Phase Space Boundary Research Articles

Plasma kinetic simulation in Eulerian approach fully respecting f≥0 requirement and total particle number conservation

Abstract Plasma kinetic simulation in Eulerian approach (PKSiEA) can yield a dynamic description on plasmas and hence is a popular method in theoretical plasmas physics. Its mathematical model consists of 8 members: 5 partial differential equations (PDEs), a phase space boundary condition and an initial condition of the distribution function (PDF) f, and a mandatory non-negativity requirement on f. Because Vlasov equation (VE) is a PDE with variable coefficients and the non-negativity requirement is mandatory, when 5 PDEs (4 Maxwell equations and the VE) are treated as an initial-value problem, the temporal evolution of f is difficult to be attacked by existing popular computational mathematics techniques, such as Fourier analysis technique and finite-difference technique, and hence these techniques do not yield efficient and reliable schemes of the PKSiEA. In a universal scheme of the PKSiEA presented here, because the phase space boundary condition f∣ r=∞,∣υ∣=c = 0 causes the VE to yield a conservation of total particle number ∫fd 3 rd 3 υ, we propose an efficient and strict expression of the f-profile in terms of two classes of functions of phase space coordinates r , υ , and the VE displays a universal relation between two classes. By this new expression, we express the initial-value problem of 5 PDEs in terms of allowed deformation modes from an initial f-profile, and ensure the PKSiEA fully respecting the non-negativity requirement on f.

Visualization of phase-space orbit topological boundary using imaging neutral particle analyzer

A newly-developed Imaging Neutral Particle Analyzer (INPA) in the DIII-D tokamak interrogates phase space occupied by fast ions on multiple different orbit topologies, including passing, stagnation, trapped and potato orbits. Depending on plasma parameters and beam injection geometries, this new INPA system is capable of visualizing distributions of fast ions on the selected orbit topology and its associated orbit topological boundaries. More importantly, the system is able to directly visualize the effective pitch angle scattering ν eff in phase space by measuring fast ions that are scattered across the trapped-passing orbit topological boundaries and from counter-passing orbits to co-passing orbits. It also enables visualization of fast ion confined-loss boundaries and resolves the change of the boundary in phase space, as plasma equilibrium evolves. The key goal of this new INPA system is to directly measure ν eff across phase space induced by drift waves and its interaction with Alfvén eigenmodes, i.e. a key issue towards a future fusion power plant.

A fragmentation approach to jet flavor

An intuitive definition of the partonic flavor of a jet in quantum chromodynamics is often only well-defined in the deep ultraviolet, where the strong force becomes a free theory and a jet consists of a single parton. However, measurements are performed in the infrared, where a jet consists of numerous particles and requires an algorithmic procedure to define their phase space boundaries. To connect these two regimes, we introduce a novel and simple partonic jet flavor definition in the infrared. We define the jet flavor to be the net flavor of the partons that lie exactly along the direction of the Winner-Take-All recombination scheme axis of the jet, which is safe to all orders under emissions of soft particles, but is not collinear safe. Collinear divergences can be absorbed into a perturbative fragmentation function that describes the evolution of the jet flavor from the ultraviolet to the infrared. The evolution equations are linear and a small modification to traditional DGLAP and we solve them to leading-logarithmic accuracy. The evolution equations exhibit fixed points in the deep infrared, we demonstrate quantitative agreement with parton shower simulations, and we present various infrared and collinear safe observables that are sensitive to this flavor definition.

Orbital Dynamics Landscape near the Most Distant Known Trans-Neptunian Objects

The most distant known trans-Neptunian objects (TNOs), those with perihelion distance above 38 au and semimajor axis above 150 au, are of interest for their potential to reveal past, external, or present but unseen perturbers. Realizing this potential requires understanding how the known planets influence their orbital dynamics. We use a recently developed Poincaré mapping approach for orbital phase space studies of the circular planar restricted three-body problem, which we have extended to the case of the 3D restricted problem with N planetary perturbers. With this approach, we explore the dynamical landscape of the 23 most distant TNOs under the perturbations of the known giant planets. We find that, counter to common expectations, almost none of these TNOs are far removed from Neptune’s resonances. Nearly half (11) of these TNOs have orbits consistent with stable libration in Neptune’s resonances; in particular, the orbits of TNOs 148209 and 474640 overlap with Neptune’s 20:1 and 36:1 resonances, respectively. Five objects can be ruled currently nonresonant, despite their large orbital uncertainties, because our mapping approach determines the resonance boundaries in angular phase space in addition to semimajor axis. Only three objects are in orbital regions not appreciably affected by resonances: Sedna, 2012 VP113 and 2015 KG163. Our analysis also demonstrates that Neptune’s resonances impart a modest (few percent) nonuniformity in the longitude of perihelion distribution of the currently observable distant TNOs. While not large enough to explain the observed clustering, this small dynamical sculpting of the perihelion longitudes could become relevant for future, larger TNO data sets.

