Phase Space Structure Research Articles

Orbital dynamics in galactic potentials under mass transfer

Context. Time-dependent potentials are common in galactic systems that undergo significant evolution, interactions, or encounters with other galaxies, or when there are dynamic processes such as star formation and merging events. Recent studies show that an ensemble approach along with the so-called snapshot framework in the theory of dynamical systems provide a powerful tool to analyze the time-dependent dynamics. Aims. In this work, we aim to explore and quantify the phase space structure and dynamical complexity in time-dependent galactic potentials consisting of multiple components. Methods. We applied the classical method of Poincaré surface of sections to analyze the phase space structure in a chaotic Hamiltonian system subjected to parameter drift. This, however, makes sense only when the evolution of a large ensemble of initial conditions is followed. Numerical simulations explore the phase space structure of such ensembles while the system undergoes a continuous parameter change. The pair-wise average distance of ensemble members allowed us to define a generalized Lyapunov exponent, which might also be time-dependent, to describe the system stability. Results. We provide a comprehensive dynamical analysis of the system under circumstances where linear mass transfer occurs between the disk and bulge components of the model.

Non-linearity and chaos in the kicked top

Classical chaos arises from the inherent non-linearity of dynamical systems. However, quantum maps are linear; therefore, the definition of chaos is not straightforward. To address this, we study a quantum system that exhibits chaotic behavior in its classical limit: the kicked top model, whose classical dynamics are governed by Hamilton’s equations on phase space, whereas its quantum dynamics are described by the Schrödinger equation in Hilbert space. We explore the critical degree of non-linearity signifying the onset of chaos in the kicked top by modifying the original Hamiltonian so that the non-linearity is parameterized by a quantity p. We find two distinct behaviors of the modified kicked top depending on the value of p. Chaos intensifies as p varies within the range of 1≤p≤2, whereas it diminishes for p>2, eventually transitioning to a purely regular oscillating system as p tends to infinity. We also comment on the complicated phase space structure for non-chaotic dynamics. Our investigation sheds light on the relationship between non-linearity and chaos in classical systems, offering insights into their dynamical behavior.

An electronic phase-space Hamiltonian approach for electronic current density and vibrational circular dichroism.

The Born-Oppenheimer framework stipulates that chemistry and physics occur on potential energy surfaces VBO(X) parameterized by a nuclear coordinate X, which are built by diagonalizing a BO Hamiltonian ĤBO(X). However, such a framework cannot recover many measurable chemical and physical features, including vibrational circular dichroism spectra. In this article, we show that a phase-space electronic Hamiltonian ĤPS(X,P), parameterized by both nuclear position X and momentum P, with a similar computational cost as solving ĤBO(X), can recover not just experimental vibrational circular dichroism signals but also a meaningful electronic current density that explains the features of the vibrational circular dichroism rotational strengths. Combined with earlier demonstrations that such Hamiltonians can also recover qualitatively correct electronic momenta with electronic densities that approximately satisfy a continuity equation, the data would suggest that, if one looks closely enough, chemistry in fact occurs on potential energy surfaces parameterized by both X and P, EPS(X, P). While the dynamical implications of such a phase-space electronic Hamiltonian are not yet known, we hypothesize that, by offering classical trajectories that explicitly offer nonzero electronic momentum while also conserving the total angular momentum (unlike Born-Oppenheimer theory), this new phase-space electronic structure Hamiltonian may well explain some fraction of the chiral-induced spin selectivity effect.

Interaction of driven ‘cold’ electron plasma wave with thermal bulk via ion spatial inhomogeneity

Abstract Using high resolution Vlasov - Poisson simulations, evolution of driven ``cold" electron plasma wave (EPW) in the presence of stationary inhomogeneous background of ions is studied. Mode coupling dynamics between ``cold'' EPW with phase velocity $v_{\phi}$ greater than thermal velocity i.e $v_{\phi} \gg v_{thermal}$ and its inhomogeneity induced sidebands is illustrated as an initial value problem. In driven cases, formation of BGK like phase space structures corresponding to sideband modes due to energy exchange from primary mode to bulk particles via wave-wave and wave-particle interactions leading to particle trapping is demonstrated for inhomogeneous plasma. Qualitative comparison studies between initial value perturbation and driven problem is presented, which examines the relative difference in energy transfer time between the interacting modes. Effect of variation in background ion inhomogeneity amplitude as well as ion inhomogeneity scale length on the driven EPWs is reported.

