Particle Phase Space Research Articles

Analysis of the nonlinear behavior of shear-Alfvén modes in tokamaks based on Hamiltonian mapping techniques

We present a series of numerical simulation experiments set up to illustrate the fundamental physics processes underlying the nonlinear dynamics of Alfvénic modes resonantly excited by energetic particles in tokamak plasmas and of the ensuing energetic particle transports. These phenomena are investigated by following the evolution of a test particle population in the electromagnetic fields computed in self-consistent MHD-particle simulation performed by the HMGC code. Hamiltonian mapping techniques are used to extract and illustrate several features of wave-particle dynamics. The universal structure of resonant particle phase space near an isolated resonance is recovered and analyzed, showing that bounded orbits and untrapped trajectories, divided by the instantaneous separatrix, form phase space zonal structures, whose characteristic non-adiabatic evolution time is the same as the nonlinear time of the underlying fluctuations. Bounded orbits correspond to a net outward resonant particle flux, which produces a flattening and/or gradient inversion of the fast ion density profile around the peak of the linear wave-particle resonance. The connection of this phenomenon to the mode saturation is analyzed with reference to two different cases: a Toroidal Alfvén eigenmode in a low shear magnetic equilibrium and a weakly unstable energetic particle mode for stronger magnetic shear. It is shown that, in the former case, saturation is reached because of radial decoupling (resonant particle redistribution matching the mode radial width) and is characterized by a weak dependence of the mode amplitude on the growth rate. In the latter case, saturation is due to resonance detuning (resonant particle redistribution matching the resonance width) with a stronger dependence of the mode amplitude on the growth rate.

Soliton dynamics in symmetric and non-symmetric complex potentials

Soliton propagation dynamics under the presence of a complex potential are investigated. A large variety of qualitatively different potentials, including periodic, semi-infinite periodic and localized potentials, is considered. Cases of both symmetric and non-symmetric potentials are studied in terms of their effect on soliton dynamics. The rich set of dynamical features of soliton propagation includes dynamical trapping, periodic and non-periodic soliton mass variation and non-reciprocal scattering dynamics. These features are systematically investigated with the utilization of an effective particle phase space approach which is shown in remarkable agreement with direct numerical simulations. The generality of the results enables the consideration of potential applications where the inhomogeneity of the gain and loss is appropriately engineered in order to provide desirable soliton dynamics.

Application of the phase space action principle to finite-size particle plasma simulations in the drift-kinetic approximation

We formulate a finite-size particle numerical model of strongly magnetized plasmas in the drift-kinetic approximation. We use the phase space action as an alternative to previous variational formulations based on Low’s Lagrangian or on a Hamiltonian with a non-canonical Poisson bracket. The useful property of this variational principle is that it allows independent transformations of particle coordinates and velocities, i.e., transformations in particle phase space. With such transformations, a finite degree-of-freedom drift-kinetic action is obtained through time-averaging of the finite degree-of-freedom fully-kinetic action. Variation of the drift-kinetic Lagrangian density leads to a self-consistent, macro-particles and fields numerical model. Since the computational particles utilize only guiding center coordinates and velocities, there is a large computational advantage in the time integration part of the algorithm. Numerical comparison between the time-averaged fully-kinetic and drift-kinetic charge and current, deposited on a computational grid, offers insight into the range of validity of the model. Being based on a variational principle, the algorithm respects the energy conserving property of the underlying continuous system. The development in this paper serves to further emphasize the advantages of using variational approaches in plasma particle simulations.

