Initial Phase Space Research Articles

Assessment of the runaway electron load distribution in ITER during 3D MHD induced beam termination

In ITER, disruption-born runaway electrons (REs), unless mitigated, are expected to form a several Mega-Ampere beam that ultimately intercepts the first wall leading to melting of the plasma facing components. Developing a successful mitigation strategy therefore requires modeling that takes into account the coupling between REs and the MHD during the formation and termination of the beam, along with estimates for the wall loads that can be compared to design values. Using the JOREK code, this work aims to provide the latter by presenting a novel model for collisions between REs and the wall and applying it to a beam termination scenario in ITER. To this end, the transport of REs is modeled by tracing particles in the fields calculated by preceding simulations of the disruption event where REs were treated in the fluid picture and self-consistently coupled to the MHD. The resulting heat loads are found to be highly localized in both the poloidal and toroidal directions, with 3D features also playing an important role in the appearance of hot spots. Peak loads were on the order of 107−108Jm2 . The load distribution was found to only be weakly sensitive to the initial phase space distribution of the REs when decoupled from the fields, while modifying properties of the underlying MHD yielded notable differences in wetted area and peak loads, in particular for cases with higher resistivity.

Contributions to Loss Across the Magnetopause During an Electron Dropout Event.

Dropout events are dramatic decreases in radiation belt electron populations that can occur in as little as 30 minutes. Loss to magnetopause due to a combination of magnetopause shadowing and outward radial transport plays a significant role in these events. We examine the dropout of relativistic electron populations during the October 2012 geomagnetic storm using simulated electron phase space density, evaluating the contribution of different processes to losses across the magnetopause. We compare loss contribution from outward transport calculated using a standard empirical radial diffusion model that assumes a dipolar geomagnetic field to an event-specific radial diffusion model evaluated with a non-dipolar geomagnetic field. We additionally evaluate the contribution of Shabansky type 1 particles, which bounce along magnetic field lines with local equatorial maxima, to the loss calculated during this event. We find that the empirical radial diffusion model with a dipolar background field underestimates the contribution of radial diffusion to this dropout event by up to 10% when compared to the event-specific, non-dipolar radial diffusion model. We additionally find that including Shabansky type 1 particles in the initial electron phase space density, that is, allowing some magnetic field lines distorted from the typical single-minima configuration in drift shell construction, increases the calculated loss by an average of 0.75%. This shows that the treatment of the geomagnetic field significantly impacts the calculation of electron losses to the magnetopause during dropout events, with the non-dipolar treatment of radial diffusion being essential to accurately quantify the loss of outer radiation belt populations.

Holographic Kolmogorov-Sinai entropy and the quantum Lyapunov spectrum

In classical chaotic systems the entropy, averaged over initial phase space distributions, follows a universal behavior. While approaching thermal equilibrium it passes through a stage where it grows linearly, while the growth rate, the Kolmogorov-Sinai entropy (rate), is given by the sum over all positive Lyapunov exponents. A natural question is whether a similar relation is valid for quantum systems. We argue that the Maldacena-Shenker-Stanford bound on quantum Lyapunov exponents implies that the upper bound on the growth rate of the entropy, averaged over states in Hilbert space that evolve towards a thermal state with temperature T, should be given by πT times the thermal state’s von Neumann entropy. Strongly coupled, large N theories with black hole duals should saturate the bound. To test this we study a large number of isotropization processes of random, spatially homogeneous, far from equilibrium initial states in large N, mathcal{N} = 4 Super Yang Mills theory at strong coupling and compute the ensemble averaged growth rate of the dual black hole’s apparent horizon area. We find both an analogous behavior as in classical chaotic systems and numerical evidence that the conjectured bound on averaged entropy growth is saturated granted that the Lyapunov exponents are degenerate and given by λi = ±2πT. This fits to the behavior of classical systems with plus/minus symmetric Lyapunov spectra, a symmetry which implies the validity of Liouville’s theorem.

