Phase-space Distribution Function Research Articles

Constraining the shape of dark matter haloes using only starlight. I. A new technique and its application to the galaxy Nube

We present a new technique to constrain the gravitational potential of a galaxy from the observed stellar mass surface density alone under a number of assumptions. It uses the classical Eddington inversion method to compute the phase-space distribution function (DF) needed for stars to reside in a given gravitational potential. In essence, each potential defines a set of density profiles, and it is the expansion of the observed profile in this database that provides the DF. If the required DF becomes negative, then the potential is inconsistent with the observed stars and can be discarded. It is particularly well suited for analyzing low-mass low surface brightness galaxies, where photometric but not spectroscopic data can be obtained. The recently discovered low surface brightness galaxy was used to showcase its application. For observed stellar core to be reproduced with a non-negative DF, cuspy NFW (Navarro, Frenk, and White) potentials are highly disfavored compared with potentials that have cores (Schuster-Plummer or ρ_230). The method assumes the stellar system to have spherical symmetry and isotropic velocity distribution; however, we discuss simple extensions that relax the need for isotropy and may help to drop the spherical symmetry assumption.

Neutrinos of magnetorotational supernovae

The neutrino dynamics in hot and dense magnetized matter corresponding to a supernova explosion is considered. It is shown that accounting for fluctuations during interaction of neutrinos with matter leads to the Fokker–Planck equation for the dynamics of the phase space distribution function. The addition to the energy transfer component of the kinetic equation is determined by straggling in neutrino collisions with a magnetized nucleon gas caused by the neutral current Gamow–Teller interaction. When accounting for the effect of fluctuations, the switching of acceleration and stopping regimes in neutrino evolution is evident for average energy. The effect of fluctuations leads to an additional increase in the hardness of the neutrino spectra. It is shown that the high-energy component of the electron antineutrino flux is enhanced in addition due to the effect of neutrino oscillations. Such a strengthening of the spectrum hardness is especially noticeable in the case of the inverted mass ordering and makes the signal more registrable by ground-based detectors. The possibilities of detecting supernova neutrinos by KM3NeT and Baikal-GVD observatories are discussed. The sensitivity of counting rate to the mass ordering is demonstrated to increase at growing difference in the hardness of energy spectra for various flavors.

On the convergence of phase space distributions to microcanonical equilibrium: dynamical isometry and generalized coarse-graining

Abstract This work explores the manner in which classical phase space distribution functions converge to the microcanonical distribution. We first prove a theorem about the lack of convergence, then define a generalization of the coarse-graining procedure that leads to convergence. We prove that the time evolution of phase space distributions is an isometry for a broad class of statistical distance metrics, implying that ensembles do not get any closer to (or farther from) equilibrium, according to these metrics. This extends the known result that strong convergence of phase space distributions to the microcanonical distribution does not occur. However, it has long been known that weak convergence can occur, such that coarse-grained distributions---defined by partitioning phase space into a finite number of cells---converge pointwise to the microcanonical distribution. We define a generalization of coarse-graining that removes the need for partitioning phase space into cells. We prove that our generalized coarse-grained distribution converges pointwise to the microcanonical distribution if the dynamics are strong mixing. As an example, we study an ensemble of triangular billiard systems.

Evolution of the mass-radius relation of expanding very young star clusters

The initial mass–radius relation of embedded star clusters is an essential boundary condition for understanding the evolution of embedded clusters in which stars form to their release into the galactic field via an open star cluster phase. The initial mass–radius relation of embedded clusters deduced by Marks & Kroupa (2012, A&A, 543, A8) is significantly different from the relation suggested by Pfalzner et al. (2016, A&A, 586, A68). Here, we use direct N-body simulations to model the early expansion of embedded clusters after the expulsion of their residual gas. The observationally deduced radii of clusters up to a few million years old, compiled from various sources, are well fitted by N-body models, implying that these observed very young clusters are most likely in an expanding state. We show that the mass–radius relation of Pfalzner et al. (2016) reflects the expansion of embedded clusters following the initial mass–radius relation of Marks & Kroupa (2012). We also suggest that even the embedded clusters in ATLASGAL clumps with HII regions are probably already in expansion. All the clusters collected here from different observations show a mass-radius relation with a similar slope, which may indicate that all clusters were born with a profile resembling that of the Plummer phase-space distribution function.

Optimization of fast-ion diagnostic sets in tokamaks and stellarators using diagnostic weight functions.

