Equilibrium Particle Distribution Research Articles

The effect of the diamagnetic drift on geodesic acoustic modes in tokamak plasmas

The effect of the radial derivative of the equilibrium particle distribution, i.e., the diamagnetic drift effect, on geodesic acoustic modes is taken into account in this work. The effect was routinely neglected in the previous studies on the geodesic acoustic mode since the dominant mode number is m/n=0/0. However, for finite electron temperatures, small m ≠ 0 side bands are present and the diamagnetic drift effect enters through these side bands. In this work, we find that the mode frequencies increase with the particle density gradient. The temperature ratio between electrons and ions, i.e., τ=Te/Ti, is a key parameter influencing this effect. The effect is more prominent for higher τ values. Another effect is the symmetry breaking of the propagating direction of the sideband potentials. In contrast to the pure standing wave form, the potential perturbation consists of a standing wave superimposed with a small amplitude traveling wave, which is nearly proportional to the density gradient.

Cluster Variation for Spatially Distributed Heterogeneous Systems

Fundamentals of cluster variation (CV) are developed for locally heterogeneous spatially distributed systems. The theory is based on the principles of homogeneous CV in which all variants of the location of the basis cluster on a heterogeneous lattice are additionally considered when it is translated over the system. The structure of the statistical sum of homogeneous CV is shown to remain upon moving to a heterogeneous or homogeneous spatially distributed lattice. However, cofactors of the statistical sum, which previously corresponded to homogeneous clusters, must now consider all arrangements of heterogeneous sites inside each cluster. The general approach is to use a layered structure of the transitional region with variable density between vapor and fluid on a planar square lattice. Explicit expressions for a heterogeneous statistical sum of the transitional region are given on the basis of a 3 × 3 cluster. Using a 2 × 2 cluster, it is shown how an explicit equation for the equilibrium particle distribution in the transitional region can be obtained from the heterogeneous statistical sum. A gradual increase in the size of the m × n basis cluster in the transitional region converges to the exact solution.

Surface Tension of the Planar Interface of a Vapor–Liquid System on a Two-Dimensional Square Lattice

In an earlier work, the authors described a theoretical approach to deriving equations for equilibrium particle distributions by means of heterogeneous cluster variation (CV) that converges to the exact solution as the basis cluster grows. In this work, they present calculations of the surface tension (ST) of the planar interface of a vapor-liquid system on a two-dimensional square lattice by means of heterogeneous CV. The transitional region of the interface is a sequence of monomolecular layers with a variable fluid density fluidа. Calculations are made for six types of clusters with different sizes inside the phases (2 × n, n = 1–4, 3 × 3, and the k1s cluster with the nearest neighbors of any central site), and for eight clusters inside the transitional region (2 × 1, 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 3 × 3, k1s) that differ by each cluster’s orientation relative to the normal to the surface. As the clusters grow, so does the accuracy of describing indirect correlations of laterally interacting particles. The temperature dependence of the ST is calculated. A monotonically growing ST is obtained as the temperature falls, starting from zero at the critical temperature. The calculation results converge to the exact Onsager solution as the clusters grow. Differences between thermodynamic requirements and ST calculations performed with the Ising model are discussed.

Cluster Variation for Spatially Distributed Heterogeneous Systems

Fundamentals of cluster variation (CV) are developed for locally heterogeneous spatially distributed systems. The theory is based on the principles of homogeneous CV in which all variants of the location of the basis cluster on a heterogeneous lattice are additionally considered when it is translated over the system. The structure of the statistical sum of homogeneous CV is shown to remain upon moving to a heterogeneous or homogeneous spatially distributed lattice. However, cofactors of the statistical sum, which previously corresponded to homogeneous clusters, must now consider all arrangements of heterogeneous sites inside each cluster. The general approach is to use a layered structure of the transitional region with variable density between vapor and fluid on a planar square lattice. Explicit expressions for a heterogeneous statistical sum of the transitional region are given on the basis of a 3 × 3 cluster. Using a 2 × 2 cluster, it is shown how an explicit equation for the equilibrium particle distribution in the transitional region can be obtained from the heterogeneous statistical sum. A gradual increase in the size of the m × n basis cluster in the transitional region converges to the exact solution.

