Relativistic Fokker-Planck Equation Research Articles

Relativistic Weibel instability in magneto‐inertial fusion using Krook collision model

AbstractIn this work, we studied the relativistic Weibel instability (RWI) in magneto‐inertial fusion experiment (MIFE); for this, the basic equation in our model is the relativistic Fokker–Planck equation. We solve the dispersion relation of low frequency and deduce the growth rate of RWI. The main results obtained are that the IBA source alone is capable to reduce the RWI, and also the coupling between it and the external magnetic field contributes strongly to achieving a reduction of the RWI, it has been shown clearly that the relativistic effect under this conditions can be decrease the values of growth rate of WI.

A physics-informed deep learning model of the hot tail runaway electron seed

A challenging aspect of the description of a tokamak disruption is evaluating the hot tail runaway electron seed that emerges during the thermal quench. This problem is made challenging due to the requirement of describing a strongly non-thermal electron distribution, together with the need to incorporate a diverse range of multiphysics processes, including magnetohydrodynamic instabilities, impurity transport, and radiative losses. This work develops a physics-informed neural network (PINN) tailored to the solution of the hot tail seed during an axisymmetric thermal quench. Here, a PINN is developed to identify solutions to the adjoint relativistic Fokker–Planck equation in the presence of a rapid quench of the plasma's thermal energy. It is shown that the PINN is able to accurately predict the hot tail seed across a range of parameters, including the thermal quench timescale, initial plasma temperature, and local current density, in the absence of experimental or simulation data. The hot tail PINN is verified by comparison with direct Monte Carlo simulations, with excellent agreement found across a broad range of thermal quench conditions.

Proses difusi relativistik melalui persamaan langevin dan fokker-planck

Brownian motion theory is always challenging how to describe diffusion phenomena with the main issue is how to extend the classical theory of Brownian motion to the special relativity framework. In this study, we formulated dynamics and distribution of a Brownian particle in relativistic framework by using Langevin and Fokker-Planck equation. By representing Brownian particle dynamics by Langevin equation, the velocity curves were dependent on the presence of viscous friction coefficient (heat bath), and were used generalized in special relativity theory, A relativistic Langevin equation reduces to the classical theory at low velocities. Likewise, the distribution of Brownian particles is represented as a stationary solution of the relativistic Fokker-Planck equation. From numerical results, we found that the probability density in the relativistic Fokker-Planck equation for was reduced to the standard Fokker-Planck equation in Netownian classical theory. For the calculation result showed that the Hanggi-Klimontovich approach has a consistent result to the relativistic Maxwell distribution. This work could open a promising interpretation to formulate the diffusion phenomena into general relativity theory.

DREAM: A fluid-kinetic framework for tokamak disruption runaway electron simulations

Avoidance of the harmful effects of runaway electrons (REs) in plasma-terminating disruptions is pivotal in the design of safety systems for magnetic fusion devices. Here, we describe a computationally efficient numerical tool, that allows for self-consistent simulations of plasma cooling and associated RE dynamics during disruptions. It solves flux-surface averaged transport equations for the plasma density, temperature and poloidal flux, using a bounce-averaged kinetic equation to self-consistently provide the electron current, heat, density and RE evolution, as well as the electron distribution function. As an example, we consider disruption scenarios with material injection and compare the electron dynamics resolved with different levels of complexity, from fully kinetic to fluid modes. Program summaryProgram Title:DreamCPC Library link to program files:https://doi.org/10.17632/vs3yvnrzg6.1Developer's repository link:https://github.com/chalmersplasmatheory/DREAMLicensing provisions: MITProgramming language: C++, PythonNature of problem: Self-consistently simulates the plasma evolution in a tokamak disruption, with specific emphasis on runaway electron dynamics. The runaway electrons can be simulated either as a fluid, fully kinetically, or as a mix of the two. Plasma temperature, current density, electric field, ion density and charge states are all evolved self-consistently, where kinetic non-thermal contributions are captured using an orbit-averaged relativistic electron Fokker-Planck equation, which couples to the plasma evolution. In the typical use case, the electrons are represented by two distinct populations: a cold fluid population and a kinetic superthermal population.Solution method: The system of equations is solved using a standard multidimensional Newton's method. Partial differential equations—most prominently the bounce-averaged Fokker–Planck and current diffusion equations—are discretized using a high-resolution finite volume scheme that preserves density and positivity.

The Influence of the Hydrodynamic Response of the Flare Plasma on the Kinetics of an Accelerated Electron Beam and Hard X-rays

The transfer of accelerated electrons in the plasma of flaring loops is accompanied by a hydrodynamic response of the heated plasma. The dynamics of the accelerated electron beam was determined from the solution of the one-dimensional, time-dependent relativistic Fokker-Planck equation, and the hydrodynamic response of the thermal plasma was studied by integrating a system of one-dimensional equations using the numerical RADYN code. Thus, the electron distribution function and thermal plasma parameters are changing self-consisting. In this problem the most important values are the energy flux, the duration of continuous injection, and the accelerated electrons spectrum. The most significant for the beam—thermal plasma system is the process of plasma evaporation. The upflows of the plasma (evaporation) increases the looptop plasma density comparing with steady-state value and thus reduces the Coulomb mean free path of fast electrons over time. For impulsive events, we consider heating plasma by electron beams with the energy flux of 3 × 1010, 1011 erg cm–2 s–1 in the energy band from 25 keV up to 10 MeV, and duration of 20 s, which is quite consistent with M-X GOES class flares. It is shown that for impulsive events, the effect of plasma “evaporation” for the specified values of the energy flux, isotropic pitch-angle distributions of the accelerated electrons, and power–low energy spectra leads to an increase of the plasma density in the magnetic loop on one-two orders near the transition region only. The effect of plasma evaporation is insignificant and does not affect the hard X-rays from the top of the loop for the considered parameters.