Theoretical and numerical analysis of radiation belt electron precipitation by coherent whistler mode waves

AbstractThe interaction between coherent whistler mode waves and energetic radiation belt electrons can result in pitch angle scattering of electrons into the bounce loss cone and subsequent precipitation. In studying the effects of VLF transmitter signals on particle precipitation, past modeling efforts have focused on the computation of diffusion coefficients for a Fokker‐Planck model. In contrast, to capture the nonlinear effects of large‐amplitude coherent waves, we evaluate particle precipitation using a Vlasov‐Liouville (VL) model which computes the phase space particle distribution function directly using a characteristic‐based solution of the Vlasov equation. Previous work has shown that in the case of large‐amplitude coherent waves, phase trapping can significantly perturb resonant particles from their adiabatic paths. We evaluate the importance of phase trapping over a range of wave amplitudes (up to 200 pT); the percentage of particles that precipitate after being phase trapped is computed over a phase space grid in the loss cone. The results demonstrate that phase trapping contributes significantly to precipitation when a large‐amplitude wave (>100 pT) is present. Additionally, linear theory can be valid over a broad range of amplitudes and the relative accuracy of linear theory in calculating the precipitated flux depends strongly on the initial particle distribution. Additionally, we demonstrate the ability of the VL model to calculate the time evolution of the precipitated flux due to short‐duration whistler mode pulses. The physical parameters used in this study are typical of those associated with the Siple Station wave injection experiment.

Prevalence of the sling effect for enhancing collision rates in turbulent suspensions

AbstractTurbulence facilitates collisions between particles suspended in a turbulent flow. Two effects have been proposed that can enhance the collision rate at high turbulence intensities: ‘preferential concentration’ (a clustering phenomenon) and the ‘sling effect’ (arising from the formation of caustic folds in the phase space of the suspended particles). We have determined numerically the collision rate of small heavy particles as a function of their size and densities. The dependence on particle densities allows us to quantify the contribution of the sling effect to the collision rate. Our results demonstrate that the sling effect provides the dominant mechanism to the enhancement of the collision rate of particles, when inertia becomes significant.

Clustering in the phase space of dark matter haloes – I. Results from the Aquarius simulations

We present a novel perspective on the clustering of dark matter in phase space by defining the particle phase space average density ($P^2SAD$) as a two-dimensional extension of the two-point correlation function averaged within a certain volume in phase space. This statistics is a sensitive measure of small scale (sub-)structure of dark matter haloes. By analysing the structure of $P^2SAD$ in Milky-Way-size haloes using the Aquarius simulations, we find it to be nearly universal at small scales, i.e. small separations in phase space, where substructures dominate. This remarkable universality occurs across time and in regions of substantially different ambient densities (by nearly four orders of magnitude), with typical variations in $P^2SAD$ of a factor of a few. The maximum variations occur in regions where substructures have been strongly disrupted. The universality is also preserved across haloes of similar mass but diverse mass accretion histories and subhalo distributions. The universality is also broken at large scales, where the smooth dark matter distribution in the halo dominates. Although at small scales the structure of $P^2SAD$ is roughly described by a subhalo model, we argue that the simulation data is better fitted by a family of superellipse contours. This functional shape is inspired by a model that extends the stable clustering hypothesis into phase space. In a companion paper, we refine this model and show its advantages as a method to obtain predictions for non-gravitational signatures of dark matter.

Clustering in the phase space of dark matter haloes – II. Stable clustering and dark matter annihilation

We present a model for the structure of the particle phase space average density ($P^2SAD$) in galactic haloes, introduced recently as a novel measure of the clustering of dark matter. Our model is based on the stable clustering hypothesis in phase space, the spherical collapse model, and tidal disruption of substructures, which is calibrated against the Aquarius simulations. Using this model, we can predict the behaviour of $P^2SAD$ in the numerically unresolved regime, down to the decoupling mass limit of generic WIMP models. This prediction can be used to estimate signals sensitive to the small scale structure of dark matter. For example, the dark matter annihilation rate can be estimated for arbitrary velocity-dependent cross sections in a convenient way using a limit of $P^2SAD$ to zero separation in physical space. We illustrate our method by computing the global and local subhalo annihilation boost to that of the smooth dark matter distribution in a Milky-Way-size halo. Two cases are considered, one where the cross section is velocity independent and one that approximates Sommerfeld-enhanced models. We find that the global boost is $\sim10-30$, which is at the low end of current estimates (weakening expectations of large extragalactic signals), while the boost at the solar radius is below the percent level. We make our code to compute $P^2SAD$ publicly available, which can be used to estimate various observables that probe the nanostructure of dark matter haloes.

