Particle Momentum Distribution Function Research Articles

THE ROLE OF HIGHER MOMENTS ON THE DISTRIBUTION OF PARTICLES IN THE SPACE OF IMPULSES AT CYCLOTRON RESONANCES

The results of the analysis of the dynamics of charged particles under conditions of cyclotron resonances in the field of an intense electromagnetic wave are presented. Particular attention is paid to regimes with dynamic chaos. It is shown that there are two qualitatively different regimes. The appearance of the first one is due to the overlap of nonlinear cyclotron resonances. The second mode is related to intermittency. The moments and spectra of each of these regimes are determined. It is shown that with an increase in the intensity of an external electromagnetic wave, the first regime appears at the beginning and only then the second regime appears. A characteristic feature of the second regime is intermittency. Steps appear on the time dynamics of pulses in the second mode. It is shown that the spectra in the second mode are narrower than in the first mode. A characteristic feature of the second regime (the regime with intermittency) is the fact that the higher moments turn out to be larger than the lower moments. In the first regime, the highest moments decrease rapidly. To find the particle momentum distribution function, the generalized Fokker-Planck equation was used. Solutions of this equation are written out for some important cases.

The wave energy density and growth rate for the resonant instability in relativistic plasmas

ABSTRACT The wave instability acts in astrophysical plasmas to redistribute energy and momentum in the absence of frequent collisions. There are many different types of waves, and it is important to quantify the wave energy density and growth rate for understanding what types of wave instabilities are possible in different plasma regimes. There are many situations throughout our Universe where plasmas contain a significant fraction of relativistic particles. Theoretical estimates for the wave energy density and growth rate are constrained to either field-aligned propagation angles or non-relativistic considerations. Based on linear theory, we derive the analytic expressions for the energy density and growth rate of an arbitrary resonant wave with an arbitrary propagation angle in relativistic plasmas. For this derivation, we calculate the Hermitian and anti-Hermitian parts of the relativistic-plasma dielectric tensor. We demonstrate that our analytic expression for the wave energy density presents an explicit energy increase in resonant waves in the wavenumber range where the analytic expression for the growth rate is positive (i.e. where a wave instability is driven). For this demonstration, we numerically analyse the loss-cone driven instability, as a specific example, in which the whistler-mode waves scatter relativistic electrons into the loss cone in the radiation belt. Our analytic results further develop the basis for linear theory to better understand the wave instability, and have the potential to combine with quasi-linear theory, which allows to study the time evolution of not only the particle momentum distribution function but also resonant wave properties through an instability.

Subdynamics of fluctuations in a many-particle quantum system

It is shown that the relevant part of an equilibrium two-time correlation function of operators related to a subsystem of S < N particles, selected from the whole system of N ≫ 1 quantum particles, obeys the exact completely closed (homogeneous) evolution equation. This equation differs from the conventional Nakajima–Zwanzig equation by the absence of the undesirable inhomogeneous term comprising all interparticle correlations at the initial time moment. The equation defines the subdynamics in a subspace of S selected particles and is obtained without any use of approximation like the Boltzmann ‘molecular chaos’, factorized initial state or Bogoliubov’s principle of weakening of initial correlations. In the second order in a small inter-particle interaction, this equation reduces to the Markovian evolution equation, which for a one-particle correlation function (S = 1) takes the form of the linear generalized quantum Boltzmann-like equation for a relevant part of a correlation function (one particle momentum distribution function) with the additional terms conditioned by initial correlations taken into account. This equation makes it possible to obtain the solution for the corresponding one-particle correlation function, which is valid for all timescales and reveals the influence of initial correlations on the evolution in time process. On a large timescale t ≫ t cor ∼ ℏ/k B T (decoherence time), the initial correlations cease to influence the damping process and the evolution enters a kinetic stage. However, the factor related to initial correlations survives on the large timescale, and can contribute to the kinetic coefficients.

A Quantum Correction to the Momentum Distribution Function of Particles

On the basis of the solution to the equation for a single-particle density matrix, the momentum distribution is obtained for a light-particle impurity placed in an ordered system of heavy particles and interacting with them with the amplitude U0. The effect of the value of U0/T0 on the functional form of the momentum distribution is investigated. It is shown that the momentum-distribution function obtained within the perturbation theory up to the terms ~(U0/T0)2 retains its form also outside the region of applicability of the perturbation theory; however, the relative magnitude of the correction is much smaller than that given by the perturbation theory.

Photon emission from a momentum-anisotropic quark-gluon plasma

We compute the photon emission rate from a quark-gluon plasma with an anisotropic particle momentum distribution induced by a non-vanishing local shear pressure tensor. Our calculation includes photon production through Compton scattering and quark-antiquark annihilation at leading order in $\alpha_s$, with all off-equilibrium corrections to leading order in the momentum anisotropy. For fermions we prove that the Kubo-Martin-Schwinger (KMS) relation holds in the hard loop regime for any particle momentum distribution function that is reflection-symmetric. This supports the equivalence, for 2 to 2 scattering processes, of the diagrammatic and kinetic approaches to calculating the photon emission rate. We compare the viscous rates from these two approaches at weak and realistic coupling strengths and provide parameterizations of the equilibrium and viscous photon emission rates for phenomenological studies in relativistic heavy-ion collisions.

