Explicit Symplectic Integrator Research Articles

Effects of Two Quantum Correction Parameters on Chaotic Dynamics of Particles near Renormalized Group Improved Schwarzschild Black Holes

A renormalized group improved Schwarzschild black hole spacetime contains two quantum correction parameters. One parameter γ represents the identification of cutoff of the distance scale, and another parameter Ω stems from nonperturbative renormalization group theory. The two parameters are constrained by the data from the shadow of M87* central black hole. The dynamics of electrically charged test particles around the black hole are integrable. However, when the black hole is immersed in an external asymptotically uniform magnetic field, the dynamics are not integrable and may allow for the occurrence of chaos. Employing an explicit symplectic integrator, we survey the contributions of the two parameters to the chaotic dynamical behavior. It is found that a small change of the parameter γ constrained by the shadow of M87* black hole has an almost negligible effect on the dynamical transition of particles from order to chaos. However, a small decrease in the parameter Ω leads to an enhancement in the strength of chaos from the global phase space structure. A theoretical interpretation is given to the different contributions. The term with the parameter Ω dominates the term with the parameter γ, even if the two parameters have same values. In particular, the parameter Ω acts as a repulsive force, and its decrease means a weakening of the repulsive force or equivalently enhancing the attractive force from the black hole. On the other hand, there is a positive Lyapunov exponent that is universally given by the surface gravity of the black hole when Ω≥0 is small and the external magnetic field vanishes. In this case, the horizon would influence chaotic behavior in the motion of charged particles around the black hole surrounded by the external magnetic field. This point can explain why a smaller value of the renormalization group parameter would much easily induce chaos than a larger value.

Application of Explicit Symplectic Integrators in the Magnetized Reissner–Nordström Spacetime

In recent works by Wu and Wang a class of explicit symplectic integrators in curved spacetimes was presented. Different splitting forms or appropriate choices of time-transformed Hamiltonians are determined based on specific Hamiltonian problems. As its application, we constructed a suitable explicit symplectic integrator for surveying the dynamics of test particles in a magnetized Reissner–Nordström spacetime. In addition to computational efficiency, the scheme exhibits good stability and high precision for long-term integration. From the global phase-space structure of Poincaré sections, the extent of chaos can be strengthened when energy E, magnetic parameter B, or the charge q become larger. On the contrary, the occurrence of chaoticity is weakened with an increase of electric parameter Q and angular momentum L. The conclusion can also be supported by fast Lyapunov indicators.

Impact of electric charges on chaos in magnetized Reissner–Nordström spacetimes

We consider the motion of test particles around a Reissner–Nordström black hole immersed into a strong external magnetic field modifying the spacetime structure. When the particles are neutral, their dynamics are nonintegrable because the magnetic field acts as a gravitational effect, which destroys the existence of a fourth motion constant in the Reissner–Nordström spacetime. A time-transformed explicit symplectic integrator is used to show that the motion of neutral particles can be chaotic under some circumstances. When test particles have electric charges, their motions are subject to an electromagnetic field surrounding the black hole as well as the gravitational forces from the black hole and the magnetic field. It is found that increasing both the magnetic field and the particle energy or decreasing the particle angular momentum can strengthen the degree of chaos regardless of whether the particles are neutral or charged. The effect of varying the black hole positive charge on the dynamical transition from order to chaos is associated with the electric charges of particles. The dynamical transition of neutral particles has no sensitive dependence on a change of the black hole charge. An increase of the black hole charge weakens the chaoticity of positive charged particles, whereas enhances the chaoticity of negative charged particles. With the magnitude of particle charge increasing, chaos always gets stronger.

Application of Explicit Symplectic Integrators in a Magnetized Deformed Schwarzschild Black Spacetime

Abstract Following the latest work of Wu et al., we construct time-transformed explicit symplectic schemes for a Hamiltonian system on the description of charged particles moving around a deformed Schwarzschild black hole with an external magnetic field. Numerical tests show that such schemes have good performance in stabilizing energy errors without secular drift. Meantime, tangent vectors are solved from the variational equations of the system with the aid of an explicit symplectic integrator. The obtained tangent vectors are used to calculate several chaos indicators, including Lyapunov characteristic exponents, fast Lyapunov indicators, a smaller alignment index, and a generalized alignment index. It is found that the smaller alignment index and generalized alignment index are the fastest indicators for distinguishing between regular and chaotic cases. The smaller alignment index is applied to explore the effects of the parameters on the dynamical transition from order to chaos. When the positive deformation factor and angular momentum decrease, or when the energy, positive magnetic parameter, and the magnitude of the negative deformation parameter increase, chaos easily occurs for the appropriate choices of initial conditions and the other parameters.