Balancing turbulent heating with radiative cooling in blazars

ABSTRACT Recently, particle-in-cell (PIC) simulations have shown that relativistic turbulence in collisionless plasmas can result in an equilibrium particle distribution function where turbulent heating is balanced by radiative cooling of electrons. Strongly magnetized plasmas are characterized by higher energy peaks and broader particle distributions. In relativistically moving astrophysical jets, it is believed that the flow is launched Poynting flux dominated and that the resulting magnetic instabilities may create a turbulent environment inside the jet, i.e. the regime of relativistic turbulence. In this paper, we extend previous PIC simulation results to larger values of plasma magnetization by linearly extrapolating the diffusion and advection coefficients relevant for the turbulent plasmas under consideration. We use these results to build a single-zone turbulent jet model that is based on the global parameters of the blazar emission region, and consistently calculate the particle distribution and the resulting emission spectra. We then test our model by comparing its predictions with the broad-band quiescent emission spectra of a dozen blazars. Our results show good agreement with observations of low synchrotron peaked (LSP) sources and find that LSPs are moderately Poynting flux dominated with magnetization 1 ≲ σ ≲ 5, have bulk Lorentz factor Γj ∼ 10–30, and that the turbulent region is located at the edge, or just beyond the broad-line region (BLR). The turbulence is found to be driven at an area comparable to the jet cross-section.

Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity.

How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.

Heating and Nonequilibrium Distributions of Ions in a Reverse Shock Wave of the SN 1987A Remnant

A hydrodynamical description of supernova remnants is based on the approximation of locally equilibrium particle distributions. Shock waves in supernova remnants at various stages of ejecta propagation are collisionless and form nonequilibrium particle distributions that relax slowly to quasiequilibrium distributions within a time longer than the hydrodynamic time. A kinetic model of the heating of ions behind the front of a shock wave in the SN 1987A remnant is considered with allowance for a complex chemical composition of the ejecta, and nonequilibrium distributions of ions in the vicinity of this shock wave are calculated. In addition to the quasi-Maxwellian peak, which determines the effective temperature of a given charge state of an ion, nonequilibriumdistributions of ions contain, in some cases, a suprethermal component, which may describe to the injection of ions in the process of cosmic-ray acceleration.

On the high frequency perpendicular propagating waves in ultra-relativistic fully degenerate electron plasma

Using Vlasov-Maxwell's equations, the spectra of the perpendicular propagating Bernstein wave and Extraordinary wave in ultra-relativistic fully degenerate electron plasma are studied. The equilibrium particle distribution function is assumed to be isotropic Fermian. The analysis of high frequency spectra of the waves is carried out in the weak propagation limit Ω≫k.v and in the weak magnetic field limit |ω−k.v|≫Ω and graphically observed.

Beam halo study on the electron storage ring

AbstractHalo distribution is a key topic for background study. This paper has developed an analytical method to give an estimation of beam halo distribution in storage rings. This is a creative new theory. As an example, the equilibrium particle distribution of the beam tail in the Accelerator Test Facility damping ring is calculated analytically with different emittance and different vacuum degree. The analytical results agree the measurements very well. This is a general method, which can be applied to any electron rings.

Lattice Boltzmann method for simulation of time-dependent neutral particle transport

In this paper, a novel model is proposed to investigate the neutron transport in scattering and absorbing medium. This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method (LBM) with the collision and streaming process. The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time. The fully implicit backward difference scheme is used to ensure the numerical stability, and relaxation time and equilibrium particle distribution function are obtained. To validate the new lattice Boltzmann model, the LBM formulation is tested for a homogenous media with different sources, and both transient and steady-state LBM results get a good agreement with the benchmark solutions.

Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids.