Multiple collisions in turbulent flows

In turbulent suspensions, collision rates determine how rapidly particles coalesce or react with each other. To determine the collision rate, many numerical studies rely on the ghost collision approximation (GCA), which simply records how often pairs of point particles come within a threshold distance. In many applications, the suspended particles stick (or in the case of liquid droplets, coalesce) upon collision, and it is the frequency of first contact which is of interest. If a pair of "ghost" particles undergoes multiple collisions, the GCA may overestimate the true collision rate. Here, using fully resolved direct numerical simulations of turbulent flows at moderate Reynolds number (Re(λ)=130), we investigate the prevalence and properties of multiple collisions. We find the probability P(N(c)) for a given pair of ghost particles to collide N(c) times to be of the form P(N(c))=βα(N(c)) for N(c)>1, where α and β are coefficients which depend upon the particle inertia. We also investigate the statistics of the times that ghost particles remain in contact. We show that the probability density function of the contact time is different for the first collision. The difference is explained by the effect of caustics in the phase space of the suspended particles. We demonstrate that, as a result of multiple collisions, the GCA leads to a small, but systematic overestimate of the collision rate, which is of the order of ∼15% when the particle inertia is small, and slowly decreases when inertia increases.

Unification of Quantum and Gravity by Non Classical Information Entropy Space

A quantum entropy space is suggested as the fundamental arena describing the quantum effects. In the quantum regime the entropy is expressed as the superposition of many different Boltzmann entropies that span the space of the entropies before any measure. When a measure is performed the quantum entropy collapses to one component. A suggestive reading of the relational interpretation of quantum mechanics and of Bohm’s quantum potential in terms of the quantum entropy are provided. The space associated with the quantum entropy determines a distortion in the classical space of position, which appears as a Weyl-like gauge potential connected with Fisher information. This Weyl-like gauge potential produces a deformation of the moments which changes the classical action in such a way that Bohm’s quantum potential emerges as consequence of the non classical definition of entropy, in a non-Euclidean information space under the constraint of a minimum condition of Fisher information (Fisher Bohm- entropy). Finally, the possible quantum relativistic extensions of the theory and the connections with the problem of quantum gravity are investigated. The non classical thermodynamic approach to quantum phenomena changes the geometry of the particle phase space. In the light of the representation of gravity in ordinary phase space by torsion in the flat space (Teleparallel gravity), the change of geometry in the phase space introduces quantum phenomena in a natural way. This gives a new force to F. Shojai’s and A. Shojai’s theory where the geometry of space-time is highly coupled with a quantum potential whose origin is not the Schrödinger equation but the non classical entropy of a system of many particles that together change the geometry of the phase space of the positions (entanglement). In this way the non classical thermodynamic changes the classical geodetic as a consequence of the quantum phenomena and quantum and gravity are unified. Quantum affects geometry of multidimensional phase space and gravity changes in any point the torsion in the ordinary four-dimensional Lorenz space-time metric.