Dynamics of Charged Particles Moving around Kerr Black Hole with Inductive Charge and External Magnetic Field

We mainly focus on the effects of small changes of parameters on the dynamics of charged particles around Kerr black holes surrounded by an external magnetic field, which can be considered as a tidal environment. The radial motions of charged particles on the equatorial plane are studied via an effective potential. It is found that the particle energies at the local maxima values of the effective potentials increase with an increase in the black hole spin and the particle angular momenta, but decrease with an increase of one of the inductive charge parameter and magnetic field parameter. The radii of stable circular orbits on the equatorial plane also increase, whereas those of the innermost stable circular orbits decrease. On the other hand, the effects of small variations of the parameters on the orbital regular and chaotic dynamics of charged particles on the non-equatorial plane are traced by means of a time-transformed explicit symplectic integrator, Poincaré sections and fast Lyapunov indicators. It is shown that the dynamics sensitivity depends on small variations in the inductive charge parameter, magnetic field parameter, energy, and angular momentum. Chaos occurs easily as each of the inductive charge parameter, magnetic field parameter, and energy increases but is weakened as the angular momentum increases. When the dragging effects of the spacetime increase, the chaotic properties are not always weakened under some circumstances.

Merlin++, a flexible and feature-rich accelerator physics and particle tracking library

Merlin++ is a C++ charged-particle tracking library developed for the simulation and analysis of complex beam dynamics within high energy particle accelerators. Accurate simulation and analysis of particle dynamics is an essential part of the design of new particle accelerators, and for the optimization of existing ones. Merlin++ is a feature-full library with focus on long-term tracking studies. A user may simulate distributions of protons or electrons in either single particle or sliced macro-particle bunches. The tracking code includes both straight and curvilinear coordinate systems allowing for the simulation of either linear or circular accelerator lattice designs, and uses a fast and accurate explicit symplectic integrator. Physics processes for common design studies have been implemented, including RF cavity acceleration, synchrotron radiation damping, on-line physical aperture checks and collimation, proton scattering, wakefield simulation, and spin-tracking. Merlin++ was written using C++ object orientated design practices and has been optimized for speed using multicore processors. This article presents an account of the program, including its functionality and guidance for use. Program summaryProgram Title: Merlin++CPC Library link to program files:https://doi.org/10.17632/4x4nsbhz37.1Developer's repository link: 10.5281/zenodo.3700155Licensing provisions: GPLv2+Programming language: C++Nature of problem: Complexity of particle accelerators beam dynamics over extensive tracking distances.Solution method: Long-term particle accelerator and tracking simulations utilizing explicit symplectic integrators.Additional comments including restrictions and unusual features: For further information see github.com/Merlin-Collaboration

Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole

In a recent work of Wu, Wang, Sun and Liu, a second-order explicit symplectic integrator was proposed for the integrable Kerr spacetime geometry. It is still suited for simulating the nonintegrable dynamics of charged particles moving around the Kerr black hole embedded in an external magnetic field. Its successful construction is due to the contribution of a time transformation. The algorithm exhibits a good long-term numerical performance in stable Hamiltonian errors and computational efficiency. As its application, the dynamics of order and chaos of charged particles is surveyed. In some circumstances, an increase of the dragging effects of the spacetime seems to weaken the extent of chaos from the global phase-space structure on Poincaré sections. However, an increase of the magnetic parameter strengthens the chaotic properties. On the other hand, fast Lyapunov indicators show that there is no universal rule for the dependence of the transition between different dynamical regimes on the black hole spin. The dragging effects of the spacetime do not always weaken the extent of chaos from a local point of view.

A slow-down time-transformed symplectic integrator for solving the few-body problem

ABSTRACTAn accurate and efficient method dealing with the few-body dynamics is important for simulating collisional N-body systems like star clusters and to follow the formation and evolution of compact binaries. We describe such a method which combines the time-transformed explicit symplectic integrator and the slow-down method. The former conserves the Hamiltonian and the angular momentum for a long-term evolution, while the latter significantly reduces the computational cost for a weakly perturbed binary. In this work, the Hamilton equations of this algorithm are analysed in detail. We mathematically and numerically show that it can correctly reproduce the secular evolution like the orbit averaged method and also well conserve the angular momentum. For a weakly perturbed binary, the method is possible to provide a few orders of magnitude faster performance than the classical algorithm. A publicly available code written in the c++ language, sdar, is available on github. It can be used either as a standalone tool or a library to be plugged in other N-body codes. The high precision of the floating point to 62 digits is also supported.

Explicit symplectic integrator for particle tracking in s -dependent static electric and magnetic fields with curved reference trajectory

We describe a method for symplectic tracking of charged particles through static electric and magnetic fields. The method can be applied to cases where the fields have a dependence on longitudinal as well as transverse position, and where the reference trajectory may have non-zero curvature. Application of the method requires analytical expressions for the scalar and vector potentials: we show how suitable expressions, in the form of series analogous to multipole expansions, can be constructed from numerical field data, allowing the method to be used in cases where only numerical field data are available.