The properties of a classical simple liquid are strongly affected by the application of an external potential that supports inhomogeneity. To understand the nature of these property changes, the equilibrium particle distribution functions of the liquid have, typically, been calculated directly using either integral equation or density functional based analyses. In this study, we develop a different approach with a focus on two distribution functions that characterize the inhomogeneous liquid: the pair direct correlation function c(r1,r2) and the pair correlation function g(r1,r2). With g(r1,r2) considered to be an experimental observable, we solve the Ornstein-Zernike equation for the inhomogeneous liquid to obtain c(r1,r2), using information about the well studied and resolved g(0)(r1,r2) and c(0)(r1,r2) for the parent homogeneous ((0)) system. In practical cases, where g(r1,r2) is available from experimental data in a discrete form, the resulting c(r1,r2) is expressed as an explicit function of g(r1,r2) in a discrete form. A weaker continuous form of solution is also obtained, in the form of an integral equation with finite integration limits. The result obtained with our formulation is tested against the exact solutions for the correlation and distribution functions of a one-dimensional inhomogeneous hard rod liquid. Following the success of that test, the formalism is extended to higher dimensional systems with explicit consideration of the two-dimensional liquid.

One-step simulation of thermoacoustic waves in two-dimensional enclosures

Abstract This paper reports on a one-step simulation study of the generation and propagation of thermoacoustic waves in a two-dimensional enclosure using a finite-difference lattice-Boltzmann-type method (FDLBM) with a single relaxation time and one equilibrium particle distribution function (EPDF), and a direct aeroacoustic simulation (DAS) technique that solves the primitive Navier–Stokes (N–S) equations. Conventional expansion of the EPDF is not adopted; instead, it is expanded in terms of the particle velocity alone, and the expansion coefficients are determined by requiring the FDLBM to fully recover the N–S equations. The expansion coefficients are found to depend on the flow and thermal properties and their nonlinear interactions. Thus formulated, physical boundary conditions can be specified for the FDLBM. This EPDF has been validated against simple aeroacoustic problems and good agreement with DAS results is obtained. In this paper, the EPDF is further validated against three thermoacoustic cases: 1) a sudden increase of temperature on the left vertical wall of the enclosure; 2) a sudden increase of temperature on the left vertical wall coupled by a sudden decrease of temperature on the right vertical wall in the enclosure; 3) a gradually heated up left vertical wall of the enclosure. Comparisons with DAS simulations show that FDLBM and DAS give essentially identical results; hence, the validity and extent of the EPDF is ascertained for the calculation of these thermoacoustic waves. The FDLBM results are also compared with known theoretical and numerical results. These known solutions are found to be less accurate compared to the FDLBM results. The discrepancy could be attributed to the partially linearized equations solved in the theoretical analysis, and the inadequacy of the numerical scheme to replicate the nonlinear effects accurately in the generation and propagation of thermoacoustic waves.

Macroscopic model and truncation error of discrete Boltzmann method

A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman–Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection–diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection–diffusion–reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier–Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.

Twin tubular pinch effect in curving confined flows.

Colloidal suspensions of buoyancy neutral particles flowing in circular pipes focus into narrow distributions near the wall due to lateral migration effects associated with fluid inertia. In curving flows, these distributions are altered by Dean currents and the interplay between Reynolds and Dean numbers is used to predict equilibrium positions. Here, we propose a new description of inertial lateral migration in curving flows that expands current understanding of both focusing dynamics and equilibrium distributions. We find that at low Reynolds numbers, the ratio δ between lateral inertial migration and Dean forces scales simply with the particle radius, coil curvature and pipe radius as . A critical value δc = 0.148 of this parameter is identified along with two related inertial focusing mechanisms. In the regime below δc, coined subcritical, Dean forces generate permanently circulating, twinned annuli, each with intricate equilibrium particle distributions including eyes and trailing arms. At δ > δc (supercritical regime) inertial lateral migration forces are dominant and particles focus to a single stable equilibrium position.

Weakly nonlinear localization for a 1-D FPU chain with clustering zones

We study weakly nonlinear spatially localized solutions of a Fermi-Pasta-Ulam model describing a unidimensional chain of particles interacting with a number of neighbors that can vary from site to site. The interaction potential contains quadratic and quartic terms, and is derived from a nonlinear elastic network model proposed by Juanico et al. [1]. The FPU model can be also derived for arbitrary dimensions, under a small angular displacement assumption. The variable interaction range is a consequence of the spatial inhomogeneity in the equilibrium particle distribution. We here study some simple one-dimensional examples with only a few, well defined agglomeration regions. These agglomerations are seen to lead to spatially localized linear modes and gaps in the linear spectrum, which in turn imply a normal form that has spatially localized periodic orbits.