The fast-ion phase-space distribution function in the magnetic fusion devices is always underdiagnosed, and every new fast-ion diagnostic should be carefully assessed before installation to minimize redundancies in measurements and maximize the information from the yet undiagnosed part of the fast-ion phase space distribution function. Here, we present a novel method of assessing the added value of a considered fast-ion diagnostic, taking actual geometry and an existing set of fast-ion diagnostics into account. The new method is based on a reformulation of the diagnostic weight functions in constants of motion (COM). We compare the proposed method with the previous approach using Monte Carlo simulations.

Time evolutions of information entropies in a one-dimensional Vlasov–Poisson system

A one-dimensional Vlasov–Poisson system is considered to elucidate how the information entropies of the probability distribution functions of the electron position and velocity variables evolve in the Landau damping process. Considering the initial condition given by the Maxwellian velocity distribution with the spatial density perturbation in the form of the cosine function of the position, we derive linear and quasilinear analytical solutions that accurately describe both early and late time behaviors of the distribution function and the electric field. The validity of these solutions is confirmed by comparison with numerical simulations based on contour dynamics. Using the quasilinear analytical solution, the time evolutions of the velocity distribution function and its kurtosis indicating deviation from the Gaussian distribution are evaluated with the accuracy of the squared perturbation amplitude. We also determine the time evolutions of the information entropies of the electron position and velocity variables and their mutual information. We further consider Coulomb collisions that relax the state in the late-time limit in the collisionless process to the thermal equilibrium state. In this collisional relaxation process, the mutual information of the position and velocity variables decreases to zero, while the total information entropy of the phase-space distribution function increases by the decrease in the mutual information and demonstrates the validity of Boltzmann's H-theorem.

A nonequilibrium statistical mechanics derivation of the hydrodynamic equations of simple fluids using a noncanonical form of the Poisson bracket

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood.

Nonseparable wave evolution equations in quantum kinetics

A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton‐Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density.

Halo densities and pericenter distances of the bright Milky Way satellites as a test of dark matter physics

ABSTRACT We provide new constraints on the dark matter halo density profile of Milky Way (MW) dwarf spheroidal galaxies (dSphs) using the phase-space distribution function (DF) method. After assessing the systematics of the approach against mock data from the Gaia Challenge project, we apply the DF analysis to the entire kinematic sample of well-measured MW dwarf satellites for the first time. Contrary to previous findings for some of these objects, we find that the DF analysis yields results consistent with the standard Jeans analysis. In particular, in this study we rediscover (i) a large diversity in the inner halo densities of dSphs (bracketed by Draco and Fornax), and (ii) an anticorrelation between inner halo density and pericenter distance of the bright MW satellites. Regardless of the strength of the anticorrelation, we find that the distribution of these satellites in density versus pericenter space is inconsistent with the results of the high-resolution N-body simulations that include a disc potential. Our analysis motivates further studies on the role of internal feedback and dark matter microphysics in these dSphs.

Orbit tomography in constants-of-motion phase-space

Tomographic reconstructions of a 3D fast-ion constants-of-motion phase-space distribution function are computed by inverting synthetic signals based on projected velocities of the fast ions along the diagnostic lines of sight. A spectrum of projected velocities is a key element of the spectrum formation in fast-ion D-alpha spectroscopy, collective Thomson scattering, and gamma-ray and neutron emission spectroscopy, and it can hence serve as a proxy for any of these. The fast-ion distribution functions are parameterised by three constants of motion, the kinetic energy, the magnetic moment and the toroidal canonical angular momentum. The reconstructions are computed using both zeroth-order and first-order Tikhonov regularisation expressed in terms of Bayesian inference to allow uncertainty quantification. In addition to this, a discontinuity appears to be present in the solution across the trapped-passing boundary surface in the three-dimensional phase space due to a singularity in the Jacobian of the transformation from position and velocity space to phase space. A method to allow for this apparent discontinuity while simultaneously penalising large gradients in the solution is demonstrated. Finally, we use our new methods to optimise the diagnostic performance of a set of six fans of sightlines by finding where the detectors contribute most complementary diagnostic information for the future COMPASS-Upgrade tokamak.