Diagnosis of physical parameters of fast particles in high power fusion plasmas with high resolution neutron and gamma-ray spectroscopy

High resolution neutron emission spectroscopy (NES) and gamma-ray spectroscopy (GRS) measurements of fast ions in high power fusion plasmas are reviewed. NES is a well established diagnostics of the velocity distribution of fast fuel ions and was recently used to investigate the interaction of energetic ions with MHD instabilities. High energy resolution GRS on fusion plasmas is a more recent application and was shown to provide information on the distribution function of fast minority ions accelerated by ICRH, such as 4He and 3He. Starting from measurements on today's high power D plasmas, fast ion measurements with NES and GRS in a DT burning plasma of next step tokamaks, such as ITER, are discussed. The enhanced neutron and gamma-ray fluxes expected on ITER will allow for time-resolved measurements of the fast fuel and minority ion dynamics in the ms time scale. The intensity of the alpha knock-on component in the 14 MeV neutron spectrum and of the 4.44 and 3.21 MeV gamma-ray peaks from the 9Be(α,nγ)12C reaction is studied as a diagnostics for the α particle slowing down distribution in a DT plasma. The results show that the two techniques are sensitive to different regions of the α particle phase space and thus provide complementary information.

Topology of collisionless relaxation.

Using extensive molecular dynamics simulations we explore the fine-grained phase space structure of systems with long-range interactions. We find that if the initial phase space particle distribution has no holes, the final stationary distribution will also contain a compact simply connected region. The microscopic holes created by the filamentation of the initial distribution function are always restricted to the outer regions of the phase space. In general, for complex multilevel distributions it is very difficult to a priori predict the final stationary state without solving the full dynamical evolution. However, we show that, for multilevel initial distributions satisfying a generalized virial condition, it is possible to predict the particle distribution in the final stationary state using Casimir invariants of the Vlasov dynamics.

Nonlinear dynamics of beta-induced Alfvén eigenmode in tokamak

The beta-induced Alfvén eigenmode (BAE) excited by energetic particles in toroidal plasmas is studied in the global gyrokinetic simulations. It is found that the nonlinear BAE dynamics depends on the deviation from the marginality. In the strongly driven case, the mode exhibits a bursting state with fast and repetitive chirping. The nonlinear saturation is determined by the thermal ion nonlinearity and has no clear dependence on the linear growth rate. In the weakly driven case, the mode reaches a nearly steady state with small frequency chirping. The nonlinear dynamics is dominated by the energetic particle nonlinearity. In both cases, the nonlinear intensity oscillation and frequency chirping are correlated with the evolution of the coherent structures in the energetic particle phase space. Due to the radial variation of the mode amplitude and the radially asymmetric guiding center dynamics, the wave-particle interaction in the toroidal geometry is much more complex than the conventional one-dimensional wave-particle interaction paradigm.

Criticality and Strong Intermittency in the Lorentz Channel

We demonstrate the emergence of criticality due to power-law cross correlations in an ensemble of noninteracting particles propagating in an infinite Lorentz channel. The origin of these interparticle long-range correlations is the intermittent dynamics associated with the ballistic corridors in the single particle phase space. This behavior persists dynamically, even in the presence of external driving, provided that the billiard's horizon becomes infinite at certain times. For the driven system, we show that Fermi acceleration permits the synchronization of the particle motion with the periodic appearance of the ballistic corridors. The particle ensemble then acquires characteristics of self-organization as the weight of the phase space regions leading to critical behavior increases with time.

SU-E-T-469: The Effects of Patient Anatomy and Parallel Magnetic Fields on Beamlet Dose Distributions.

Beamlets are generated in a patient geometry in the presence of a magnetic field to investigate the effects of tissue density and magnetic field on beamlet dose distributions, which is important for the optimization of photon fluence to be delivered by a linac-MR system. 50×50 mm2 fields were placed with isocenter in the middle of a patient's right lung. Each treatment field was decomposed into 100 beamlets (each 5×5 mm2 ). BEAMnrc scored the particle phase space at 100.2 cm from the source in the linac-MR geometry (isocentre at 126 cm) with parallel magnetic fields of 0, 0.56, and 3T. DOSXYZnrc was modified to score the energy deposited by particles from this phase space as a function of the beamlet the particle passed through. The calculation volume of 70×46×64 voxels encompassed the patient with a voxel size of 3×3×3 mm3 . Each beamlet was normalized to the dose calculated to a 3×3×3 mm3 voxel with isocenter at 5cm depth in a flat water tank without a magnetic field. Beamlet files were calculated on Western Canada's high performance computing cluster (Westgrid) using 100 processors, enabling simulation of 109 histories in less than 3 hours. The resulting files, which contained 3D dose distributions for all 100 beamlets, were 81 MB per field. The Monte Carlo uncertainty was also stored. The gyroradii for 1 MeV electron traversing field lines at 20 degrees are 2.9mm and 0.5mm for 0.56 and 3T fields respectively. The 0.56T parallel magnetic field has a small effect compared to the distortion of the beamlet introduced by the presence of lung. The effect of tissue heterogeneities is more significant than the effect of a 0.56T parallel magnetic field. A 3T field refocuses the dose in lung to the beamlet path and significantly reduces the lateral electron scatter.

Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields

We introduce natural differential geometric structures underlying thePoisson-Vlasov equations in momentum variables. First, we decompose thespace of all vector fields over particle phase space into a semi-directproduct algebra of Hamiltonian vector fields and its complement. The latteris related to dual space of the Lie algebra. We identify generators ofhomotheties as dynamically irrelevant vector fields in the complement. Liealgebra of Hamiltonian vector fields is isomorphic to the space of allLagrangian submanifolds with respect to Tulczyjew symplectic structure. Thisis obtained as tangent space at the identity of the group of canonicaldiffeomorphisms represented as space of sections of a trivial bundle. Weobtain the momentum-Vlasov equations as vertical equivalence, orrepresentative, of complete cotangent lift of Hamiltonian vector fieldgenerating particle motion. Vertical representatives can be described byholonomic lift from a Whitney product to a Tulczyjew symplectic space. Weshow that vertical representatives of complete cotangent lifts form anintegrable subbundle of this Tulczyjew space. We exhibit dynamical relationsbetween Lie algebras of Hamiltonian vector fields and of contact vectorfields, in particular; infinitesimal quantomorphisms on quantization bundle.Gauge symmetries of particle motion are extended to tensorial objectsincluding complete lift of particle motion. Poisson equation is thenobtained as zero value of momentum map for the Hamiltonian action of gaugesymmetries for kinematical description.

Anomalous dimension in three-dimensional semiclassical gravity

The description of the phase space of relativistic particles coupled to three-dimensional Einstein gravity requires momenta which are coordinates on a group manifold rather than on ordinary Minkowski space. The corresponding field theory turns out to be a non-commutative field theory on configuration space and a group field theory on momentum space. Using basic non-commutative Fourier transform tools we introduce the notion of non-commutative heat-kernel associated with the Laplacian on the non-commutative configuration space. We show that the spectral dimension associated to the non-commutative heat kernel varies with the scale reaching a non-integer value smaller than three for Planckian diffusion scales.

Courant-Snyder invariant density screening method for emittance analysis

Emittance is an important characteristic of describing charged particle beams. In hadron accelerators, we often meet irregular beam distributions that are not appropriately described by a single rms emittance or 95% emittance or total emittance. In this paper, it is pointed out that in many cases a beam halo should be described with very different Courant-Snyder parameters from the ones used for the beam core. A new method — the Courant-Snyder invariant density screening method — is introduced for analyzing emittance data clearly and accurately. The method treats the emittance data from both measurements and numerical simulations. The method uses the statistical distribution of the beam around each particle in phase space to mark its local density parameter, and then uses the density distribution to calculate the beam parameters such as the Courant-Snyder parameters and emittance for different beam boundary definitions. The method has been used in the calculations for beams from different sources, and shows its advantages over other methods. An application code based on the method including the graphic interface has also been designed.

