Thermal Distribution Function Research Articles

A ballistic transport model for electronic excitation following particle impact

Abstract We present a ballistic model for the transport of electronic excitation energy induced by keV particle bombardment onto a solid surface. Starting from a free electron gas model, the Boltzmann transport equation (BTE) is employed to follow the evolution of the temporal and spatial distribution function f ( r → , k → , t ) describing the occupation probability of an electronic state k → at position r → and time t. Three different initializations of the distribution function are considered: i) a thermal distribution function with a locally and temporally elevated electron temperature, ii) a peak excitation at a specific energy above the Fermi level with a quasi-isotropic distribution in k-space and iii) an anisotropic peak excitation with k-vectors oriented in a specific transport direction. While the first initialization resembles a distribution function which may, for instance, result from electronic friction of moving atoms within an ion induced collision cascade, the peak excitation can in principle result from an autoionization process after excitation in close binary collisions. By numerically solving the BTE, we study the electronic energy exchange along a one dimensional transport direction to obtain a time and space resolved excitation energy distribution function, which is then analyzed in view of general transport characteristics of the chosen model system.

Fixed points and flow analysis on off-equilibrium dynamics in the boson Boltzmann equation

We consider fixed points of steady solutions and flow directions using the boson Boltzmann equation that is a one-dimensionally reduced kinetic equation after the angular integration. With an elastic collision integral of the two-to-two scattering process, in the dense (dilute) regime where the distribution function is large (small), the boson Boltzmann equation has approximate fixed points with a power-law spectrum in addition to the thermal distribution function. We argue that the power-law fixed point can be exact in special cases. We elaborate a graphical presentation to display evolving flow directions similarly to the renormalization group flow, which explicitly exhibits how fixed points are connected and parameter space is separated by critical lines. We discuss that such a flow diagram contains useful information on thermalization processes out of equilibrium.

Mesoscopic pore-scale simulations of natural convection of porous media in closed square cavity by using LBM

A lattice Boltzmann model is applied to simulate natural convection of porous media in closed square cavity. The numerical simulation is conducted to investigate the influences of porosity on fluid flows and the effect of thermal conductivity ratio of solid matrix to fluid on heat transfer. Micro X-CT has been used to obtain 2D optical image of various slice with different location. A construction porous model has been established accurately by 2D optical micro X-CT image of porous media. A fluid –solid conjugate heat transfer model has been applied to the established construction porous model by using Lattice Boltzmann method.This paper confirms the reliability and the computational efficiency of the lattice Boltzmann method in simulating natural convection in porous media at mesoscopic pore-scale.The temperature field is simulated by a simplified thermal energy distribution function which neglects the compression work done by the pressure and the viscous heat dissipation. A comprehensive parametric study of natural convective flows is carried out for various values of Rayleigh number.

Beyond the thermal model in relativistic heavy-ion collisions

Deviations from thermal distribution functions of produced particles in relativistic heavy-ion collisions are discussed as indicators for nonequilibrium processes. The focus is on rapidity distributions of produced charged hadrons as functions of collision energy and centrality which are used to infer the fraction of produced particles from a central fireball as compared to the one from the fragmentation sources that are out of equilibrium with the rest of the system. Overall thermal equilibrium would only be reached for large times t -> infinity.

Electrostatic analyzer measurements of ionospheric thermal ion populations

AbstractWe define the observational parameter regime necessary for observing low‐altitude ionospheric origins of high‐latitude ion upflow/outflow. We present measurement challenges and identify a new analysis technique which mitigates these impediments. To probe the initiation of auroral ion upflow, it is necessary to examine the thermal ion population at 200–350 km, where typical thermal energies are tenths of eV. Interpretation of the thermal ion distribution function measurement requires removal of payload sheath and ram effects. We use a 3‐D Maxwellian model to quantify how observed ionospheric parameters such as density, temperature, and flows affect in situ measurements of the thermal ion distribution function. We define the viable acceptance window of a typical top hat electrostatic analyzer in this regime and show that the instrument's energy resolution prohibits it from directly observing the shape of the particle spectra. To extract detailed information about measured particle population, we define two intermediate parameters from the measured distribution function, then use a Maxwellian model to replicate possible measured parameters for comparison to the data. Liouville's theorem and the thin‐sheath approximation allow us to couple the measured and modeled intermediate parameters such that measurements inside the sheath provide information about plasma outside the sheath. We apply this technique to sounding rocket data to show that careful windowing of the data and Maxwellian models allows for extraction of the best choice of geophysical parameters. More widespread use of this analysis technique will help our community expand its observational database of the seed regions of ionospheric outflows.

