Fokker-Planck Coefficients Research Articles

The correct cutoff for Rutherford scattering cross-section

The correct cutoff variable for the integrals with Rutherford scattering cross-section is established in this paper. The traditional cutoff variables in plasma physics, such as scattering angle θ and impact parameter b, are incorrect or just partial cutoff variables for lacking the necessary cutoff on relative speed g. The correct cutoff variable is the same variable that correctly describing the singularity of the integrals. The difference of the partial cutoff variables and the correct one is compared through their contour lines in the b − g plane. With the correct cutoff, many physical results become more simplified and structured for both rigid-sphere interaction and Coulomb interaction. These physical results include the arbitrary higher order of Fokker-Planck coefficients, transition moments, and energy transfer rates for both rigid-sphere and Coulomb interactions. All the physical results depending on velocity are expressed by a set of functions . A useful integral formula∫0∞qn(k)(ymin,u)uk+2e−u2ξdu=ξk+3221+ξk−n2∑j=0⌊k/2⌋k!Γn+k+12−j,ymin21+ξj!(k−2j)!4ξj,which associates with incomplete gamma functions is also proved. This integral formula is the key to show that the correct cutoff constants for both rigid-sphere and Coulomb interactions are all in the form of incomplete gamma functions of different orders. In particular, the so called Coulomb logarithm should be replaced by the exact form—the zeroth order incomplete Gamma function.

Fokker-Planck coefficients for a magnetized ion-electron plasma

The Fokker-Planck coefficients are developed for an ion-electron plasma in an ambient uniform magnetic field. The calculation is based on the binary collision model where a single encounter between an ion and an electron is first considered. The collision is taken as a perturbation to the free helical motion of the two particles involved, and the terms are kept up to the second order. The collision process is reformulated in terms of the changes of the perpendicular and parallel speeds and gyrophase shift. This new approach allows us to properly take into account the persisting helical motion in the magnetic field when the collisional interaction ceases.

Derivation of the magnetized Balescu-Lenard-Guernsey collision term based on the Fokker-Planck approach

The Fokker-Planck approach is employed to derive the kinetic equation for spatially uniform magnetized plasmas. By using the fluctuating electrostatic field for quiescent plasmas, the magnetized Fokker-Planck coefficients are calculated explicitly based on the wave theory which takes into account the collective effects in a proper manner. Manipulating the magnetized Fokker-Planck collision term into the Landau form, the magnetized Balescu-Lenard-Guernsey collision term is obtained, which is identical to the results derived by using the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations and the quasilinear method. The H theorem is proved in the end.

Adaptive time-stepping Monte Carlo integration of Coulomb collisions

We report an accessible and robust tool for evaluating the effects of Coulomb collisions on a test particle in a plasma that obeys Maxwell–Jüttner statistics. The implementation is based on the Beliaev–Budker collision integral which allows both the test particle and the background plasma to be relativistic. The integration method supports adaptive time stepping, which is shown to greatly improve the computational efficiency. The Monte Carlo method is implemented for both the three-dimensional particle momentum space and the five-dimensional guiding center phase space.Detailed description is provided for both the physics and implementation of the operator. The focus is in adaptive integration of stochastic differential equations, which is an overlooked aspect among existing Monte Carlo implementations of Coulomb collision operators. We verify that our operator converges to known analytical results and demonstrate that careless implementation of the adaptive time step can lead to severely erroneous results.The operator is provided as a self-contained Fortran 95 module and can be included into existing orbit-following tools that trace either the full Larmor motion or the guiding center dynamics. The adaptive time-stepping algorithm is expected to be useful in situations where the collision frequencies vary greatly over the course of a simulation. Examples include the slowing-down of fusion products or other fast ions, and the Dreicer generation of runaway electrons as well as the generation of fast ions or electrons with ion or electron cyclotron resonance heating. Program summaryProgram Title: AMCC –(A)daptive (M)onte-(C)arlo (C)oulomb collisionsProgram Files doi:http://dx.doi.org/10.17632/wd3t3t6dy7.1Licensing provisions: MIT licenceProgramming language: Fortran 95Nature of problem: Test-particle tracing is a common feat within existing fusion applications. While efficient adaptive methods exist for integrating the incompressible Hamiltonian flow, the effects of Coulomb collisions are commonly implemented with far less sophisticated algorithms.Solution method: The relativistic Fokker–Planck equation for test-particles in Maxwell–Jüttner background plasmas is converted into a stochastic differential equation. The stochastic differential equation is solved using adaptive Monte Carlo techniques. Methods to evaluate the effect of Coulomb collisions for both the three-dimensional particle momentum space and the five-dimensional reduced guiding center phase space are included.Additional comments including restrictions and unusual features: The package includes optionality to evaluate the relativistic Fokker–Planck coefficients, a feature useful for constructing accurate orbit averaged collision operators. The package also provides explicit one-step symplectic integrator for the relativistic Lorentz force that can be used for tracing test-particles in given electromagnetic backgrounds.