Ambiguity resolution for integrable gravitational charges

Recently, Ciambelli, Leigh, and Pai (CLP) [arXiv:2111.13181] have shown that nonzero charges integrating Hamilton’s equation can be defined for all diffeomorphisms acting near the boundary of a subregion in a gravitational theory. This is done by extending the phase space to include a set of embedding fields that parameterize the location of the boundary. Because their construction differs from previous works on extended phase spaces by a covariant phase space ambiguity, the question arises as to whether the resulting charges are unambiguously defined. Here, we demonstrate that ambiguity-free charges can be obtained by appealing to the variational principle for the subregion, following recent developments on dealing with boundaries in the covariant phase space. Resolving the ambiguity produces corrections to the diffeomorphism charges, and also generates additional obstructions to integrability of Hamilton’s equation. We emphasize the fact that the CLP extended phase space produces nonzero diffeomorphism charges distinguishes it from previous constructions in which diffeomorphisms are pure gauge, since the embedding fields can always be eliminated from the latter by a choice of unitary gauge. Finally, we show that Wald-Zoupas charges, with their characteristic obstruction to integrability, are associated with a modified transformation in the extended phase space, clarifying the reason behind integrability of Hamilton’s equation for standard diffeomorphisms.

Dynamical landscape of transitional pipe flow.

The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge-states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge. We also discuss alternatives to the suggested bifurcation diagram, which could be relevant for wall-bounded flows other than straight pipes.

Two-parameter non-smooth bifurcations of period-one motions in a plastic impacting oscillator

A 2-DOF plastic impacting oscillator is considered. The discontinuity boundaries of the system phase space are defined as well as the local maps and Poincaré map. Four cases of sliding bifurcation of limit cycles in impact system are described schematically. The period-one motion modes and occurrence regions of the system are identified in the two-parameter space by numerical simulation. One-parameter bifurcation diagrams and trajectories are computed to explain the evolutions of adjacent periodic motions and the non-smooth bifurcation behaviors including grazing and four types of sliding bifurcation. Due to the discontinuous grazing bifurcation there appear a narrow hysteresis domain in the two-parameter evolution between neighboring non-impacting and impacting motions. Grazing, switching–sliding and multi-sliding bifurcations of periodic sticking motions are all manifested as rising phenomenon of the time trajectory, but the different locations of the rises in the sticking phase correspond to the different types of non-smooth bifurcations. The grazing induced dynamics and the characteristics of rising events are discussed. Notably, the codimension-2 bifurcation points are detected and investigated by numerical continuation algorithm.

Transition to the classical regime in quantum mechanics on a lattice and implications of discontinuous space

It is well known that, due to the uncertainty principle, the Planck constant sets a resolution boundary in phase space and the resulting trade-off in resolution between incompatible measurements has been thoroughly investigated. It is also known that, in the classical regime, sufficiently coarse measurements of position and momentum can simultaneously be determined. However, the picture of how the uncertainty principle gradually disappears as we transition from the quantum to the classical regime is not so vivid. In the present work we clarify this picture by studying the associated probabilities that quantify the effects of the uncertainty principle in the framework of finite-dimensional quantum mechanics on a lattice. We also study how these probabilities are perturbed by the granularity of the lattice and show that they can signal the discontinuity of the underlying space.

How to GAN LHC events

Event generation for the LHC can be supplemented by generative adversarial networks, which generate physical events and avoid highly inefficient event unweighting. For top pair production we show how such a network describes intermediate on-shell particles, phase space boundaries, and tails of distributions. In particular, we introduce the maximum mean discrepancy to resolve sharp local features. It can be extended in a straightforward manner to include for instance off-shell contributions, higher orders, or approximate detector effects.

Electron trapping in the coma of a weakly outgassing comet

Measurements from the Rosetta mission have shown a multitude of nonthermal electron distributions in the cometary environment, challenging the previously assumed plasma interaction mechanisms near a cometary nucleus. In this paper, we discuss electron trapping near a weakly outgassing comet from a fully kinetic (particle-in-cell) perspective. Using the electromagnetic fields derived from the simulation, we characterize the trajectories of trapped electrons in the potential well surrounding the cometary nucleus and identify the distinguishing features in their respective velocity and pitch angle distributions. Our analysis allows us to define a clear boundary in velocity phase space between the distributions of trapped and passing electrons.