The effects of zonal fields on energetic-particle excitations of reversed-shear Alfvén eigenmode: simulation and theory

Abstract By employing both nonlinear gyrokinetic simulation and analytical theory, we have investigated the effects of zonal (electromagnetic) fields on the energetic particle’s (EPs) drive of reversed-shear Alfvén eigenmodes (AEs) in tokamak plasmas. Contrary to the conventional expectation, simulations with zonal fields that are turned on and off in the EP dynamics while keeping the full nonlinear dynamics of the thermal plasma indicate that zonal fields further enhance the instability drive and thus lead to a higher saturation level. These puzzling simulation results can be understood analytically in terms of the general fishbone-like dispersion relation with the correspondingly different EP phase-space structures induced by the zonal fields. Analytical expressions for the zonal fields that are beat driven by the reversed-shear AEs are also derived, and shown to be in good agreement with the simulation results.

Eccentric mergers in AGN Discs: Influence of the supermassive black-hole on three-body interactions

Abstract There are indications that stellar-origin black holes (BHs) are efficiently paired up in binary black holes (BBHs) in Active Galactic Nuclei (AGN) disc environments, which can undergo interactions with single BHs in the disc. Such binary-single interactions can potentially lead to an exceptionally high fraction of gravitational-wave mergers with measurable eccentricity in LIGO/Virgo/KAGRA. We here take the next important step in this line of studies by performing post-Newtonian N-body simulations between migrating BBHs and single BHs set in an AGN disc-like configuration, with a consistent inclusion of the central supermassive black hole (SMBH) in the equations of motion. With this setup, we study how the fraction of eccentric mergers varies in terms of the initial size of the BBH semi-major axis relative to the Hill sphere, as well as how it depends on the angle between the BBH and the incoming single BH. We find that the fraction of eccentric mergers is still relatively large, even when the interactions are notably influenced by the gravitational field of the nearby SMBH. However, the fraction as a function of the BBH semi-major axis does not follow a smooth functional shape, but instead shows strongly varying features that originate from the underlying phase-space structure. The phase-space further reveals that many of the eccentric mergers are formed through prompt scatterings. Finally, we present the first analytical solution to how the presence of an SMBH in terms of its Hill sphere affects the probability for forming eccentric BBH mergers through chaotic three-body interactions.

Detection of bifurcation in phase-space perturbative structures across transient wave–particle interaction in laboratory plasmas

A key ingredient for realizing a magnetically confined tritium-deuterium plasma fusion reactor is plasma heating by fusion-born high-energy helium ions, as a chained cycle of "nuclear burning." Efficient collisionless plasma heating by high-energy particles is anticipated when their energy is directly transferred to the plasma through waves. Those processes often involve nonlinear structure formations in phase-space, spanned by real-space and velocity-space coordinates, that significantly influence heating efficiency. To date, experimental knowledge of phase-space structure formation physics is severely limited, due to lack of experimental diagnostic schemes. Here, state-of-the-art hardware and software approaches for phase-space perturbative structure detection are introduced. A bifurcation in the phase-space structure formation is found, which affects the plasma heating efficiency. A confinement magnetic field structure is shown to be a potential knob for nuclear-burning control through phase-space manipulation.

Jet-induced enhancement of deuteron production in pp and p-Pb collisions at the LHC

Jet-associated deuteron production in pp collisions at s=13 TeV and p-Pb collisions at sNN=5.02 TeV is studied in the coalescence model by using the phase-space information of proton and neutron pairs from a multiphase transport (AMPT) model at the kinetic freezeout. In the low transverse momentum (pT) region pT/A<1.5 GeV/c, where A is the mass number of a nucleus, the in-jet coalescence factor B2In-jet for deuteron production, given by the ratio of the in-jet deuteron number to the square of the in-jet proton number, is found to be larger than the coalescence factor B2 in the medium perpendicular to the jet by a factor of about 10 in pp collisions and of 25 in p−Pb collisions, which are consistent with the ALICE measurements at the LHC. Such large low-momentum enhancements mainly come from coalescence of nucleons inside the jet with the medium nucleons. Coalescence of nucleons inside the jet dominates deuteron production only at the higher pT region of pT/A≳4 GeV/c, where both the yield ratio d/p of deuteron to proton numbers and the B2 are also significantly larger in the jet direction than in the direction perpendicular to the jet due to the strong collinear correlation among particles produced from jet fragmentation. Studying jet-associated deuteron production in relativistic nuclear collisions thus opens up a new window to probe the phase-space structure of nucleons inside jets.