Solutions of the cosmic ray velocity diffusion equation

In order to describe the propagation and acceleration of cosmic rays, one usually uses a diffusive transport equation. The most fundamental equation is the pitch-angle dependent diffusion equation which is usually called the Fokker-Planck equation. In the current paper we solve the position integrated equation numerically and analytically. For a constant pitch-angle Fokker-Planck coefficient we derive an exact solution of the corresponding transport equation and compare it with numerical solutions. We show that even if the scattering coefficient is assumed to be constant, the solution behaves well. As a second example we consider the case of a linear pitch-angle Fokker-Planck coefficient. Again we solve the corresponding transport equation numerically and analytically. In all cases considered, we find similar distribution functions. We also compute the corresponding velocity correlation functions and parallel diffusion coefficients. Our results are relevant for improving analytical theories of perpendicular diffusion, for code tests, and for different astrophysical applications where a pitch-angle dependent description of the particle motion is required.

Numerical analysis of the Fokker-Planck equation with adiabatic focusing: Realistic pitch-angle scattering

We solve the focused transport equation of cosmic rays numerically to investigate non-isotropic models of the pitch-angle scattering coefficient. In previous work, the Fokker-Planck equation was solved either analytically by using approximations, or by using a numerical approach together with simple models for the pitch-angle scattering coefficient. It is the purpose of the current article so compute particle distribution functions as well as the parallel diffusion coefficient by solving numerically the focused transport equation for a more realistic Fokker-Planck coefficient of pitch-angle scattering. Our analytical form for the scattering parameter is based on non-linear diffusion theory that takes into account realistic scattering at pitch-angles close to 90°. This general form contains the isotropic form as well as the quasi-linear limit as special cases. We show that the ratio of the diffusion coefficients with and without focusing sensitively depends on the ratio of the turbulent magnetic field and the mean field. The assumed form of the pitch-angle Fokker-Planck coefficient has an influence on the parallel diffusion coefficient. In all considered cases we found a reduction of the ratio of the diffusion coefficients if the ratio of magnetic fields is reduced.

Fokker-Planck equation in the presence of a uniform magnetic field

The Fokker-Planck equation in the presence of a uniform magnetic field is derived which has the same form as the case of no magnetic field but with different Fokker-Planck coefficients. The coefficients are calculated explicitly within the binary collision model, which are free from infinite sums of Bessel functions. They can be used to investigate relaxation and transport phenomena conveniently. The kinetic equation is also manipulated into the Landau form from which it is straightforward to compare with previous results and prove the conservation laws.