MHD‐Test Particles Simulations of Moderate CME and CIR‐Driven Geomagnetic Storms at Solar Minimum

AbstractAs part of the Whole Heliosphere and Planetary Interactions initiative, contrasting drivers of radiation belt electron response at solar minimum have been investigated with MHD‐test particle simulations for the May 13–14, 2019 Coronal Mass Ejection (CME)‐shock event and the August 30–September 3, 2019 high speed solar wind interval. Both solar wind drivers produced moderate geomagnetic storms characterized by a minimum Dst = −65 nT and −52 nT, respectively, with the August ‐ September event accompanied by prolonged substorm activity. The latter, with characteristic features of a Corotating Interaction Region (CIR)‐driven storm, produced the hardest relativistic electron spectrum observed by Van Allen Probes during the last two years of the mission, ending in October 2019. MHD simulations were performed using both the Lyon‐Fedder‐Mobarry global MHD code and recently developed GAMERA model coupled to the Rice Convection Model, run with measured L1 solar wind input for both events studied, and coupled with test particle simulations, including an initial trapped and injected population. Initial electron Phase Space Density (PSD) profiles used measurements from the Relativistic Electron Proton Telescope and MagEIS energetic particle instruments on Van Allen Probes for test particle weighting and updating of the injected population at apogee. Results were compared directly with measurements and found to reproduce magnetopause loss for the CME‐shock event and increased PSD for the CIR event. The two classes of events are contrasted for their impact on outer zone relativistic electrons near the end of Solar Cycle 24.

Efficient uncertainty quantification for Monte Carlo dose calculations using importance (re-)weighting

Objective. To present an efficient uncertainty quantification method for range and set-up errors in Monte Carlo (MC) dose calculations. Further, we show that uncertainty induced by interplay and other dynamic influences may be approximated using suitable error correlation models. Approach. We introduce an importance (re-)weighting method in MC history scoring to concurrently construct estimates for error scenarios, the expected dose and its variance from a single set of MC simulated particle histories. The approach relies on a multivariate Gaussian input and uncertainty model, which assigns probabilities to the initial phase space sample, enabling the use of different correlation models. Through modification of the phase space parameterization, accuracy can be traded between that of the uncertainty or the nominal dose estimate. Main results. The method was implemented using the MC code TOPAS and validated for proton intensity-modulated particle therapy (IMPT) with reference scenario estimates. We achieve accurate results for set-up uncertainties (γ 2 mm/2% ≥ 99.01% (E[ d ]), γ 2 mm/2% ≥ 98.04% (σ( d ))) and expectedly lower but still sufficient agreement for range uncertainties, which are approximated with uncertainty over the energy distribution. Here pass rates of 99.39% (E[ d ])/ 93.70% (σ( d )) (range errors) and 99.86% (E[ d ])/ 96.64% (σ( d )) (range and set-up errors) can be achieved. Initial evaluations on a water phantom, a prostate and a liver case from the public CORT dataset show that the CPU time decreases by more than an order of magnitude. Significance. The high precision and conformity of IMPT comes at the cost of susceptibility to treatment uncertainties in particle range and patient set-up. Yet, dose uncertainty quantification and mitigation, which is usually based on sampled error scenarios, becomes challenging when computing the dose with computationally expensive but accurate MC simulations. As the results indicate, the proposed method could reduce computational effort while also facilitating the use of high-dimensional uncertainty models.

Charged-particle chaotic dynamics in rotational discontinuities.

The interplanetary plasma is characterized by a high level of complexity over a broad range of spatial scales. Spacecraft have detected a large variety of embedded structures that have been identified as discontinuities in the magnetic field vector. They can be either generated within the solar corona and advected by the plasma flow or locally generated as a result of the turbulent cascade of the solar wind turbulence. Since magnetic field fluctuations and structures influence the energetic particle propagation, here we set up a numerical model to study the interaction between charged particles and an ideal magnetohydrodynamics rotational discontinuity. This interaction is strongly influenced by the model parameters, such as the rotation angle of the discontinuity, the orientation of the mean-field direction with respect to the normal to the discontinuity direction, the initial particle pitch angle, and the initial particle gyrophase. Numerical results clearly show that the motion of particles crossing the discontinuity is extremely complex and highly sensitive to the initial conditions of the system, with transitions to a chaotic behavior. We find that particles can be temporarily trapped in rotational discontinuity and that the trapping times have a nearly power-law distribution. Also, the separatrix in the initial conditions phase space between crossing and noncrossing trajectories has a fractal structure. Implications for energetic particle propagation in space plasmas are discussed.

