of recently developed symplectic integrators in canonicalized magnetic flux coordinates and early classification into regular and chaotic orbit types. Only chaotic orbits have to be traced up to the end, as their behaviour is unpredictable. An implementation of this technique is provided in the code SIMPLE (symplectic integration methods for particle loss estimation, Albert et al., 2020b, doi:10.5281/zenodo.3666820). Reliable results were obtained for an ensemble of 1000 orbits in a quasi-isodynamic, a quasi-helical and a quasi-axisymmetric configuration. Overall, a computational speed up of approximately one order of magnitude is achieved compared to direct integration via adaptive Runge–Kutta methods. This reduces run times to the range of typical magnetic equilibrium computations and makes direct alpha particle loss computation adequate for use within a stellarator optimization loop.

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