DIFFUSIVE COSMIC-RAY ACCELERATION AT RELATIVISTIC SHOCK WAVES WITH MAGNETOSTATIC TURBULENCE

The analytical theory of diffusive cosmic ray acceleration at parallel stationary shock waves with magnetostatic turbulence is generalized to arbitrary shock speeds $V_s=\beta_1c$, including in particular relativistic speeds. This is achieved by applying the diffusion approximation to the relevant Fokker-Planck particle transport equation formulated in the mixed comoving coordinate system. In this coordinate system the particle's momentum coordinates $p$ and $\mu =p_{\parallel }/p$ are taken in the rest frame of the streaming plasma, whereas the time and space coordinates are taken in the observer's system. For magnetostatic slab turbulence the diffusion-convection transport equation for the isotropic (in the rest frame of the streaming plasma) part of the particle's phase space density is derived. For a step-wise shock velocity profile the steady-state diffusion-convection transport equation is solved. For a symmetric pitch-angle scattering Fokker-Planck coefficient $D_{\mu \mu }(-\mu )=D_{\mu \mu }(\mu )$ the steady-state solution is independent of the microphysical scattering details. For nonrelativistic mono-momentum particle injection at the shock the differential number density of accelerated particles is a Lorentzian-type distribution function which at large momenta approaches a power law distribution function $N(p\ge p_c)\propto p^{-\xi }$ with the spectral index $\xi (\beta_1) =1+[3/(\Gamma_1\sqrt{r^2-\beta_1^2}-1)(1+3\beta_1^2)]$. For nonrelativistic ($\beta_1\ll 1$) shock speeds this spectral index agrees with the known result $\xi (\beta_1\ll 1)\simeq (r+2)/(r-1)$, whereas for ultrarelativistic ($\Gamma_1\gg 1$) shock speeds the spectral index value is close to unity.

PITCH-ANGLE SCATTERING OF ENERGETIC PARTICLES WITH ADIABATIC FOCUSING

Understanding turbulent transport of charged particles in magnetized plasmas often requires a model for the description of random variations in the particle's pitch angle. The Fokker-Planck coefficient of pitch-angle scattering, which is used to describe scattering parallel to the mean magnetic field, is therefore of central importance. Whereas quasi-linear theory assumes a homogeneous mean magnetic field, such a condition is often not fulfilled, especially for high-energy particles. Here, a new derivation of the quasi-linear approach is given that is based on the unperturbed orbit found for an adiabatically focused mean magnetic field. The results show that, depending on the ratio of the focusing length and the particle's Larmor radius, the Fokker-Planck coefficient is significantly modified but agrees with the classical expression in the limit of a homogeneous mean magnetic field.

Detailed numerical investigation of 90∘ scattering of energetic particles interacting with magnetic turbulence

In the present paper, we re-visit a well-known problem in diffusion theory, namely the 90∘ scattering problem. We use a test-particle code to compute the pitch-angle Fokker-Planck coefficient at 90∘ for different values of the turbulent magnetic field strength and the magnetic rigidity. We consider a slab model and compare our numerical findings with the analytical result provided by second-order quasilinear theory. We show that the latter theory accurately describes 90∘ scattering. We also replace the slab model by a more realistic two-component model to explore the influence of the turbulence model on 90∘ scattering.

Diffusion Theory Applied to Tool-Life Stochastic Modeling Under a Progressive Wear Process

In this paper, a novel approach to the derivation of the tool-life distribution, when the tool useful life ends after a progressive wear process, is presented. It is based on the diffusion theory and exploits the Fokker–Planck equation. The Fokker–Planck coefficients are derived on the basis of the injury theory assumptions. That is, tool-wear occurs by detachment of small particles from the tool working surfaces, which are assumed to be identical and time-independent. In addition, they are supposed to be small enough to consider the detachment process as continuous. The tool useful life ends when a specified total volume of material is thus removed. Tool-life distributions are derived in two situations: (i) both Fokker–Planck coefficients are time-dependent only and (ii) the diffusion coefficient is neglected and the drift is wear-dependent. Theoretical results are finally compared to experimental data concerning flank wear land in continuous turning of a C40 carbon steel bar adopting a P10 type sintered carbide insert. The adherence to the experimental data of the tool-life distributions derived exploiting the Fokker–Planck equation is satisfactory. Moreover, the tool-life distribution obtained, when the diffusion coefficient is neglected and the drift is wear-dependent, is able to well-represent the wear behavior at intermediate and later times.

