Space Distribution Function Research Articles

Spin relaxation rate for heavy quarks in weakly coupled QCD plasma

We compute the relaxation rate of the spin density of heavy quarks in a perturbative QCD plasma to leading-log order in the coupling constant g. The spin relaxation rate Γs in spin hydrodynamics is shown to be Γs ~ g4 log(1/g)T(T/M)2 in the heavy-quark limit T/M ≪ 1, which is smaller than the relaxation rate of other non- hydrodynamic modes by additional powers of T/M. We demonstrate three different methods to evaluate the spin relaxation rate: 1) the Green-Kubo formula in the spin hydrodynamic regime, 2) the spin density correlation function in the strict hydrodynamic limit, and 3) quantum kinetic theory of the spin distribution function in momentum space. We highlight the interesting differences between these methods, while they are ultimately connected to each other by the underlying Ward-Takahashi identity for the non-conserved spin density.

Geometric goodness of fit measure to detect patterns in data point clouds

In this work, we derive a geometric goodness-of-fit index similar to \(R^{2}\) using geometric data analysis techniques. We build the alpha shape complex from the data-cloud projected onto each variable and estimate the area of the complex and its domain. We create an index that measures the difference of area between the alpha shape and the smallest squared window of observation containing the data. By applying ideas similar to those found in the closest neighbor distribution and empty space distribution functions, we can establish when the characterizing geometric features of the point set emerge. This allows for a more precise application for our index. We provide some examples with anomalous patterns to show how our algorithm performs.

A Novel Approach for Data Analysis Based on Visualization of Phase Space Distribution Function in Plasma Turbulence Simulations

A Novel Approach for Data Analysis Based on Visualization of Phase Space Distribution Function in Plasma Turbulence Simulations

Fourth-order moment of the light field in the atmosphere for moderate and strong turbulence

Collisionless Boltzmann equation is used to describe the intensity correlations in partially saturated and fully saturated regimes in terms of photon distribution function in the phase space. Explicit expression for fourth moment of the light fields is obtained for the case of moderate and strong turbulence. Such expression consists of two terms accounting for two regions in the phase space that independently contribute to the correlation function. It is shown that present solution agrees with previous results for fully saturated regime. Additionally it embodies the effect of partially saturated radiation where the correlations of photon trajectories are important and the magnitude of the scintillation index is well above the unity. Fourth moment is used to study the fluctuations of transmittance which consider the effect of finite detector aperture.

Pressure-induced local structural crossover in a high-entropy metallic glass

Achieving effective tuning of glass structures between distinct states is intriguing for fundamental studies and applications, but has previously turned out to be challenging in practice. High-entropy metallic glasses (HEMGs), as an emerging type of metallic glasses (MGs) based on the high-entropy effect, are expected to have more disordered and frustrated chemical short-range structure compared with conventional MGs. Therefore, HEMGs may offer possibilities for structure and properties tuning in glasses. In this work, we employ pressure as a tuning parameter and monitor the atomic structural evolution of a senary HEMG, ${\mathrm{Ti}}_{16.7}{\mathrm{Zr}}_{16.7}{\mathrm{Hf}}_{16.7}{\mathrm{Cu}}_{16.7}{\mathrm{Ni}}_{16.7}{\mathrm{Be}}_{16.7}$, up to \ensuremath{\sim}40 GPa using in situ synchrotron x-ray diffraction. Analysis of its structure factor in reciprocal space and reduced pair distribution function in real space both reveal a pressure-induced structural crossover at \ensuremath{\sim}20 GPa with a dramatic change in short-range order (SRO), while no similar phenomenon is observed in a conventional MG, ${\mathrm{Cu}}_{36}{\mathrm{Zr}}_{64}$, as a control sample, suggesting the pressure-induced highly tunable SRO in HEMGs originates from the local chemical complexity, namely, the high-entropy effect. These results confirm that enhanced flexibility and tunability of atomic structures could be achieved by introducing the high-entropy effect into MGs. Therefore, configurational entropy could be another dimension for exploring MGs with highly tunable structures and properties for various potential applications.

Effect of Protoneutron Star Magnetized Envelops in Neutrino Energy Spectra

The neutrino dynamics in hot and dense magnetized matter, which corresponds with protoneutron star envelopes in the core collapse supernova explosions, is considered. The kinetic equation for a neutrino phase space distribution function is obtained, taking into account inelastic scattering by nuclear particles. The transfer component in a momentum space using transport properties is studied. The energy transfer coefficient is shown to change from positive to negative values when the neutrino energy exceeds four times the matter temperature. In the vicinity of a neutrino sphere, such effects are illustrated to lead to the energy strengthening in the neutrino spectra. As this paper demonstrates, such a property is favorable for the possibility of observing supernova neutrino fluxes using Large Volume Neutrino Telescopes.

Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

ABSTRACT We investigate the evolution of the phase–space distribution function around slightly perturbed stationary states and the process of violent relaxation in the context of the dissipationless collapse of an isolated spherical self-gravitating system. By means of the recently introduced symplectic coarse graining technique, we obtain an effective evolution equation that allows us to compute the scaling of the frequencies around a stationary state, as well as the damping times of Fourier modes of the distribution function, with the magnitude of the Fourier k −vectors themselves. We compare our analytical results with N-body simulations.

Статистический метод частиц для решения фононного уравнения Больцмана

The propagation of heat in a crystal is considered as a process of transport of phonons – quasi-particles with quasi-momentum and energy. The Boltzmann kinetic equation is constructed for the phonon’s distribution function in the phase space. The scattering of phonons is modeled in the approximation of the time of relaxation of their distribution to the equilibrium state. The numerical algorithm for solving the kinetic equation is based on the statistical method of particles, which combines the solution of the phonons motion equations with stochastic modeling of their creation and annihilation. The results of the numerical solution of the problems of temperature relaxation in a crystal during heating of its surface and energy release in the volume are considered.

Simulation study of energetic-particle driven off-axis fishbone instabilities in tokamak plasmas

Kinetic-magnetohydrodynamic hybrid simulations were performed to investigate the linear growth and the nonlinear evolution of off-axis fishbone mode (OFM) destabilized by trapped energetic ions in tokamak plasmas. The spatial profile of OFM is mainly composed of m/n = 2/1 mode inside the q = 2 magnetic flux surface while the m/n = 3/1 mode is predominant outside the q = 2 surface, where m and n are the poloidal and toroidal mode numbers, respectively, and q is the safety factor. The spatial profile of the OFM is a strongly shearing shape on the poloidal plane, suggesting the nonperturbative effect of the interaction with energetic ions. The frequency of the OFM in the linear growth phase is in good agreement with the precession drift frequency of trapped energetic ions, and the frequency chirps down in the nonlinear phase. Two types of resonance conditions between trapped energetic ions and OFM are found. For the first type of resonance, the precession drift frequency matches the OFM frequency, while for the second type, the sum of the precession drift frequency and the bounce frequency matches the OFM frequency. The first type of resonance is the primary resonance for the destabilization of OFM. The resonance frequency which is defined based on precession drift frequency and bounce frequency of the nonlinear orbit for each resonant particle is analyzed to understand the frequency chirping. The resonance frequency of the particles that transfer energy to the OFM chirps down, which may result in the chirping down of the OFM frequency. A detailed analysis of the energetic ion distribution function in phase space shows that the gradient of the distribution function along the E′ = const. line drives or stabilizes the instability, where E′ is a combination of energy and toroidal canonical momentum and conserved during the wave–particle interaction. The distribution function is flattened along the E′ = const. line in the nonlinear phase leading to the saturation of the instability.

Dynamics in condensed phase for systems involving phase functions obeying Gaussian statistics

The study of non-equilibrium processes in condensed phase has been fascinating and has attracted considerable attention of experimentalists as well as theoreticians during the last few decades. Since the theoretical treatment using Liouville equation involves multi-dimensional phase space functions, it is very difficult to visualize and also to obtain analytical expressions for the time-dependent distribution functions in Liouville space, which has led to formulation of various reduced space descriptions. The bottleneck in describing the non-equilibrium processes in the reduced space lies in the appearance of coupled velocity and position correlation functions in the equations, which are difficult to evaluate in general. In this work, we have developed a scheme to decouple the complicated multi-point coupled correlation function into two-point velocity and position correlation functions for situations characterized by a time-dependent phase function obeying Gaussian distribution, by making use of the Novikov's theorem. As an illustrative application, we have investigated the problem of optically controlled solvation dynamics and obtained exact expressions for the non-equilibrium solvation time correlation functions showing explicit dependence on the excitation frequency, as observed experimentally. We have shown that the linear response theory result is valid as long as the energy gap obeys the Gaussian distribution, rescuing the linear response theory based result from the domain of validity for small fluctuations in solvation coordinate.