The two-state Bose-Hubbard model in the hard-core boson limit: Non-ergodicity and the Bose-Einstein condensation

The Bose-Einstein condensation in the hard-core boson limit (HCB) of the Bose-Hubbard model with two local states and the particle hopping in the excited band only is investigated. For the purpose of considering the non-ergodicity, a single-particle spectral density is calculated in the random phase approximation by means of the temperature boson Green functions. The non-ergodic contribution to the momentum distribution function of particles (connected with the static density fluctuations) increases significantly and becomes comparable with the ergodic contribution in the superfluid phase near the tricritical point.

Fractional approach, quantum statistics, and non-crystalline solids at very low temperatures

We consider nonlocal effects, obtained by incorporating fractional derivatives in the kinetic energy of a conventional Hamiltonian, to analyze physical properties of non-crystalline solids at very low temperature. By using thermal Green functions, we deduce some experimentally observable quantities such as the particle momentum distribution function, the particle energy distribution function, and the specific heat. The agreement between the results obtained for the specific heat and the experimental data suggests that the approach presented here may be useful as a phenomenological model to investigate thermal properties of non-crystalline solids at low temperature.

One-particle density matrix of liquid 4He

Using the expression for the total density matrix for a system of N interacting Bose particles found in our previous papers, we calculate the one-particle density matrix in the coordinate representation. At low temperatures, the leading approximation of this matrix reproduces the results of the Bogoliubov theory. In the classical limit, the proposed theory reproduces the results of the theory of the classical liquid in the approximation of chaotic phases. From the one-particle density matrix, we find the particle momentum distribution function and the mean kinetic energy of the Bose liquid and investigate the phenomenon of Bose-Einstein condensation.

Test particle acceleration in three‐dimensional Hall MHD turbulence

Numerical experiments of test particle acceleration are performed using turbulent magnetic and electric fields obtained from direct numerical solutions of the compressible three‐dimensional Hall magnetohydrodynamic (MHD) equations. Comparisons are made of the results for the test particle momentum distribution function with and without the Hall term in the MHD solution. Electrons and protons are considered for the test particles at short times before particles leave the simulation box (with length size of the order of a few turbulent correlation lengths). The particles momentum distribution functions develop long tails in a short time. For electrons, no substantial difference it is found between results with and without the Hall term in the MHD solution, while for protons differences are seen in the low momentum part of the distributions. When a background uniform magnetic field (guide field) is added, the particle acceleration is anisotropic. Electrons develop large parallel momentum, while protons develop large perpendicular momentum. A discussion of the basic particle acceleration mechanisms in this system is made.

Quantum corrections to the particle distribution function and reaction rates in dense media

Quantum mechanics predicts the existence of power-law tails in the momentum distribution function of particles in dense media even under conditions of thermodynamic equilibrium. The generalized expressions allowing for the effect of the medium density show that quantum corrections lead to a sharp increase in the reaction rates of threshold exothermic processes (such as fusion and chemical reactions and vibrational-translational relaxation). The accompanying modification of the distribution function changes the wings of the emission and absorption lines. The profiles of the absorption lines in dense gaseous media are shown to be asymmetric with respect to the line center.

Analysis of the Synchrotron Emission from the M87 Jet

We propose that the intensity changes and spectral evolution along the M87 jet can be explained by adiabatic changes to the particle momentum distribution function and the magnetic field. This is supported by the lack of any significant variation in the radio-to-optical spectral index along the jet and the moderate changes in radio brightness. Assuming a simple scaling law between magnetic field and density, we use the deprojection of a 2 cm VLA intensity map by Sparks, Biretta, & Macchetto (1996) to predict the spectral evolution along the jet. We derive limits for the magnetic field and the total pressure by comparing our results with the spatially resolved fit to spectral data by Neumann, Meisenheimer, & Roeser (1997) of a model spectrum that cuts off at approx 10^15 Hz. To explain the weakness of synchrotron cooling along the jet, the magnetic field strength must lie below the equipartition value. Although the inferred pressure in the limit of nonrelativistic bulk flow lies far above the estimated pressure of the interstellar matter in the center of M87, bulk Lorentz factors Gamma_jet in the range of 3 - 5 and inclination angles approx less than 25 deg lead to pressure estimates close to the ISM pressure. The average best fit magnetic fields we derive fall in the range of 20 - 40 microG, departing from equipartition by a factor approx 1.5 - 5. This model is consistent with the proposal by Bicknell & Begelman (1996) that the knots in the M87 jet are weak, oblique shocks. First-order Fermi acceleration will then have a minimal effect on the slope of the radio-- to-optical spectrum while possibly accounting for the X-ray spectrum.

Drift instabilities of a relativistic plasma. Part 2. Kinetic theory of low-frequency drift instabilities of a relativistic finite-pressure plasma

The generalization of the kinetic theory of low-frequency long-wavelength drift instabilities (instabilities due to the number density and temperature gradients and an instability of the Kelvin–Helmholtz type) is given for the case of a relativistic plasma. The role of relativistic effects, taking into account the anisotropy of the particle momentum distribution function, is clarified. Particular attention is paid to the investigation of instabilities of these perturbations in a relativistic electron-positron plasma.