Pitch-angle scattering in magnetostatic turbulence

Context. Spacecraft observations have motivated the need for a refined description of the phase-space distribution function. Of particular importance is the pitch-angle diffusion coefficient that occurs in the Fokker-Planck transport equation. Aims. Simulations and analytical test-particle theories are compared to verify the diffusion description of particle transport, which does not allow for non-Markovian behavior. Methods. A Monte-Carlo simulation code was used to trace the trajectories of test particles moving in turbulent magnetic fields. From the ensemble average, the pitch-angle Fokker-Planck coefficient is obtained via the mean square displacement. Results. It is shown that, while excellent agreement with analytical theories can be obtained for slab turbulence, considerable deviations are found for isotropic turbulence. In addition, all Fokker-Planck coefficients tend to zero for high time values.

Pitch-angle scattering in magnetostatic turbulence

Aims. The process of pitch-angle isotropization is important for many applications ranging from diffusive shock acceleration to large-scale cosmic-ray transport. Here, the basic analytical description is revisited on the basis of recent simulation results. Methods. Both an analytical and a numerical investigation were undertaken of the Fokker-Planck equation for pitch-angle scattering. Additional test-particle simulations obtained with the help of a Monte-Carlo code were used to verify the conclusions. Results. It is shown that the usual definition of the pitch-angle Fokker-Planck coefficient via the mean-square displacement is flawed. The reason can be traced back to the assumption of homogeneity in time which does not hold for pitch-angle scattering. Conclusions. Calculating the mean free path via the Fokker-Planck coefficient has often proven to give an accurate description. For numerical purposes, accordingly, it is the definition that has to be exchanged in favor of the pitch-angle correlation function.

Cosmic-ray diffusion modeling: Solutions using variational methods

The diffusion of energetic particles in turbulent magnetic fields is usually described via the two-point, two-time velocity correlation function. A variational principle is used to determine the characteristic function that results from the Fourier-transformed correlation function. Both for a linear approximation and for the wave vector set to zero, explicit solutions are derived that depend on the Fokker-Planck coefficient of pitch-angle scattering. It is shown that, for an isotropic form of the Fokker-Planck coefficient, the characteristic function is divergent, which can be remedied only by using a Fokker-Planck coefficient that is finite at all pitch angles.

COSMIC-RAY TRANSPORT THEORY IN PARTIALLY TURBULENT SPACE PLASMAS WITH COMPRESSIBLE MAGNETIC TURBULENCE

Recently a new transport theory of cosmic rays in magnetized space plasmas extending the quasilinear approximation to the particle orbit has been developed for the case of an axisymmetric incompressible magnetic turbulence. Here we generalize the approach to the important physical case of a compressible plasma. As previously obtained in the case of an incompressible plasma we allow arbitrary gyrophase deviations from the unperturbed spiral orbits in the uniform magnetic field. For the case of quasi-stationary and spatially homogeneous magnetic turbulence we derive in the small Larmor radius approximation gyro-phase averaged cosmic ray Fokker-Planck coefficients. Upper limits for the perpendicular and pitch-angle Fokker-Planck coefficients and for the perpendicular and parallel spatial diffusion coefficients are presented.

A NEW COSMIC RAY TRANSPORT THEORY IN PARTIALLY TURBULENT SPACE PLASMAS: EXTENDING THE QUASILINEAR APPROACH

A new transport theory of cosmic rays in magnetized space plasmas with axisymmetric incompressible magnetic turbulence is developed extending the quasilinear approximation to the particle orbit. Arbitrary gyrophase deviations from the unperturbed spiral orbits in the uniform magnetic field are allowed. For quasi-stationary and spatially homogeneous magnetic turbulence, we derive the small Larmor radius approximation gyrophase-averaged cosmic ray Fokker-Planck coefficients. The generalized Fokker-Planck coefficients correctly reduce to their known quasilinear values in the corresponding limit. New forms of the quasilinear Fokker-Planck coefficients in axisymmetric turbulence are derived which no longer involve infinite sums of products of Bessel functions, which facilitate their numerical computation for specified turbulence field correlation tensors. The Fokker-Planck coefficients for arbitrary phase orbits of the cosmic ray particles provide strict upper limits for the perpendicular and pitch-angle Fokker-Planck coefficients, which in turn yield strict upper and lower limits for the perpendicular and parallel spatial diffusion coefficients, respectively, describing the spatial diffusion of the isotropic part of the cosmic ray phase space density. For the associated mean free paths, we find for this general case that the product of the minimum parallel mean free path with the sum of the maximum perpendicular mean free paths equals R 2 L , where RL denotes the cosmic ray gyroradius.