Energetic particle phase space densities at Saturn: Cassini observations and interpretations

[1] Saturn's magnetosphere has been studied extensively by the Cassini spacecraft during the last 6 years. We present mission-averaged energetic proton and electron measurements obtained by the MIMI/LEMMS instrument onboard Cassini in an energy range from several 10 keV to several 10 MeV separated by equatorial pitch angle. We discuss the resulting radial profiles and energy spectra. The measured intensities are converted to phase space densities. The distribution of energetic particles is governed by a large variety of processes. For instance, moons absorb energetic particles, creating macrosignatures or microsignatures. We have found that the moon Rhea is partly responsible for a change in gradient of electron phase space densities. We show that, in contrast to larger distances, the particle distribution for L < 8 is not driven by radial diffusion alone. There, the particle profiles are significantly modified due to Saturn's Neutral Torus, plasma environment, E ring, injection events, and cosmic ray albedo neutron decay. Large parts of our analysis are focused near L = 7. There, protons are lost within the Neutral Torus and not the E ring. For electrons, we find that these two losses are of comparable rate but have discovered that neither process is the dominant driver of loss. We point out that intensity measured by a energy channel, such as in a particle instrument, can actually increase in the region of ring and torus instead of decrease. The importance of injection events is shown to be at least of similar importance as radial diffusion.

A gyrokinetic collision operator for magnetized Lorentz plasmas

A gyrocenter collision operator for magnetized Lorentz plasmas is derived using the Fokker–Plank method. The gyrocenter collision operator consists of drift and diffusion terms in the gyrocenter coordinates, including the diffusion of the gyrocenter, which does not exist for the collision operator in the particle phase space coordinates. The gyrocenter collision operator also depends on the transverse electric field explicitly, which is crucial for the correct treatment of collisional effects and transport in the gyrocenter coordinates. The gyrocenter collision operator derived is applied to calculate the particle and heat transport fluxes in a magnetized Lorentz plasma with an electric field. The particle and heat transport fluxes calculated from our gyrocenter collision operator agree exactly with the classical Braginskii’s result [S. I. Braginskii, Reviews of Plasma Physics (Consultants Bureau, New York, 1965), Vol. 1, p. 205: P. Helander and D. J. Sigmar, Collisional Transport in Magnetized Plasmas (Cambridge University, Cambridge, 2002), p. 65], which validates the correctness of our collision operator. To calculate the transport fluxes correctly, it is necessary to apply the pullback transformation associated with gyrocenter coordinate transformation in the presence of collisions, which also serves as a practical algorithm for evaluating collisional particle and heat transport fluxes in the gyrocenter coordinates.

Effects of velocity-dependent dark matter annihilation on the energy spectrum of the extragalactic gamma-ray background

We calculate the effects of velocity-dependent dark matter annihilation cross sections on the intensity of the extragalactic gamma-ray background. Our formalism does not assume a locally thermal distribution of dark matter particles in phase space, and is valid for arbitrary velocity-dependent annihilation. As concrete examples, we calculate the effects of p-wave annihilation (with the $v$-weighted cross section of $\sigma v=a+bv^2$) on the mean intensity of extragalactic gamma rays produced in cosmological dark matter halos. This velocity variation makes the shape of the energy spectrum harder, but this change in the shape is too small to see unless $b/a\agt 10^6$. While we find no such models in the parameter space of the Minimal Supersymmetric Standard Model (MSSM), we show that it is possible to find $b/a\agt 10^6$ in the extension MSSM$\otimes U(1)_{B-L}$. However, we find that the most dominant effect of the p-wave annihilation is the suppression of the amplitude of the gamma-ray background. A non-zero $b$ at the dark matter freeze-out epoch requires a smaller value of $a$ in order for the relic density constraint to be satisfied, suppressing the amplitude by a factor as low as $10^{-6}$ for a thermal relic. Non-thermal relics will have weaker amplitude suppression. As another velocity-dependent effect, we calculate the spectrum for s-wave annihilation into fermions enhanced by the attractive Sommerfeld effect. Resonances associated with this effect result in significantly enhanced intensities, with a slightly softer energy spectrum.