Space-Charge Aberrations in Single-Shot Time-Resolved Transmission Electron Microscopy

In this paper, we discuss effects of space charge fields on imaging performance in a single shot time-resolved electron microscope (TEM). Using a Green's function perturbation method, we derive analytical estimates for the effects of space charge nonlinearity on the image formation process and the associated aberration coefficients. The dependence of these coefficients on the initial beam phase space distribution is elucidated. The results are validated by particle tracking simulations and provide fundamental scaling laws for the trade-off between temporal and spatial resolution in single-shot time-resolved TEM.

Excitation of Internally Driven ULF Waves by the Drift‐Bounce Resonance With Ring Current Ions Based on the Drift‐Kinetic Simulation

AbstractWe investigated the excitation mechanism and global distribution of internally driven Pc3‐Pc5 ultralow‐frequency waves in the inner magnetosphere using a kinetic and self‐consistent simulation (Geospace Environment Modeling System for Integrated Studies‐Ring Current model). These waves can be excited through the drift‐bounce resonance with ring current ions injected into the inner magnetosphere during substorms. Initial phase space density of ring current ions is the butterfly‐like distribution with the asymmetry in pitch angle direction. We find two kinds of internally driven ultralow‐frequency waves. First, the dynamic power spectra of the field fluctuations show the excitation of poloidal, toroidal, and compressional Pc5 waves. We find that the excited waves are fundamental modes propagating westward with the azimuthal wave number of m ~ − 50. These waves are excited by the drift resonance with ions having the energy of 50–150 keV. Second, we can reproduce the excitation of poloidal Pc3 waves in the dusk region with m ~ − 30, which are driven by the bounce resonance with ions having energies of 20–50 keV for the first time. We investigate the energy transfer between ions and waves and find that poloidal Pc5 and Pc3 waves are excited mainly by positive energy gradient of the phase space density, while we find the inward gradient of the phase space density at L > 6. The velocity distribution of the growth rate suggests that Pc5 waves are driven by ions with pitch angle of around 90°, while Pc3 waves are excited by ions at oblique pitch angle.

Thermalization of Local Observables in the $$\alpha $$-FPUT Chain

Most studies on the problem of equilibration of the Fermi-Pasta-Ulam-Tsingou (FPUT) system have focused on equipartition of energy being attained amongst the normal modes of the corresponding harmonic system. In the present work, we instead discuss the equilibration problem in terms of local variables, and consider initial conditions corresponding to spatially localized energy. We estimate the time-scales for equipartition of space localized degrees of freedom and find significant differences with the times scales observed for normal modes. Measuring thermalization in classical systems necessarily requires some averaging, and this could involve one over initial conditions or over time or spatial averaging. Here we consider averaging over initial conditions chosen from a narrow distribution in phase space. We examine in detail the effect of the width of the initial phase space distribution, and of integrability and chaos, on the time scales for thermalization. We show how thermalization properties of the system, quantified by its equilibration time, defined in this work, can be related to chaos, given by the maximal Lyapunov exponent. Somewhat surprisingly we also find that the ensemble averaging can lead to thermalization of the integrable Toda chain, though on much longer time scales.

Hyperbolic Spin Ruijsenaars-Schneider Model from Poisson Reduction

We derive a Hamiltonian structure for the N-particle hyperbolic spin Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double of a factorisable Lie group with another symplectic manifold that is a certain deformation of the standard canonical relations for Nl conjugate pairs of dynamical variables. We show that the model enjoys the Poisson-Lie symmetry of the spin group GLl(ℂ), which explains its superintegrability. Our results are obtained in the formalism of the classical r-matrix, and they are compatible with the recent findings on the different Hamiltonian structure of the model established in the framework of the quasi-Hamiltonian reduction applied to a quasi-Poisson manifold.

Isolated Stable Initial Perturbations of Unstable Equilibria of Some Mechanical Systems

The unstable equilibrium positions of four mechanical systems are considered. The following systems are chosen: the system with one degree of freedom (an inverted mathematical pendulum), the systems with two degrees of freedom (an inverted spherical pendulum), the system with a countable number of degrees of freedom (an elastic beam loaded by a compressive force greater than the Eulerian critical load), and a continual system (a gravitationally unstable two-layer system of incompressible inviscid liquids). For each of these cases, we find an isolated class of perturbations leading to an exponential tendency to the unstable equilibrium or to an exponential tendency to an oscillatory regime in the initial phase space. The conditions characterizing such a class of perturbations are similar for the four considered systems.