Velocity correlation functions of charged particles derived from the Fokker–Planck equation

An important ingredient in theories for diffusion of charged particles across a mean magnetic field are velocity correlation functions along and across that field. In the current article we present an analytical study of these functions by investigating the two-dimensional Fokker–Planck equation. We show that for an isotropic pitch-angle Fokker–Planck coefficient, the parallel velocity correlation function is an exponential function in agreement with the standard model used previously. For other forms of the pitch-angle diffusion coefficient, however, we find non-exponential forms. Also a new, velocity correlation function based, approach for deriving the so-called Earl-relation is presented. This new derivation is more systematic and simpler than previous derivations. We also discuss higher-order velocity correlations and the applicability of the quasi-normal hypothesis in particle diffusion theory. Furthermore, we compute velocity correlation functions across the mean field and develop an alternative theory for perpendicular diffusion.

Analytical reduction of pitch-angle scattering in isotropic turbulence

The scattering of energetic particles in stochastic magnetic fields cannot be described using quasi-linear theories except for simple turbulence geometries. A promising candidate for the analytical description of such scattering processes is the second-order quasi-linear theory [A. Shalchi, Phys. Plasmas 12 (2005) 052905]. For isotropic turbulence, the theory has been evaluated only semi-numerically but the results are in good agreement with numerical simulations [R.C. Tautz, A. Shalchi, R. Schlickeiser, Astrophys. J. 685 (2008) L165]. Here, using well-established methods from the summation of Bessel functions, the equation for the pitch-angle Fokker–Planck coefficient is reduced to a few single and double integrals that are left to numerical evaluation. In doing so, the evaluation of transport parameters is facilitated, thus enabling one to conduct parameter studies.

Cosmic ray transport in non-uniform magnetic fields: consequences of gradient and curvature drifts

AbstractLarge-scale spatial variations of the guide magnetic field of interplanetary and interstellar plasmas give rise to the mirror force −(p⊥2/2mγB)∇B). The parallel component of this mirror force causes adiabatic focusing of the cosmic ray guiding center whereas the perpendicular component of the mirror force gives rise to the gradient and curvature drifts of the cosmic ray guiding center. Adiabatic focusing and the gradient and curvature drift terms additionally enter the Fokker–Planck transport equation for the gyrotropic cosmic ray particle phase space density in partially turbulent non-uniform magnetic fields. For magnetohydrodynamic turbulence with dominating magnetic fluctuations, the diffusion approximation is justified, which results in a modification of the diffusion–convection transport equation for the isotropic part of the gyrotropic phase space density from the additional focusing and drift terms. For axisymmetric undamped slab Alfvenic turbulence we show that all perpendicular spatial diffusion coefficients are caused by the non-vanishing gradient and curvature drift terms. For a specific (symmetric in μ) choice of the pitch-angle Fokker–Planck coefficients we show that the ratio of the perpendicular to parallel spatial diffusion coefficients apart from a constant is determined by the spatial first derivatives of the non-constant cosmic ray Larmor radius in the non-uniform magnetic field.

PITCH-ANGLE DIFFUSION COEFFICIENTS OF CHARGED PARTICLES FROM COMPUTER SIMULATIONS

Pitch-angle diffusion is a key process in the theory of charged particle scattering by turbulent magnetic plasmas. This process is usually assumed to be diffusive and can, therefore, be described by a pitch-angle diffusion or Fokker–Planck coefficient. This parameter controls the parallel spatial diffusion coefficient as well as the parallel mean free path of charged particles. In the present paper, we determine pitch-angle diffusion coefficients from numerical computer simulations. These results are then compared with results from analytical theories. Especially, we compare the simulations with quasilinear, second-order, and weakly nonlinear diffusion coefficients. Such a comparison allows the test of previous theories and will lead to an improved understanding of the mechanism of particle scattering.