Modelling uncertainties in phase-space boundary integral models of ray propagation

A recently proposed phase-space boundary integral model for the stochastic propagation of ray densities is presented and, for the first time, explicit connections between this model and parametric uncertainties arising in the underlying physical model are derived. In particular, an asymptotic analysis for a weak noise perturbation of the propagation speed is used to derive expressions for the probability distribution of the phase-space boundary coordinates after transport along uncertain, and in general curved, ray trajectories. Furthermore, models are presented for incorporating geometric uncertainties in terms of both the location of an edge within a polygonal domain, as well as small scale geometric fluctuations giving rise to rough boundary reflections. Uncertain source terms are also considered in the form of stochastically distributed point sources and uncertain boundary data. A series of numerical experiments is then performed to illustrate these uncertainty models in two-dimensional convex polygonal domains.

Zero-flux approximations for multivariate quadrature-based moment methods

The evolution of polydisperse systems is governed by population balance equations. A group of efficient solution approaches are the moment methods, which do not solve for the number density function (NDF) directly but rather for a set of its moments. While this is computationally efficient, a specific challenge arises when describing the fluxes across a boundary in phase space for the disappearance of elements, the so-called zero-flux. The main difficulty is the missing NDF-information at the boundary, which most moment methods cannot provide. Relevant physical examples are evaporating droplets, soot oxidation or particle dissolution.In general, this issue can be solved by reconstructing the NDF close to the boundary. However, this was previously only achieved with univariate approaches, i.e. considering only a single internal variable. Many physical problems are insufficiently described by univariate population balance equations, e.g. droplets in sprays often require the temperature or the velocity to be internal coordinates in addition to the size.In this paper, we propose an algorithm, which provides an efficient multivariate approach to calculate the zero-fluxes. The algorithm employs the Extended Quadrature Method of Moments (EQMOM) with Beta and Gamma kernel density functions for the marginal NDF reconstruction and a polynomial or spline for the other conditional dimensions. This combination allows to reconstruct the entire multivariate NDF and based on this, expressions for the disappearance flux are derived. An algorithm is proposed for the whole moment inversion and reconstruction process. It is validated against a suite of test cases with increasing complexity. The influence of the number of kernel density functions and the configuration of the polynomials and splines on the accuracy is discussed. Finally, the associated computational costs are evaluated.

Active flux schemes on moving meshes with applications to geometric optics

Active flux schemes are finite volume schemes that keep track of both point values and averages. The point values are updated using a semi-Lagrangian step, making active flux schemes highly suitable for geometric optics problems on phase space, i.e., to solve Liouville's equation. We use a semi-discrete version of the active flux scheme. Curved optics lead to moving boundaries in phase space. Therefore, we introduce a novel way of defining the active flux scheme on moving meshes. We show, using scaling arguments as well as numerical experiments, that our scheme outperforms the current industry standard, ray tracing. It has higher accuracy as well as a more favourable time scaling. The numerical experiments demonstrate that the active flux scheme is orders of magnitude more accurate and faster than ray tracing.

Experimental validation of phase space conduits of transition between potential wells

A phase space boundary between transition and non-transition, similar to those observed in chemical reaction dynamics, is shown experimentally in a macroscopic system. We present a validation of the phase space flux across rank one saddles connecting adjacent potential wells and confirm the underlying phase space conduits that mediate the transition. Experimental regions of transition are found to agree with the theory to within 1\%, suggesting the robustness of phase space conduits of transition in a broad array of two or more degree of freedom experimental systems, despite the presence of small dissipation.

Uncertainty quantification for phase-space boundary integral models of ray propagation

Vibrational and acoustic energy distributions of wave fields in the high-frequency regime are often modeled using flow transport equations. This study concerns the case when the flow of rays or non-interacting particles is driven by an uncertain force or velocity field and the dynamics are determined only up to a degree of uncertainty. A boundary integral equation description of wave energy flow along uncertain trajectories in finite two-dimensional domains is presented, which is based on the truncated normal distribution, and interpolates between a deterministic and a completely random description of the trajectory propagation. The properties of the Gaussian probability density function appearing in the model are applied to derive expressions for the variance of a propagated initial Gaussian density in the weak noise case. Numerical experiments are performed to illustrate these findings and to study the properties of the stationary density, which is obtained in the limit of infinitely many reflections at the boundary.

A boundary integral method for modelling vibroacoustic energy distributions in uncertain built up structures

A phase-space boundary integral method is developed for modelling stochastic high-frequency acoustic and vibrational energy transport in both single and multi-domain problems. The numerical implementation is carried out using the collocation method in both the position and momentum phase-space variables. One of the major developments of this work is the systematic convergence study, which demonstrates that the proposed numerical schemes exhibit convergence rates that could be expected from theoretical estimates under the right conditions. For the discretisation with respect to the momentum variable, we employ spectrally convergent basis approximations using both Legendre polynomials and Gaussian radial basis functions. The former have the advantage of being simpler to apply in general without the need for preconditioning techniques. The Gaussian basis is introduced with the aim of achieving more efficient computations in the weak noise case with near-deterministic dynamics. Numerical results for a series of coupled domain problems are presented, and demonstrate the potential for future applications to larger scale problems from industry.