The Symplectic Camel and Poincaré Superrecurrence: Open Problems

Poincaré’s Recurrence Theorem implies that any isolated Hamiltonian system evolving in a bounded Universe returns infinitely many times arbitrarily close to its initial phase space configuration. We discuss this and related recurrence properties from the point of view of recent advances in symplectic topology which have not yet reached the Physics community. These properties are closely related to Emergent Quantum Mechanics since they belong to a twilight zone between classical (Hamiltonian) mechanics and its quantization.

Interactions between energetic electrons and realistic whistler mode waves in the Jovian magnetosphere

AbstractThe role of plasma waves in shaping the intense Jovian radiation belts is not well understood. In this study we use a realistic wave model based on an extensive survey from the Plasma Wave Investigation on the Galileo spacecraft to calculate the effect of pitch angle and energy diffusion on Jovian energetic electrons due to upper and lower band chorus. Two Earth‐based models, the Full Diffusion Code and the Versatile Electron Radiation Belt code, are adapted to the case of the Jovian magnetosphere and used to resolve the interaction between chorus and electrons at L = 10. We also present a study of the sensitivity to the latitudinal wave coverage and initial electron distribution. Our analysis shows that the contribution to the electron dynamics from upper band chorus is almost negligible compared to that from lower band chorus. For 100 keV electrons, we observe that diffusion leads to redistribution of particles toward lower pitch angles with some particle loss, which could indicate that radial diffusion or interchange instabilities are important. For energies above >500 keV, an initial electron distribution based on observations is only weakly affected by chorus waves. Ideally, we would require the initial electron phase space density before transport takes place to assess the importance of wave acceleration, but this is not available. It is clear from this study that the shape of the electron phase space density and the latitudinal extent of the waves are important for both electron acceleration and loss.

Cosmic neutrinos: A dispersive and nonlinear fluid

We present a description of cosmic neutrinos as a dispersive fluid. In this approach, the neutrino phase space is reduced to density and velocity fields alongside a scale-dependent sound speed. This sound speed depends on redshift, the initial neutrino phase space density and the cold dark matter gravitational potential. The latter is a new coupling between neutrinos and large scale structure not described by previous fluid approaches. We compute the sound speed in linear theory and find that it asymptotes to constants at small and large scales regardless of the gravitational potential. By comparing with neutrino N-body simulations, we measure the small scale sound speed and find it to be lower than linear theory predictions. This allows for an explanation of the discrepency between N-body and linear response predictions for the neutrino power spectrum: neutrinos are still driven predominantly by the cold dark matter, but the sound speed on small scales is not stable to perturbations and decreases. Finally, we present a calibrated model for the neutrino power spectrum that requires no additional integrations outside of standard Boltzmann codes.

Phase space manipulations for optimization of an inverse free electron laser

The Inverse Free Electron Laser (IFEL) has developed into a mature advanced accelerator with the ability to produce mono energetic high brightness beams accelerated at gradients beyond those of conventional RF technology. By copropagating a relativistic electron beam and a high power laser in a strongly tapered undulator, resonant energy exchange can be maintained over large distances. To optimize the number of electrons accelerated in an IFEL, one must prepare the initial electron beam phase space such that a majority of particles are injected at the accelerating phase of the ponderomotive potential. We present a scheme for optimization of the Rubicon IFEL accelerator trapping by preparing the initial electron beam phase space. Utilizing two cascaded modulator-chicane pre-bunchers simulations predict 95% capture of the injected electron beam.

Deconstructing field-induced ketene isomerization through Lagrangian descriptors.

The time-dependent geometrical separatrices governing state transitions in field-induced ketene isomerization are constructed using the method of Lagrangian descriptors. We obtain the stable and unstable manifolds of time-varying transition states as dynamic phase space objects governing configurational changes when the ketene molecule is subjected to an oscillating electric field. The dynamics of the isomerization reaction are modeled through classical trajectory studies on the Gezelter-Miller potential energy surface and an approximate dipole moment model which is coupled to a time-dependent electric field. We obtain a representation of the reaction geometry, over varying field strengths and oscillation frequencies, by partitioning an initial phase space into basins labeled according to which product state is reached at a given time. The borders between these basins are in agreement with those obtained using Lagrangian descriptors, even in regimes exhibiting chaotic dynamics. Major outcomes of this work are: validation and extension of a transition state theory framework built from Lagrangian descriptors, elaboration of the applicability for this theory to periodically- and aperiodically-driven molecular systems, and prediction of regimes in which isomerization of ketene and its derivatives may be controlled using an external field.