Adaptive time-stepping Monte Carlo integration of Coulomb collisions

We report an accessible and robust tool for evaluating the effects of Coulomb collisions on a test particle in a plasma that obeys Maxwell–Jüttner statistics. The implementation is based on the Beliaev–Budker collision integral which allows both the test particle and the background plasma to be relativistic. The integration method supports adaptive time stepping, which is shown to greatly improve the computational efficiency. The Monte Carlo method is implemented for both the three-dimensional particle momentum space and the five-dimensional guiding center phase space.Detailed description is provided for both the physics and implementation of the operator. The focus is in adaptive integration of stochastic differential equations, which is an overlooked aspect among existing Monte Carlo implementations of Coulomb collision operators. We verify that our operator converges to known analytical results and demonstrate that careless implementation of the adaptive time step can lead to severely erroneous results.The operator is provided as a self-contained Fortran 95 module and can be included into existing orbit-following tools that trace either the full Larmor motion or the guiding center dynamics. The adaptive time-stepping algorithm is expected to be useful in situations where the collision frequencies vary greatly over the course of a simulation. Examples include the slowing-down of fusion products or other fast ions, and the Dreicer generation of runaway electrons as well as the generation of fast ions or electrons with ion or electron cyclotron resonance heating. Program summaryProgram Title: AMCC –(A)daptive (M)onte-(C)arlo (C)oulomb collisionsProgram Files doi:http://dx.doi.org/10.17632/wd3t3t6dy7.1Licensing provisions: MIT licenceProgramming language: Fortran 95Nature of problem: Test-particle tracing is a common feat within existing fusion applications. While efficient adaptive methods exist for integrating the incompressible Hamiltonian flow, the effects of Coulomb collisions are commonly implemented with far less sophisticated algorithms.Solution method: The relativistic Fokker–Planck equation for test-particles in Maxwell–Jüttner background plasmas is converted into a stochastic differential equation. The stochastic differential equation is solved using adaptive Monte Carlo techniques. Methods to evaluate the effect of Coulomb collisions for both the three-dimensional particle momentum space and the five-dimensional reduced guiding center phase space are included.Additional comments including restrictions and unusual features: The package includes optionality to evaluate the relativistic Fokker–Planck coefficients, a feature useful for constructing accurate orbit averaged collision operators. The package also provides explicit one-step symplectic integrator for the relativistic Lorentz force that can be used for tracing test-particles in given electromagnetic backgrounds.

New adaptive time step symplectic integrator: an application to the elliptic restricted three-body problem

The time-transformed leapfrog scheme of Mikkola & Aarseth was specifically designed for a second-order differential equation with two individually separable forms of positions and velocities. It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios, but the non-time-transformed one does not work well. Following this idea, we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian. Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical conjugate variables in the extended phase space. In addition, the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts. When this method is applied to the elliptic restricted three-body problem, its numerical precision is explicitly higher by several orders of magnitude than the nonadaptive one's, and its numerical stability is also better. In particular, it can eliminate the overestimation of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body. The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics, and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system.

Momentum Conservation Law in Explicit Symplectic Integrators for Nonlinear Wave Equations

The momentum conservation law in an explicit symplectic integrator for nonlinear wave equations is discussed. We show that the total momentum is approximately conserved within a numerical error on the order of the rounding error. We also perform numerical simulations to demonstrate the validity and numerical convergence of our expression for the total momentum. As a result, we show that a symplectic integrator for nonlinear wave equations possesses desirable properties of numerical methods, explicitness, and substantial conservation of the energy and total momentum.

A symplectic integrator for the symmetry reduced and regularised planar 3-body problem with vanishing angular momentum

We construct an explicit reversible symplectic integrator for the planar 3-body problem with zero angular momentum. We start with a Hamiltonian of the planar 3-body problem that is globally regularised and fully symmetry reduced. This Hamiltonian is a sum of 10 polynomials each of which can be integrated exactly, and hence a symplectic integrator is constructed. The performance of the integrator is examined with three numerical examples: The figure eight, the Pythagorean orbit, and a periodic collision orbit.

Self-Trapping in Discrete Nonlinear Schrödinger Equation with Next-Nearest Neighbor Interaction

The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with next-nearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using localized initial conditions, the time-averaged occupation probability of the initial site is investigated which is a function of the degree of nonlinearity and the linear coupling strengths. The self-trapping transition occurs at larger values of the nonlinearity parameter as the NNN coupling strength of the lattice increases for fixed size. Furthermore, given NNN coupling strength, the self-trapping properties for different sizes are considered which are some different from the case with general nearest neighbor (NN) interaction.