Fractional Einstein–Gauss–Bonnet Scalar Field Cosmology

Our paper introduces a new theoretical framework called the Fractional Einstein–Gauss–Bonnet scalar field cosmology, which has important physical implications. Using fractional calculus to modify the gravitational action integral, we derived a modified Friedmann equation and a modified Klein–Gordon equation. Our research reveals non-trivial solutions associated with exponential potential, exponential couplings to the Gauss–Bonnet term, and a logarithmic scalar field, which are dependent on two cosmological parameters, m and α0=t0H0 and the fractional derivative order μ. By employing linear stability theory, we reveal the phase space structure and analyze the dynamic effects of the Gauss–Bonnet couplings. The scaling behavior at some equilibrium points reveals that the geometric corrections in the coupling to the Gauss–Bonnet scalar can mimic the behavior of the dark sector in modified gravity. Using data from cosmic chronometers, type Ia supernovae, supermassive Black Hole Shadows, and strong gravitational lensing, we estimated the values of m and α0, indicating that the solution is consistent with an accelerated expansion at late times with the values α0=1.38±0.05, m=1.44±0.05, and μ=1.48±0.17 (consistent with Ωm,0=0.311±0.016 and h=0.712±0.007), resulting in an age of the Universe t0=19.0±0.7 [Gyr] at 1σ CL. Ultimately, we obtained late-time accelerating power-law solutions supported by the most recent cosmological data, and we proposed an alternative explanation for the origin of cosmic acceleration other than ΛCDM. Our results generalize and significantly improve previous achievements in the literature, highlighting the practical implications of fractional calculus in cosmology.

Low-dimensional projection of reactivity classes in chemical reaction dynamics using supervised dimensionality reduction.

Transition state theory (TST) provides a framework to estimate the rate of chemical reactions. Despite its great success with many reaction systems, the underlying assumptions such as local equilibrium and nonrecrossing do not necessarily hold in all cases. Although dynamical systems theory can provide the mathematical foundation of reaction tubes existing in phase space that enables us to predict the fate of reactions free from the assumptions of TST, numerical demonstrations for large systems have been yet one of the challenges. Here, we propose a dimensionality reduction algorithm to demonstrate structures in phase space (called reactive islands) that predict reactivity in systems with many degrees of freedom. The core of this method is the application of supervised principal component analysis, where a coordinate transformation is performed to preserve the dynamical information on reactivity (i.e., to which potential basin the system moves from a region of interest) as much as possible. The reactive island structures are expected to be reflected in the transformed, low-dimensional phase space. As an illustrative example, the algorithm is scrutinized using a modified Hénon-Heiles Hamiltonian system extended to many degrees of freedom, which has three channels leading to three different products from one stable potential basin. It is shown that our algorithm can predict the reactivity in the transformed, low-dimensional coordinate system better than a naïve coordinate system and that the reactivity distribution in the transformed low-dimensional space is considered to reflect the underlying reactive islands.

Semi-analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified semi-analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits, quasihalo orbits, Axial orbits, and their invariant manifolds, as well as transit and non-transit orbits. Based on classical in-plane and out-of-plane frequency resonance mechanisms, the Lindstedt–Poincaré method could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. In this paper, by introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt–Poincaré method. Analyzing the bifurcation equation obtained from different coupling forms reveals bifurcation conditions for all kinds of orbits near collinear libration points. When η = 0, the series solution describes non-bifurcated orbits, while when η ≠ 0, the solution describes bifurcated orbits, including quasihalo orbits, Axial orbits, and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified semi-analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

Spin–Orbit Coupling of the Ellipsoidal Secondary in a Binary Asteroid System

In our solar system, spin–orbit coupling is a common phenomenon in binary asteroid systems, where the mutual orbits are no longer invariant due to exchange of angular momentum between translation and rotation. In this work, dynamical structures in phase space are explored for the problem of spin–orbit coupling by taking advantage of analytical and numerical methods. In particular, the technique of Poincaré sections is adopted to reveal numerical structures, which are dependent on the total angular momentum, the Hamiltonian, mass ratio, and asphericity parameter. Analytical study based on perturbative treatments shows that high-order and/or secondary spin–orbit resonances are responsible for numerical structures arising in Poincaré sections. Analytical solutions are applied to (65803) Didymos, (80218) VO123 and (4383) Suruga to reveal their phase-space structures, showing that there is a high possibility for them to locate inside secondary 1:1 spin–orbit resonance.

Towards unbiased recovery of cosmic filament properties: the role of spine curvature and optimized smoothing

Cosmic filaments, the most prominent features of the cosmic web, possibly hold untapped potential for cosmological inference. While it is natural to expect the structure of filaments to show universality similar to that seen in dark matter halos, the lack of agreement between different filament finders on what constitutes a filament has hampered progress on this topic. We initiate a programme to systematically investigate and uncover possible universal features in the phase space structure of cosmic filaments, by generating particle realizations of mock filaments with a priori known properties. Using these, we identify an important source of bias in the extraction of radial density profiles, which occurs when the local curvature κ of the spine exceeds a threshold determined by the filament thickness. This bias exists even for perfectly determined spines, thus affecting all filament finders. We show that this bias can be nearly eliminated by simply discarding the regions with the highest κ, with little loss of precision. An additional source of bias is the noise generated by the filament finder when identifying the spine, which depends on both the finder algorithm as well as intrinsic properties of the individual filament. We find that to mitigate this bias, it is essential not only to smooth the estimated spine, but to optimize this smoothing separately for each filament. We propose a novel optimization based on minimizing the estimated filament thickness, along with Fourier space smoothing. We implement these techniques using two tools, FilGen which generates mock filaments and FilAPT which analyses and processes them. We expect these tools to be useful in calibrating the performance of filament finders, thereby enabling searches for filament universality.