Phase space distribution functions and energy distributions of dark matter particles in haloes

ABSTRACT For a spherical dark matter halo with isotropic velocity distribution, the phase space distribution function (DF), the energy distribution, and the density profile form a set of self-consistent description of its equilibrium state, and knowing one is sufficient to determine the other two. The Navarro–Frenk–White density profile (NFW profile) is known to be a good approximation to the spherically averaged density distribution in simulated haloes. The DARKexp energy distribution is also known to compare well with the simulated energy distribution. We present a quantitative assessment of the NFW and DARKexp fits to the simulated DF and energy distribution for a wide range of haloes in a dark-matter-only simulation from the IllustrisTNG Project. As expected, we find that the NFW fits work well except at low energy when the density at small radii deviates from the NFW profile. Further, the NFW and DARKexp fits have comparable accuracy in the region where both fit well, but the DARKexp fits are better at low energy because they require matching of the central gravitational potential. We also find an approximate relation between the energy scale parametrizing the DARKexp energy distribution and that defined by the characteristic density and radius of the NFW profile. This relation may be linked to the relaxation process during halo formation.

Diagnostic weight functions in constants-of-motion phase-space

The fast-ion phase-space distribution function in axisymmetric tokamak plasmas is completely described by the three constants of motion: energy, magnetic moment and toroidal canonical angular momentum. In this work, the observable regions of constants-of-motion phase-space, given a diagnostic setup, are identified and explained using projected velocities of the fast ions along the diagnostic lines-of-sight as a proxy for several fast-ion diagnostics, such as fast-ion Dα spectroscopy, collective Thomson scattering, neutron emission spectroscopy and gamma-ray spectroscopy. The observable region in constants-of-motion space is given by a position condition and a velocity condition, and the diagnostic sensitivity is given by a gyro-orbit and a drift-orbit weighting. As a practical example, 3D orbit weight functions quantifying the diagnostic sensitivity to each point in phase-space are computed and investigated for the future COMPASS-Upgrade and MAST-Upgrade tokamaks.

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

Forbidden dark matter annihilation into leptons with full collision terms

The standard approach of calculating the relic density of thermally produced dark matter based on the assumption of kinetic equilibrium is known to fail for forbidden dark matter models since only the high momentum tail of the dark matter phase space distribution function contributes significantly to dark matter annihilations. Furthermore, it is known that the computationally less expensive Fokker-Planck approximation for the collision term describing elastic scattering processes between non-relativistic dark matter particles and the Standard Model thermal bath breaks down if both scattering partners are close in mass. This, however, is the defining feature of the forbidden dark matter paradigm. In this paper, we therefore include the full elastic collision term in the full momentum-dependent Boltzmann equation as well as in a set of fluid equations that couple the evolution of the number density and dark matter temperature for a simplified model featuring forbidden dark matter annihilations into muon or tau leptons through a scalar mediator. On the technical side, we perform all angular integrals in the full collision term analytically and take into account the effect of dark matter self-interactions on the relic density. The overall phenomenological outcome is that the updated relic density calculation results in a significant reduction of the experimentally allowed parameter space compared to the traditional approach, which solves only for the abundance. In addition, almost the entire currently viable parameter space can be probed with CMB-S4, next-generation beam-dump experiments or at a future high-luminosity electron-position collider, except for the resonant region where the mediator corresponds to approximately twice the muon or tau mass.

Self-consistent dynamical models with a finite extent – III. Truncated power-law spheres

ABSTRACT Fully analytical dynamical models usually have an infinite extent, while real star clusters, galaxies, and dark matter haloes have a finite extent. The standard method for generating dynamical models with a finite extent consists of taking a model with an infinite extent and applying a truncation in binding energy. This method, however, cannot be used to generate models with a preset analytical mass density profile. We investigate the self-consistency and dynamical properties of a family of power-law spheres with a general tangential Cuddeford (TC) orbital structure. By varying the density power-law slope γ and the central anisotropy β0, these models cover a wide parameter space in density and anisotropy profiles. We explicitly calculate the phase–space distribution function for various parameter combinations, and interpret our results in terms of the energy distribution of bound orbits. We find that truncated power-law spheres can be supported by a TC orbital structure if, and only if, γ ≥ 2β0, which means that the central density slope–anisotropy inequality is both a sufficient and a necessary condition for this family. We provide closed expressions for structural and dynamical properties such as the radial and tangential velocity dispersion profiles, which can be compared against more complex numerical modelling results. This work significantly adds to the available suite of self-consistent dynamical models with a finite extent and an analytical description.