A public code for general relativistic, polarised radiative transfer around spinning black holes

Ray tracing radiative transfer is a powerful method for comparing theoretical models of black hole accretion flows and jets with observations. We present a public code, grtrans, for carrying out such calculations in the Kerr metric, including the full treatment of polarised radiative transfer and parallel transport along geodesics. The code is written in Fortran 90 and efficiently parallelises with OpenMP, and the full code and several components have Python interfaces. We describe several tests which are used for verifiying the code, and we compare the results for polarised thin accretion disc and semi-analytic jet problems with those from the literature as examples of its use. Along the way, we provide accurate fitting functions for polarised synchrotron emission and transfer coefficients from thermal and power law distribution functions, and compare results from numerical integration and quadrature solutions of the polarised radiative transfer equations. We also show that all transfer coefficients can play an important role in predicted images and polarisation maps of the Galactic center black hole, Sgr A*, at submillimetre wavelengths.

The dark matter distribution function and halo thermalization from the Eddington equation in galaxies

We find the distribution function [Formula: see text] for dark matter (DM) halos in galaxies and the corresponding equation of state from the (empirical) DM density profiles derived from observations. We solve for DM in galaxies the analogous of the Eddington equation originally used for the gas of stars in globular clusters. The observed density profiles are a good realistic starting point and the distribution functions derived from them are realistic. We do not make any assumption about the DM nature, the methods developed here apply to any DM kind, though all results are consistent with warm dark matter (WDM). With these methods we find: (i) Cored density profiles behaving quadratically for small distances [Formula: see text] produce distribution functions which are finite and positive at the halo center while cusped density profiles always produce divergent distribution functions at the center. (ii) Cored density profiles produce approximate thermal Boltzmann distribution functions for [Formula: see text] where [Formula: see text] is the halo radius. (iii) Analytic expressions for the dispersion velocity and the pressure are derived yielding at each halo point an ideal DM gas equation of state with local temperature [Formula: see text]. [Formula: see text] turns out to be constant in the same region where the distribution function is thermal and exhibits the same temperature within the percent. The self-gravitating DM gas can thermalize despite being collisionless because it is an ergodic system. (iv) The DM halo can be consistently considered at local thermal equilibrium with: (a) a constant temperature [Formula: see text] for [Formula: see text], (b) a space dependent temperature [Formula: see text] for [Formula: see text], which slowly decreases with [Formula: see text]. That is, the DM halo is realistically a collisionless self-gravitating thermal gas for [Formula: see text]. (v) [Formula: see text] outside the halo radius nicely follows the decrease of the circular velocity squared.

Electromagnetic fluctuations in magnetized plasmas II: Extension of the theory for parallel wave vectors

Starting from the recently derived general expressions for the electromagnetic fluctuation spectra (electric and magnetic field) from uncorrelated plasma particles in plasmas with an uniform magnetic field, the case of strictly parallel (k⊥=0) oriented wave vectors with the respect to the uniform magnetic field direction is investigated. To derive fluctuation spectra valid in the entire complex frequency plane, the relevant dispersion functions and form factors are analytically continued to negative values of the imaginary part of the frequency for arbitrary gyrotropic plasma particle distribution functions. The generalized fluctuation-dissipation theorems for non-collective fluctuations in isotropic equal-temperature thermal distribution functions for general complex values of the frequency of the fluctuations with parallel wave vectors are derived.

Bernstein instability driven by thermal ring distribution

The classic Bernstein waves may be intimately related to banded emissions detected in laboratory plasmas, terrestrial, and other planetary magnetospheres. However, the customary discussion of the Bernstein wave is based upon isotropic thermal velocity distribution function. In order to understand how such waves can be excited, one needs an emission mechanism, i.e., an instability. In non-relativistic collision-less plasmas, the only known Bernstein wave instability is that associated with a cold perpendicular velocity ring distribution function. However, cold ring distribution is highly idealized. The present Brief Communication generalizes the cold ring distribution model to include thermal spread, so that the Bernstein-ring instability is described by a more realistic electron distribution function, with which the stabilization by thermal spread associated with the ring distribution is demonstrated. The present findings imply that the excitation of Bernstein waves requires a sufficiently high perpendicular velocity gradient associated with the electron distribution function.