Metal-insulator transition of spinless fermions coupled to dispersive optical bosons

Including the previously ignored dispersion of phonons we revisit the metal-insulator transition problem in one-dimensional electron-phonon systems on the basis of a modified spinless fermion Holstein model. Using matrix-product-state techniques we determine the global ground-state phase diagram in the thermodynamic limit for the half-filled band case, and show that in particular the curvature of the bare phonon band has a significant effect, not only on the transport properties characterized by the conductance and the Luttinger liquid parameter, but also on the phase space structure of the model as a whole. While a downward curved (convex) dispersion of the phonons only shifts the Tomonaga-Luttinger-liquid to charge-density-wave quantum phase transition towards stronger EP coupling, an upward curved (concave) phonon band leads to a new phase-separated state which, in the case of strong dispersion, can even completely cover the charge-density wave. Such phase separation does not occur in the related Edwards fermion-boson model.

Statistical description of interacting multistream quantum systems

In this research, the electrostatically coupled multistream quasiparticle excitations are studied in the framework of the Wigner distribution function. It is remarked that the Wigner distribution of coupled multistream collective quantum excitations satisfies a simple Liouville-like evolution equation from which a generalized distribution function for multistream quasiparticle excitations is deduced. The phase-space structure of collective quantum excitations in counter-stream electron and two-stream electron–positron gas with their evolution is calculated and electron/positron hole formation due to the onset of quantum stream instability is studied in connection with the energy band structure of the multistream quantum system, for the first time. The quantum stream instabilities in symmetric and asymmetric stream systems are studied and compared. It is found that the presence of opposite-charge streams leads to overall stability due to lowering the interaction potential effect. The generalized Wigner theory is also applied to study the electron transport in a one-dimensional periodic lattice using the concept of virtual streams. Current generalized statistical formalism may be used to model different quantum phenomena in the linear excitations limit with collective electrostatic interactions. The applications extend to the stream instability in quantum charge transport in metals, semiconductors, plasmonic devices, phase-space structure of charge carriers in periodic lattices interacting with the external potential of arbitrary shape and the dynamic evolution of dense electron–positron jets in active galactic nuclei or within the extremely dense astrophysical objects.

Linear operator theory of phase mixing

ABSTRACT We study solutions of the collisionless Boltzmann equation (CBE) in a functional Koopman representation. This facilitates the use of linear spectral techniques characteristic of the analysis of Schrödinger-type equations. For illustrative purposes, we consider the classical phase mixing of a non-interacting distribution function in a quartic potential. Solutions are determined perturbatively relative to a harmonic oscillator. We impose a form of coarse-graining by choosing a finite-dimensional basis to represent the distribution function and time evolution operators, which sets a minimum length-scale on phase space structure. We observe a relationship between the dimension of the representation and the multiplicity of the harmonic oscillator eigenvalues. System dynamics are understood in terms of degenerate subspaces of the linear operator spectra. Each subspace is associated with a mode of the harmonic oscillator, the first two being bending and breathing structures. The quartic potential splits the degenerate eigenvalues within each subspace. This facilitates the formation of spiral structure as deformations from the harmonic oscillator modes. We ultimately argue that this construction provides a promising avenue for study of self-interacting systems experiencing phase mixing, which is an outstanding problem in the context of the Gaia DR2 vertical phase space spirals.

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.

Analysis of double-resonance crossing in adiabatic trapping phenomena for quasi-integrable area-preserving maps with time-dependent exciters.

In this paper we analyze the adiabatic crossing of a resonance for Hamiltonian systems when a double-resonance condition is satisfied by the linear frequency at an elliptic fixed point. We discuss in detail the phase-space structure on a class of Hamiltonians and area-preserving maps with an elliptic fixed point in the presence of a time-dependent exciter. Various regimes have been identified and carefully studied. This study extends results obtained recently for the trapping and transport phenomena for periodically perturbed Hamiltonian systems, and it could have relevant applications in the adiabatic beam splitting in accelerator physics.

The debris of the ‘last major merger’ is dynamically young

ABSTRACT The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the ‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space, because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago. We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data 1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’ did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within the last few Gyr, consistent with the body of work surrounding the VRM.

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.