Analytical estimation of ATF beam halo distribution

In order to study the backgrounds in the ATF2 beam line and the interaction point (IP), this paper has developed an analytical method to give an estimation of the ATF beam halo distribution based on K. Hirata and K. Yokoya's theory. The equilibrium particle distribution of the beam tail in the ATF damping ring is presented, with each electron affected by several stochastic processes such as beam-gas scattering, beam-gas bremsstrahlung and intra-beam scattering, in addition to the synchrotron radiation damping effects. This is a general method which can also be applied to other electron rings.

Equilibrium distributions for straight, curved, and immersed boundary conditions in the lattice Boltzmann method

A versatile velocity boundary condition treatment for the lattice Boltzmann method is proposed. The method can handle straight and curved boundaries, whether immersed or not. It is based on the streaming of equilibrium particle distributions emanating from ghost nodes. The fundamental idea is the location of these ghost nodes, specifically chosen on the lattice to avoid the evaluation of non equilibrium distributions. The methodology considers that the particle collision process time on the ghost nodes is proportional to the distance from the fluid nodes missing distributions. The linear nature of the BGK operator, which relaxes distributions exactly to their equilibrium state when the collision time is equal to the relaxation time, is also a key element. The method is compared to two other classical approaches for boundary condition treatment in the lattice Boltzmann method, on four test cases for which results are available in the scientific literature. Overall, the proposed method is second order accurate, and, when compared to the other two approaches, smaller errors were observed, especially for flows with a shorter relaxation time.

Forces acting on a permanent magnet placed in a rectangular cavity with a magnetic fluid

A boundary value problem of forces acting on a permanent magnet placed in a rectangular cavity with concentrated magnetic fluid is solved by the control-volume method. The solutions available in the current literature have been obtained for dilute solutions, in which the inter-particle interactions (steric, magnetodipole and hydrodynamic) and magnetic fields generated by the magnetic fluid are inessential. Moreover, these studies neglect the magnetophoresis of colloidal particles and diffusion processes, which strongly restrict the applicability range of the obtained results. The inter-particle interactions can significantly increase the intensity of the fluid magnetization, and disregard of the magnetophoresis and particle diffusion implies that the known solutions are valid only over a limited time interval, which is short compared to the time of the onset of equilibrium particle distribution in the cavity. The main objective of this study is to estimate quantitatively the contributions of all these factors. The selected problem geometry corresponds to a simple uniaxial magnetofluidic accelerometer. The solution is searched for a two-dimensional problem using the dynamic equation of mass transfer recently introduced in J. Chem. Phys., 2011, vol. 134, 184508. The calculations have been performed to evaluate the magnetic field generated by the fluid and the field of colloidal particle concentration. The plots of the resultant force acting on the magnet versus its displacement from the equilibrium, the energy of magnetodipole interactions and volume-averaged particle concentration are presented. The basic finding of this work is the establishment of a very strong dependence of the computation results on the selection of an adequate theoretical model. In particular, it has been shown that disregard of inter-particle interactions in the fluid can lead to a computational error exceeding hundred percents.

From fractional exclusion statistics back to Bose and Fermi distributions

Fractional exclusion statistics (FES) is a generalization of the Bose and Fermi statistics. Typically, systems of interacting particles are described as ideal FES systems and the properties of the FES systems are calculated from the properties of the interacting systems. In this paper I reverse the process and I show that a FES system may be described in general as a gas of quasiparticles which obey Bose or Fermi distributions; the energies of the newly defined quasiparticles are calculated starting from the FES equations for the equilibrium particle distribution. In the end I use a system in the effective mass approximation as an example to show how the procedure works.

Ion cyclotron resonance heating with consistent finite orbit widths and anisotropic equilibria

Minority ion cyclotron resonance heating is studied using the self-consistent numerical model SCENIC. This model includes 3D geometries with full shaping and anisotropic pressure effects, warm contributions to the dielectric tensor and full orbit effects. It evolves the equilibrium, wave field and hot particle distribution function iteratively until a self-consistent solution is found. We will show applications to JET-like two-dimensional equilibria with minority heating scenarios. The effects due to different heating locations on the hot particle distribution function, the hot dielectric tensor and the equilibrium will be studied for symmetric wave injection. Finally, the RF-induced particle pinch is investigated using asymmetric wave injection.