Use of transfer maps for modeling beam dynamics in a nonscaling fixed-field alternating-gradient accelerator

Transfer maps for magnetic components are fundamental to studies of beam dynamics in accelerators. In the work presented here, transfer maps are computed in Taylor form for a particle moving through any specified magnetostatic field by applying an explicit symplectic integrator in a differential algebra code. The techniques developed are illustrated by their application to study the beam dynamics in the electron model for many applications (EMMA), the first nonscaling fixed-field alternating-gradient accelerator ever built. The EMMA lattice has 4 degrees of freedom (strength and transverse position of each of the two quadrupoles in each periodic cell). Transfer maps may be used to predict efficiently the dynamics in any lattice configuration. The transfer map is represented by a mixed variable generating function, obtained by interpolation between the maps for a set of reference configurations: use of mixed variable generating functions ensures the symplecticity of the map. An optimization routine uses the interpolation technique to look for a lattice defined by four constraints on the time of flight at different beam energies. This provides a way to determine the lattice configuration required to produce the desired dynamical characteristics. These tools are benchmarked against data from the recent EMMA commissioning.

Conservative Algorithm for an Adaptive Change of Resolution in Mixed Atomistic / Coarse-Grained Multiscale Simulations

Understanding complex materials often requires investigating multiple, tightly coupled time and length scales. Neither atomistic nor coarse-grained simulations are often able to adequately capture all the relevant scales. To combine the efficiency of coarse-grained models with the accuracy of atomistic models for systems that require atomistic resolution only locally, for example at an interface, mixed-resolution models have been developed. These models use a coarse-grained description for the part of the system distant from an active site and atomistic description for the active site and its direct environment. Since the active zone may diffuse during a simulation, the simulation algorithm needs to permit an on-the-fly reclassication of atoms as they transition between the high- and low-resolution regimes. In this paper, we derive a conservative Hamiltonian and present an explicit symplectic integrator for mixed-resolution systems that allows for such a change in resolution of selected groups of atoms during a MD simulation.

Vlasov and Drift Kinetic Simulation Methods Based on the Symplectic Integrator

Vlasov and drift kinetic simulation methods based on the symplectic integrators are benchmarked for test problems on the linear and nonlinear Landau dampings and the Kelvin‐Helmholtz (K‐H) instability. The explicit symplectic integrator for the separable Hamiltonian leads to straightforward generalization of the splitting scheme for the Vlasov‐Poisson system. The Nth‐order version improves the total energy conservation decreasing the error as ∝Δt N where Δt denotes the time step size. An Eulerian drift kinetic simulation scheme derived from the implicit symplectic integrator for the nonseparable Hamiltonian exactly satisfies the conservation of the energy and the enstrophy in the K‐H instability, and results in successful application to the plasma echo.

Explicit Adaptive Symplectic (Easy) Integrators: A Scaling Invariant Generalisation of the Levi-Civitaand KS Regularisations

We present a generalisation of the Levi-Civita and Kustaanheimo-Stiefel regularisation. This allows the use of more general time rescalings. In particular, it is possible to find a regularisation which removes the singularity of the equations and preserves scaling invariance. In addition, these equations can, in certain cases, be integrated with explicit symplectic Runge-Kutta-Nystrom methods. The combination of both techniques gives an explicit adaptive symplectic (EASY) integrator. We apply those methods to some perturbations of the Kepler problem and illustrate, by means of some numerical examples, when scaling invariant regularisations are more efficient that the LC/KS regularisation.

Long-term evolution of disposal orbits beyond the geostationary ring

The effect of long term gravitational influence of the Earth, Moon and Sun, together with the Sun radiation pressure was studied for a randomly generated population of 10000 orbits with the perigees taken from the interval 50–3000 km above the geostationary ring. The evolution of each orbit on a time span of 200 years was studied by means of an explicit sixth-order symplectic integrator of the Wisdom-Holman type. 389 orbits crossed the ‘security zone’ of 35 km above the geostationary orbit, most of them during the first 10 years. The simulation indicates a strong correlation between the semi-major axes and the eccentricities of ‘unsafe orbits’. Some less significant effects in the distribution of elements for such orbits are also discussed.

Explicit Symplectic Integrator for Rotating Satellites

An explicit symplectic integrator is constructed for the problem of a rotating planetary satellite on a Keplerian orbit. The spin vector is fixed perpendicularly to the orbital plane. The integrator is constructed according to the Wisdom-Holman approach: the Hamiltonian is separated in two parts so that one of them is multiplied by a small parameter. The parameter depends on the satellite’s shape or the eccentricity of its orbit. The leading part of the Hamiltonian for small eccentricity orbits is similar to the simple pendulum and hence integrable; the perturbation does not depend on angular momentum which implies a trivial ‘kick’ solution. In spite of the necessity to evaluate elliptic function at each step, the explicit symplectic integrator proves to be quite efficient.