Application of Approximate Maximum-Entropy Moment Closure to the Wigner Equation

This paper investigates the potential of moment closures as a possible path to future quantum hydrodynamics models. Moment closures are known to produce accurate predictions of continuum and non-equilibrium classical gas flows while offering modeling and numerical advantages over other commonly used methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations, however, the predicted distribution function in phase space is positive by design, which is in disagreement with the quasi-distribution function described by the Wigner equation. In this paper, predictions made by an interpolative one-dimensional five-moment system are compared to direct numerical solutions of the Wigner equation for a simple low-energy quantum state in unbounded double-well potentials. Numerical solutions for a particle in various potentials described by quartic polynomials are presented. The capacity of moment methods to provide accurate and efficient models for quantum systems is explored.

4D and 5D phase-space tomography using slowing-down physics regularization

We compute reconstructions of 4D and 5D fast-ion phase-space distribution functions in fusion plasmas from synthetic projections of these functions. The fast-ion phase-space distribution functions originating from neutral beam injection (NBI) at TCV and Wendelstein 7-X (W7-X) at full, half, and one-third injection energies can be distinguished and particle densities of each component inferred based on 20 synthetic spectra of projected velocities at TCV and 680 at W7-X. Further, we demonstrate that an expansion into a basis of slowing-down distribution functions is equivalent to regularization using slowing-down physics as prior information. Using this technique in a Tikhonov formulation, we infer the particle density fractions for each NBI energy for each NBI beam from synthetic measurements, resulting in six unknowns at TCV and 24 unknowns at W7-X. Additionally, we show that installing 40 LOS in each of 17 ports at W7-X, providing full beam coverage and almost full angle coverage, produces the highest quality reconstructions.

Relativistic hydrodynamics with spin in the presence of electromagnetic fields

We extend the classical phase-space distribution function to include the spin and electromagnetic fields coupling and derive the modified constitutive relations for charge current, energy-momentum tensor, and spin tensor. Because of the coupling, the new tensors receive corrections to their perfect fluid counterparts and make the background and spin fluid equations of motion communicate with each other. We investigate special cases which are relevant in high-energy heavy-ion collisions, including baryon free matter and large mass limit. Using Bjorken symmetries, we find that spin polarization increases with increasing magnetic field for an initially positive baryon chemical potential. The corrections derived in this framework may help to explain the splitting observed in Lambda hyperons spin polarization measurements.

Maximum entropy states of collisionless positron–electron plasma in a dipole magnetic field

We are developing a positron–electron plasma trap based on a dipole magnetic field generated by a levitated superconducting magnet to investigate the physics of magnetized plasmas with mass symmetry as well as antimatter components. Such laboratory magnetosphere is deemed essential for the understanding of pair plasmas in astrophysical environments, such as magnetars and blackholes, and represents a novel technology with potential applications in antimatter confinement and the development of coherent gamma-ray lasers. The design of the device requires a preemptive analysis of the achievable self-organized steady states. In this study, we construct a theoretical model describing maximum entropy states of a collisionless positron–electron plasma confined by a dipole magnetic field and demonstrate efficient confinement of both species under a wide range of physical parameters by analyzing the effect of the three adiabatic invariants on the phase space distribution function. The theory is verified by numerical evaluation of spatial density, electrostatic potential, and toroidal rotation velocity for each species in correspondence with the maximum entropy state.

Vlasov–Maxwell equations with spin effects

We present a numerical method to solve the Vlasov–Maxwell equations for spin-1/2 particles, in a semiclassical approximation where the orbital motion is treated classically while the spin variable is fully quantum. Unlike the spinless case, the phase-space distribution function is a $2\times 2$ matrix, which can also be represented, in the Pauli basis, as one scalar function $f_0$ and one three-component vector function $\boldsymbol f$ . The relationship between this ‘vectorial’ representation and the fully scalar representation on an extended phase space first proposed by Brodin et al. (Phys. Rev. Lett., vol. 101, 2008, p. 245002) is analysed in detail. By means of suitable approximations and symmetries, the vectorial spin-Vlasov–Maxwell model can be reduced to two-dimensions in the phase space, which is amenable to numerical solutions using a high-order grid-based Eulerian method. The vectorial model enjoys a Poisson structure that paves the way to accurate Hamiltonian split-time integrators. As an example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, and compare the results with those obtained earlier with the scalar spin-Vlasov–Maxwell model and a particle-in-cell code.