On the Physical Basis of Self-Organization

Experiments performed with the aim to explain pattern formation in plasma devices offer, as I will show in this survey, a new insight into the mechanism by which locally matter transits spontaneously from a disordered state into an ordered one. The essential news revealed by these experiments is the identification of a population of electrons that, driven at a critical distance from thermal equilibrium, is able to act as the organizer of the emergence and the survival of a complexity starting from chaos, i.e., from electric sparks the appearance of which is controlled by deterministic chaos. Supplied at a constant rate with thermal energy extracted by electrons from plasma, the complexity survives in a dynamical state performing operations in agreement with a code directly related to electrons thermal energy distribution function. Acting as a constituent of the matter, the population of electrons intrinsically controls the emergence and the survival of the complexity. Performing operations directly related to electron’s thermal energy distribution function, the complexity evolves stepwise in more advanced self-organized dynamical states, when this function is changed by an additional injection of energy. A set of nonlinear phenomena, not explainable by classical processes is involved in the mechanism by which the complexity emerges, survives and evolves. Thus, phenomena like Bose-Einstein condensation, macroscopic quantum coherence, direct and alternate Josephson effects, electron tunneling, negative differential impedance and others, potentially explain the emergence, functionality and vitality, i.e., the dynamical state of the complexity.

Effective nucleon mass and thermodynamics in the quark-meson coupling model

In this work, we study the quark energy spectrum in an MIT bag using the quark-meson coupling model. In particular, thermal distribution functions of quarks at finite temperature are explicitly incorporated into the calculations. It is found that when temperature is less than 50 MeV, the probability for a quark to occupy an exciting state is very rare and negligible, therefore the corresponding contributions to the thermodynamic potential undermean-field theory are strongly suppressed. In this context, if one calculates other physical quantities derived from thermodynamic potential, effects due to quark distribution functions are also likely to be unimportant. DOI: 10.1103/PhysRevC.86.068201

Instability Analysis of the Bipolar Plate for Welding Fuel Cell

The instability of the bipolar plate for welding fuel cell was analyzed taking the bipolar plate as thin plate. The minimum compressive stress is calculated based on thin plate instability theory caused by longitudinal compressive stress. The simplified thermal stress distribution function is determined using the analytical solution of the plane heat conduction equation. The maximum compressive stress is determined through the analysis of the longitudinal stress distribution of the cross section passing through the solder point during heating and cooling process. The results are used to obtain the pre-tensile stress controlling the instability of the bipolar plate for welding fuel cell, and the minimum pre-tensile stress is the difference between the maximum compressive stress and the minimum compressive stress as the thin plate loses instability.

Study of End Surfaces’ Deformation and Thermal Stress in Solid State Laser

High power solid state lasers have wide applications; they have extensively attracted many attentions. When laser pumped, a lot of heat produces in it causes thermal effects such as stresses end effect and variations of refraction index. In this paper, we analytically study the thermal effects in an Nd: YAG solid state laser crystal which is pumped in longitude with a second order Gaussian (super Gaussian) profile. In this survey we neglect z derivatives of temperature distribution and also consider a statement in which thermal condition of cooler system is much more than thermal condition of laser crystal. Then we apply proper boundary conditions to probe thermal distribution function and related thermal effects. It will be shown that use of second order Gaussian profile for output beam of diode lasers is a better approximation for calculating the thermal distribution and estimating the thermal lensing than the Gaussian profile.

Mixed Boundary Conditions for Two-Dimensional Transient Heat Transfer Conduction under Lattice Boltzmann Simulations

In this paper, lattice Boltzmann implementations of several types of boundary conditions are introduced and numerically demonstrated. A thermal lattice BGK model is used to simulate thermal fields for flows. The unknown thermal distribution functions at the boundary are subjected to the bounce back concept which is determined consistently with Dirichlet and/or Neumann and/or convective boundary conditions. A consistency analysis using heat transfer conduction is given and the algorithms are numerically tested in two space dimensions with respect to accuracy, numbers of iterations and CPU time. The method is used to simulate conduction transfer problems; numerical results and reference's solutions are found in satisfactory agreement for thermal fields.

A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

AbstractThe objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number(Kn);thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation. For aerodynamic flows, it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.

Novel Hamiltonian method for collective dynamics analysis of an intense charged particle beam propagating through a periodic focusing quadrupole lattice

Identifying regimes for quiescent propagation of intense beams over long distances has been a major challenge in accelerator research. In particular, the development of systematic theoretical approaches that are able to treat self-consistently the applied oscillating force and the nonlinear self-field force of the beam particles simultaneously has been a major challenge of modern beam physics. In this paper, the recently developed Hamiltonian averaging technique [E. A. Startsev, R. C. Davidson, and M. Dorf, Phys. Rev. ST Accel. Beams 13, 064402 (2010)] which incorporates both the applied periodic focusing force and the self-field force of the beam particles, is generalized to the case of time-dependent beam distributions. The new formulation allows not only a determination of quasi-equilibrium solutions of the non-linear Vlasov-Poison system of equations but also a detailed study of their stability properties. The corrections to the well-known “smooth-focusing” approximation are derived, and the results are applied to a matched beam with thermal equilibrium distribution function. It is shown that the corrections remain small even for moderate values of the vacuum phase advance συ. Nonetheless, because the corrections to the average self-field potential are non-axisymmetric, the stability properties of the different beam quasi-equilibria can change significantly.