Apparent topologically forbidden interchange of energy surfaces under slow variation of a Hamiltonian.

In this paper we consider the motion of point particles in a particular type of one-degree-of-freedom, slowly changing, temporally periodic Hamiltonian. Through most of the time cycle, the particles conserve their action, but when a separatrix is approached and crossed, the conservation of action breaks down, as shown in previous theoretical studies. These crossings have the effect that the numerical solution shows an apparent contradiction. Specifically we consider two initial constant energy phase space curves H=E(A) and H=E(B) at time t=0, where H is the Hamiltonian and E(A) and E(B) are the two initial energies. The curve H=E(A) encircles the curve H=E(B). We then sprinkle many initial conditions (particles) on these curves and numerically follow their orbits from t=0 forward in time by one cycle period. At the end of the cycle the vast majority of points initially on the curves H=E(A) and H=E(B) now appear to lie on two new constant energy curves H=E(A)' and H=E(B)', where the B' curve now encircles the A' curve (as opposed to the initial case where the A curve encircles the B curve). Due to the uniqueness of Hamilton dynamics, curves evolved under the dynamics cannot cross each other. Thus the apparent curves H=E(A)' and H=E(B)' must be only approximate representations of the true situation that respects the topological exclusion of curve crossing. In this paper we resolve this apparent paradox and study its consequences. For this purpose we introduce a "robust" numerical simulation technique for studying the complex time evolution of a phase space curve in a Hamiltonian system. We also consider how a very tiny amount of friction can have a major consequence, as well as what happens when a very large number of cycles is followed. We also discuss how this phenomenon might extend to chaotic motion in higher dimensional Hamiltonian systems.

Suppression of the emittance growth induced by coherent synchrotron radiation in triple-bend achromats**Supported by National Natural Science Foundation of China (11475202, 11405187) and Youth Innovation Promotion Association of Chinese Academy of Sciences (2015009)

The coherent synchrotron radiation (CSR) effect in a bending path plays an important role in transverse emittance dilution in high-brightness light sources and linear colliders, where the electron beams are of short bunch length and high peak current. Suppression of the emittance growth induced by CSR is critical to preserve the beam quality and help improve the machine performance. It has been shown that the CSR effect in a double-bend achromat (DBA) can be analyzed with the two-dimensional point-kick analysis method. In this paper, this method is applied to analyze the CSR effect in a triple-bend achromat (TBA) with symmetric layout, which is commonly used in the optics designs of energy recovery linacs (ERLs). A condition of cancelling the CSR linear effect in such a TBA is obtained, and is verified through numerical simulations. It is demonstrated that emittance preservation can be achieved with this condition, and to a large extent, has a high tolerance to the fluctuation of the initial transverse phase space distribution of the beam.

A scheme design of collimator for gantry in proton therapy facility

In proton radiotherapy, a round dose profile at the isocenter is beneficial to the treatment planning in the active scanning mode. Based on the theory of the round-beam method, a collimator was designed to be placed at the entrance of the gantry. A simulation of Monte Carlo method was done to validate the design scheme. With the constraining function of the collimator, the dose profile at the isocenter would maintain a round shape despite of an initial asymmetric phase space in extraction of synchrotron. Furthermore, the shape of dose profile is irrelevant to the rotating angle of the gantry. The mechanical error caused by the axis deviation of the gantry could also be partly eliminated with the collimator.

An approximation method for dispersive wave propagation

A phase space approximation method that extends a previous single-mode approximation [Loughlin, Cohen, J. Acoust. Soc. Am. 118, 1268 (2005)] for linear dispersive wave propagation is developed. We show that each mode is governed by a Schrodinger-type equation, where the corresponding Hamiltonian operator is non-Hermitian if the dispersion relation is complex, for which case there is absorption. The propagated wave is obtained by evolving each mode according to its respective Schrodinger equation. We show how to obtain the initial modes from the initial conditions on the wave. We then formulate the propagation problem in phase space and obtain the exact equation of motion for the phase space function. We also obtain an approximate solution for the phase space evolution of the wave, which involves a simple substitution into the initial phase space function. Examples are given for a parallel plate wave guide and the beam equation. [Work supported by ONR, code 321US.]