Analysis of light-wave nonstaticity in the coherent state

The characteristics of nonstatic quantum light waves in the coherent state in a static environment is investigated. It is shown that the shape of the wave varies periodically as a manifestation of its peculiar properties of nonstaticity like the case of the Fock-state analysis for a nonstatic wave. A belly occurs in the graphic of wave evolution whenever the wave is maximally displaced in the quadrature space, whereas a node takes place every time the wave passes the equilibrium point during its oscillation. In this way, a belly and a node appear in turn successively. Whereas this change of wave profile is accompanied by the periodic variation of electric and magnetic energies, the total energy is conserved. The fluctuations of quadratures also vary in a regular manner according to the wave transformation in time. While the resultant time-varying uncertainty product is always larger than (or, at least, equal to) its quantum-mechanically allowed minimal value (hbar /2), it is smallest whenever the wave constitutes a belly or a node. The mechanism underlying the abnormal features of nonstatic light waves demonstrated here can be interpreted by the rotation of the squeezed-shape contour of the Wigner distribution function in phase space.

Multiphase flow simulation with three-dimensional weighted-orthogonal multiple-relaxation-time pseudopotential lattice Boltzmann model

In this paper, based on two lattice models (D3Q19 and D3Q27), two three-dimensional weighted-orthogonal multiple-relaxation-time pseudopotential lattice Boltzmann (WMRT-PLB) models with tunable thermodynamic consistency and surface tension are developed in which the high-order terms of the equilibrium density distribution function and discrete forcing term in moment space are eliminated, and thus, the implementation of the collision process is simplified. The Chapman–Enskog analysis shows that the WMRT-PLB models can correctly recover the macroscopic Navier–Stokes equations in the low Mach number limit. Then, six classical multiphase flows benchmark cases are performed to validate the performance of the proposed model. The numerical results of the first three cases indicate that the developed WMRT-PLB models effectively weaken the non-physical coupling between kinetic viscosity and density, enhance the numerical stability because of the low spurious velocity, improve the computational efficiency by about 25% because of the simplification of the collision process, and increase the numerical accuracy in the dynamic problems. Meanwhile, the numerical results of the last three cases with the density ratio of 857.7 and the kinetic viscosity ratio of 1/15 agree well with the analytical solutions and experimental results reported in the literature. Note that it is also found that the simulation of droplet bouncing is still stable even when the Reynolds number is more than 3000, which shows the good numerical stability of the proposed model. It has the potential to be applied to the simulation of the complex multiphase flows with large density ratio and large Reynolds number.

A deterministic Wigner approach for superposed states

The Wigner formalism is a convenient way of describing quantum mechanical effects through a framework of distribution functions in phase space. Currently, there are stochastic and deterministic approaches in use. In our deterministic method, the critical discretization of the diffusion term is done through the utilization of an integral formulation of the Wigner equation. This deterministic method is studied in the context of superposed quantum states as a precursor to simulations of entangled states.

Evolution of electron cyclotron waves in a Hall-type plasma

A theoretical study of high-frequency electron waves in the plasma of a Hall-type discharge with crossed electric and magnetic fields is presented. The analysis of dispersion relations for a homogeneous weakly collisional plasma with the electron velocity distribution function obtained in the relaxation approximation is performed. It is shown that the nonequilibrium character of the electron distribution function can lead to kinetic instability of cyclotron waves. This instability occurs when the thermal velocity of electrons is comparable with the drift velocity in crossed fields. In the quasilinear approximation, a compact form of the diffusion coefficient of the distribution function in the velocity space is obtained, describing unstable modes' evolution and saturation. The analytical solutions are compared with the results of numerical simulation by the particle-in-cell method.

Open quantum dynamics theory on the basis of periodical system-bath model for discrete Wigner function

Discretizing a distribution function in a phase space for an efficient quantum dynamics simulation is a non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we employ a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths. This model is an ideal platform not only for a periodic system but also for a non-periodic system confined by a potential. We then derive the numerically ''exact'' hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. As demonstrations, we numerically integrate the discrete QFPE for a 2D free rotor and harmonic potential systems in a high-temperature Markovian case using a coarse mesh with initial conditions that involve singularity.

Inferring time-dependent distribution functions from kinematic snapshots

ABSTRACT We propose a method for constructing the time-dependent phase space distribution function (DF) of a collisionless system from an isolated kinematic snapshot. In general, the problem of mapping a single snapshot to a time-dependent function is intractable. Here, we assume a finite series representation of the DF, constructed from the spectrum of the system’s Koopman operator. This reduces the original problem to one of mapping a kinematic snapshot to a discrete spectrum rather than to a time-dependent function. We implement this mapping with a convolutional neural network. The method is demonstrated on two example models: the quantum simple harmonic oscillator and a self-gravitating isothermal plane. The latter system exhibits phase space spiral structure similar to that observed in Gaia Data Release 2.