Boltzmann Limit for a Homogeneous Fermi Gas with Dynamical Hartree-Fock Interactions in a Random Medium

We study the dynamics of the thermal momentum distribution function for an interacting, homogeneous Fermi gas on ℤ3 in the presence of an external weak static random potential, where the pair interactions between the fermions are modeled in dynamical Hartree-Fock theory. We determine the Boltzmann limits associated to different scaling regimes defined by the size of the random potential, and the strength of the fermion interactions.

Azimuthal anisotropy: Transition from hydrodynamic flow to jet suppression

Measured 2nd and 4th azimuthal anisotropy coefficients v_{2,4}(N_{part}), p_T) are scaled with the initial eccentricity \varepsilon_{2,4}(N_{part}) of the collision zone and studied as a function of the number of participants N_{part} and the transverse momenta p_T. Scaling violations are observed for $p_T \alt 3$ GeV/c, consistent with a $p_T^2$ dependence of viscous corrections and a linear increase of the relaxation time with $p_T$. These empirical viscous corrections to flow and the thermal distribution function at freeze-out constrain estimates of the specific viscosity and the freeze-out temperature for two different models for the initial collision geometry. The apparent viscous corrections exhibit a sharp maximum for $p_T \agt 3$ GeV/c, suggesting a breakdown of the hydrodynamic ansatz and the onset of a change from flow-driven to suppression-driven anisotropy.

Identification of prominence ejecta by the proton distribution function and magnetic fine structure in interplanetary coronal mass ejections in the inner heliosphere

This work presents in situ solar wind observations of three magnetic clouds (MCs) that contain cold high‐density material when Helios 2 was located at 0.3 AU on 9 May 1979, 0.5 AU on 30 March 1976, and 0.7 AU on 24 December 1978. In the cold high‐density regions embedded in the interplanetary coronal mass ejections we find (1) that the number density of protons is higher than in other regions inside the magnetic cloud, (2) the possible existence of He+, (3) that the thermal velocity distribution functions are more isotropic and appear to be colder than in the other regions of the MC, and the proton temperature is lower than that of the ambient plasma, and (4) that the associated magnetic field configuration can for all three MC events be identified as a flux rope. This cold high‐density region is located at the polarity inversion line in the center of the bipolar structure of the MC magnetic field (consistent with previous solar observation work that found that a prominence lies over the neutral line of the related bipolar solar magnetic field). Specifically, for the first magnetic cloud event on 8 May 1979, a coronal mass ejection (CME) was related to an eruptive prominence previously reported as a result of the observation of Solwind (P78‐1). Therefore, we identify the cold and dense region in the MC as the prominence material. It is the first time that prominence ejecta were identified by both the plasma and magnetic field features inside 1 AU, and it is also the first time that the thermal ion velocity distribution functions were used to investigate the microstate of the prominence material. Moreover, from our three cases, we also found that this material tended to fall behind the magnetic cloud and become smaller as it propagated farther away from the Sun, which confirms speculations in previous work. Overall, our in situ observations are consistent with three‐part CME models.

Condensation phase diagram of cavity polaritons in GaN-based microcavities: Experiment and theory

The evolution of the polariton condensation threshold $({P}_{thr})$ under incoherent optical pumping is investigated both theoretically and experimentally over a wide range of temperatures (4--340 K) and exciton-cavity photon detunings $(\ensuremath{-}120--0\text{ }\text{meV})$ in a multiple quantum-well GaN-based microcavity. The condensation phase diagram of these bosonic quasiparticles is first theoretically described within the framework of Bose-Einstein condensation of polaritons in the thermodynamic limit. Then a qualitative picture of cavity polariton relaxation kinetics including the impact of detuning and temperature is given before introducing a modeling of cavity polariton relaxation kinetics with semiclassical Boltzmann equations. The results of the theoretical modeling are finally compared with systematic measurements of ${P}_{thr}$. At low temperature and negative detunings, the polariton gas is far from thermal equilibrium and the condensation threshold is governed by the efficiency of the relaxation kinetics of the particles. Conversely, at high temperature and for less negative detunings, the relaxation kinetics is efficient enough to allow the achievement of a thermal polariton distribution function with a critical density for condensation given by the thermodynamic theory of Bose-Einstein condensation. For temperatures ranging between $\ensuremath{\sim}140$ and 340 K, an optimum detuning is found experimentally, where the condensation threshold power is minimized. At high temperatures, polariton detrapping effects from the bottom of the trap formed in ${k}_{\ensuremath{\parallel}}$ space by the lower polariton branch are found to play a supplementary role among the processes governing ${P}_{thr}$.