Competition between injunctive social norms and conservation priorities gives rise to complex dynamics in a model of forest growth and opinion dynamics

Human and environmental systems are often treated as existing in isolation from one another, whereas in fact they are often two parts of a single, coupled human-environment system. Developing theoretical models of coupled human-environment systems is a continuing area of research, although relatively few of these models are based on differential equations. Here we develop a simple differential equation coupled human-environment system model of forest growth dynamics and conservationist opinion dynamics in a human population. The model assumes logistic growth and harvesting in the forest. Opinion spread in the human population is based on the interplay between conservation values stimulated by forest rarity, and injunctive social norms that tend to support population conformity. We find that injunctive social norms drive the system to the boundaries of phase space, whereas rarity-based conservation priorities drive the system to the interior. The result is complex dynamics including limit cycles and alternative stable states that do not occur if injunctive social norms are absent. We found that the model with injunctive social norms had five possible observable outcomes, whereas the model without social norms had only two stable states. Thus social norms and have dramatic influence in conservation dynamics. We also find that increasing the conservation value of forests is the best way to boost and stabilize forest cover while also boosting conservationist opinion in the population, although for some parameter regimes it can also give rise to long-term oscillations in opinions and forest cover. We conclude that simple models can provide insights and reveal patterns that might be difficult to see with high-dimensional computational models, and therefore should be used more often in research on coupled human-environment systems.

Detecting kinematic boundary surfaces in phase space: particle mass measurements in SUSY-like events

We critically examine the classic endpoint method for particle mass determination, focusing on difficult corners of parameter space, where some of the measurements are not independent, while others are adversely affected by the experimental resolution. In such scenarios, mass differences can be measured relatively well, but the overall mass scale remains poorly constrained. Using the example of the standard SUSY decay chain tilde{q}to {tilde{chi}}_2^0to tilde{ell}to {tilde{chi}}_1^0 , we demonstrate that sensitivity to the remaining mass scale parameter can be recovered by measuring the two-dimensional kinematical boundary in the relevant three-dimensional phase space of invariant masses squared. We develop an algorithm for detecting this boundary, which uses the geometric properties of the Voronoi tessellation of the data, and in particular, the relative standard deviation (RSD) of the volumes of the neighbors for each Voronoi cell in the tessellation. We propose a new observable, overline{Sigma} , which is the average RSD per unit area, calculated over the hypothesized boundary. We show that the location of the overline{Sigma} maximum correlates very well with the true values of the new particle masses. Our approach represents the natural extension of the one-dimensional kinematic endpoint method to the relevant three dimensions of invariant mass phase space.

Unified treatment of tidal disruption by Schwarzschild black holes

Stars on orbits with pericenters sufficiently close to the supermassive black hole at the center of their host galaxy can be ripped apart by tidal stresses. Some of the resulting stellar debris becomes more tightly bound to the hole and can potentially produce an observable flare called a tidal-disruption event (TDE). We provide a self-consistent, unified treatment of TDEs by non-spinning (Schwarzschild) black holes, investigating several effects of general relativity including changes to the boundary in phase space that defines the loss-cone orbits on which stars are tidally disrupted or captured. TDE rates decrease rapidly at large black-hole masses due to direct stellar capture, but this effect is slightly countered by the widening of the loss cone due to the stronger tidal fields in general relativity. We provide a new mapping procedure that translates between Newtonian gravity and general relativity, allowing us to better compare predictions in both gravitational theories. Partial tidal disruptions in relativity will strip more material from the star and produce more tightly bound debris than in Newtonian gravity for a stellar orbit with the same angular momentum. However, for deep encounters leading to full disruption in both theories, the stronger tidal forces in relativity imply that the star is disrupted further from the black hole and that the debris is therefore less tightly bound, leading to a smaller peak fallback accretion rate. We also examine the capture of tidal debris by the horizon and the relativistic pericenter precession of tidal debris, finding that black holes of $10^6$ solar masses and above generate tidal debris precessing by $10^\circ$ or more per orbit.

Optimal joint remote state preparation in the presence of various types of noises

A main obstacle faced by any quantum information processing protocol is the noise that degrades the desired coherence/entanglement. In this work we study by means of Kraus operators the effect of four typical types of noises on the quality of joint remote state preparation of a single-qubit state using a three-qubit Greenberger–Horne–Zeilinger-type state as the initial quantum channel. Assuming that two of the three involved qubits independently suffer a type of noise, we derive analytical expressions not only for the optimal averaged fidelities but also for the boundaries in phase space of the domains in which the joint remote state preparation protocol outperforms the classical one. Detailed discussion is given for each of the total 16 noisy scenarios. We also provide physical interpretation for the obtained results and outline possible future topics.