Recovery of the Two-Dimensional Ion Distribution Function in a Magnetic Mirror from Measurements of Collective Thomson Scattering Spectra

A method is proposed for tomography of the distribution function of energetic ions that are adiabatically trapped in an open magnetic trap, according to the diagnostic data by the method of collective Thomson scattering. This method is based on measurements of the scattering spectra from successive plasma cross sections corresponding to different values of the magnetic-field strength along a single line of force. It is shown that the problem of restoring the ion distribution function in the velocity space from the measurement data in this situation is reduced to an integral equation of the first kind that allows an analytical solution. Several ways to construct exact and approximate solutions of the resulting integral equation are considered.

Momentum transfer driven by fluctuations in relativistic counter-propagating electron beams

A semi-Lagrangian relativistic Vlasov–Maxwell solver is used to describe the non-linear momentum transfer between electron beams, which is driven by spontaneous fluctuations in the oblique filamentation instability (a type of Weibel mode) in two-dimensional plasmas. For counter-streaming plasmas, these filamentation instabilities may strengthen the generation of a fine structure of the distribution function in phase space. Properties of such counter-streaming beams are investigated using full-kinetic simulations and a Hamiltonian reduction technique based on the invariance of the transverse canonical momentum, the so-called multi-stream model. In the regime where , being the wavevector and d e the electron skin depth, a momentum transfer takes place when a mesoscale current filamentation mode couples with microscopic phase-space fluctuations. This regime is characterized by the appearance of filamentation modes with large wavenumbers and large amplitude fluctuations, which induce variations of the L 2-norm of the plasma in link with the self-organization of counter-streaming electron beams. We also discuss some theoretical aspects of information theory that highlight the role of the (reversible) entropy production in momentum transfer, thus by clarifying the connections between (microscopic) fluctuations of the distribution function and the fundamental aspects of the filamentation in Vlasov theory. The momentum transfer between beams is quantitatively analyzed by means of both theoretical estimations and numerical experiments.

The stellar distribution function and local vertical potential from Gaia DR2

ABSTRACT We develop a novel method to simultaneously determine the vertical potential, force, and stellar z−vz phase space distribution function (DF) in our local patch of the Galaxy. We assume that the Solar Neighbourhood can be treated as a one-dimensional (1D) system in dynamical equilibrium and directly fit the number density in the z−vz plane to what we call the rational linear distribution function (RLDF) model. This model can be regarded as a continuous sum of isothermal DFs though it has only one more parameter than the isothermal model. We apply our method to a sample of giant stars from Gaia Data Release 2 and show that the RLDF provides an excellent fit to the data. The well-known phase space spiral emerges in the residual map of the z−vz plane. We use the best-fitting potential to plot the residuals in terms of the frequency and angle of vertical oscillations and show that the spiral maps into a straight line. From its slope, we estimate that the phase spirals were generated by a perturbation ∼540 Myr years ago. We also determine the differential surface density as a function of vertical velocity dispersion, a.k.a. the vertical temperature distribution. The result is qualitatively similar to what was previously found for SDSS/SEGUE G dwarfs. Finally, we address parameter degeneracies and the validity of the 1D approximation. Particularly, the mid-plane density derived from a cold sub-sample, where the 1D approximation is more secure, is closer to literature values than that derived from the sample as a whole.

Magnetotail‐Inner Magnetosphere Transport Associated With Fast Flows Based on Combined Global‐Hybrid and CIMI Simulation

AbstractUsing the combined Auburn global hybrid simulation code in 3‐D (ANGIE3D) and the Comprehensive Inner Magnetosphere‐Ionosphere (CIMI) model, we investigate the self‐consistent connection of fast flow injections from the tail plasma sheet and the inner magnetosphere under a southward IMF. In the dynamic run, the hybrid results provide the CIMI model with 3‐D magnetic field and the electric potential at the high latitude ionosphere boundary as well as the full ion phase space distribution function at the CIMI outer boundary at the equator. The simulation shows that magnetotail reconnection, which has a dawn‐dusk size of ∼1–5 RE, recurs with a period of several minutes, resulting in recurring localized fast flow injections. Strong ion temperature anisotropy and non‐Maxwellian distributions are present in the fast flows, and the braking of fast flows due to the dipole‐like field results in further perpendicular heating in the injection sources. Multiple fast flow injections lead to multiple peaks in the particle fluxes in the inner magnetosphere as well as layers of upward and downward field‐aligned currents at the ionosphere, and low energy particles penetrate deeper radially than the high energy particles. The combined ANGIE3D‐CIMI model can be used to calculate the global kinetic physics that contains both Region‐1 and Region‐2 field‐aligned currents.