X-ray emission reduction and photon dose lowering by energy loss of fast electrons induced by return current during the interaction of a short-pulse high-intensity laser on a metal solid target

During the interaction of a short-pulse high-intensity laser with the preplasma produced by the pulse's pedestal in front of a high-Z metal solid target, high-energy electrons are produced, which in turn create an X-ray source by interacting with the atoms of the converter target. The current brought by the hot electrons is almost completely neutralized by a return current j→ driven by the background electrons of the conductive target, and the force exerted on the hot electrons by the electric field E→ which induces Ohmic heating j→.E→, produced by the background electrons, reduces the energy of the hot electrons and thus lowers the X-ray emission and photon dose. This effect is analyzed here by means of a simple 1-D temperature model which contains the most significant terms of the relativistic Fokker-Planck equation with electron multiple scattering, and the energy equations of ions, hot, and cold electrons are then solved numerically. This Ohmic heating energy loss fraction τOh is introduced as a corrective term in an improved photon dose model. For instance, for a ps laser pulse with 10 μm spot size, the dose obtained with a tantalum target is reduced by less than about 10% to 40% by the Ohmic heating, depending upon the plasma scale length, target thickness, laser parameters, and in particular its spot size. The laser and plasma parameters may be optimized to limit the effect of Ohmic heating, for instance at a small plasma scale length or small laser spot size. Conversely, others regimes not suitable for dose production are identified. For instance, the resistive heating is enhanced in a foam target or at a long plasma scale length and high laser spot size and intensity, as the mean emission angle θ0 of the incident hot electron bunch given by the ponderomotive force is small; thus, the dose produced by a laser interacting in a gas jet may be inhibited under these circumstances. The resistive heating may also be maximized in order to reduce the X-ray emission to lower the radiation level for instance in a safety radiological goal.

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.

A study on combined effects of stochastic magnetic fluctuations and synchrotron radiation on the production of runaway electrons

The dynamics of relativistic runaway electrons are analyzed using the relativistic Fokker–Planck equation including deceleration due to the synchrotron radiation and radial diffusion loss caused by stochastic magnetic fluctuations (SMFs). SMFs are treated as friction force in (Martín-Solís et al 1999 Phys. Plasmas 6 3925). However, we think SMFs act as a ‘porter’ in configuration space, but do not directly affect the runaway electrons (REs) in momentum space. Both critical electric fields for sustainment of the existing REs and for avalanche onset are enhanced, and the modified avalanche growth rate is reduced by the combined effects of SMFs and synchrotron radiation as compared to the case with only synchrotron radiation (Aleynikov et al 2015 Phys. Rev. Lett. 114 155001).

Stabilization effect of Weibel modes in relativistic laser fusion plasma

In this work, the Weibel instability (WI) due to inverse bremsstrahlung (IB) absorption in a laser fusion plasma has been investigated. The stabilization effect due to the coupling of the self-generated magnetic field by WI with the laser wave field is explicitly shown. In this study, the relativistic effects are taken into account. Here, the basic equation is the relativistic Fokker-Planck (F-P) equation. The main obtained result is that the coupling of self-generated magnetic field with the laser wave causes a stabilizing effect of excited Weibel modes. We found a decrease in the spectral range of Weibel unstable modes. This decreasing is accompanied by a reduction of two orders in the growth rate of instable Weibel modes or even stabilization of these modes. It has been shown that the previous analysis of the Weibel instability due to IB has overestimated the values of the generated magnetic fields. Therefore, the generation of magnetic fields by the WI due to IB should not affect the experiences of an inertial confinement fusion.

Fully QED/relativistic theory of light pressure on free electrons by isotropic radiation

A relativistic/QED theory of light pressure on electrons by an isotropic, in particular blackbody, radiation predicts thermalization rates of free electrons over the entire span of energies available in the lab and the nature. The calculations based on the QED Klein–Nishina theory of electron-photon scattering and relativistic Fokker–Planck equation, show that the transition from classical (Thompson) to QED (Compton) thermalization determined by the product of electron energy and radiation temperature, is reachable under conditions for controlled nuclear fusion, and predicts large acceleration of electron thermalization in the Compton domain and strong damping of plasma oscillations at the temperatures near plasma nuclear fusion.

Simulation of hard X-ray time delays in solar flares

Hard X-ray radiation (HXR) time delays have been studied. HXR was recorded during solar flares using the BATSE spectrometer. Time-delay energy spectra were obtained for 82 flares; thе spectra were classified under three species: decreasing spectra, increasing spectra and U-shaped ones with increasing quantum energy.The spectra were derived from HXR integral over the active region. They were interpreted on the basis of a model of kinetics of accelerated electrons propagating in the flaring loop with the given plasma concentration distribution and magnetic field configuration. The kinetics in question is governed by the processes of Coulomb scattering, reflecting in the converging magnetic field, and with the return current factored in. Solving the time-dependent relativistic Fokker–Planck equation for the given initial conditions allowed to find the time-dependent electron distribution function along the loop. The brightness distribution of the bremsstrahlung of HXR derived from the electron distribution functions was calculated for different quantum energies along the flaring loop and used to plot the time-delays spectra. The calculated data showed that decreasing time-delay spectra were tractable assuming regions of electrons acceleration and injection were separated. The distinction between time-delays spectra from the looptop and footpoints was established. Hence the measurements with high resolving power may produce comprehensive data on the processes of electron transport and acceleration during solar flares.

Newtonian limit and trend to equilibrium for the relativistic Fokker-Planck equation

The relativistic Fokker-Planck equation, in which the speed of light c appears as a parameter, is considered. It is shown that in the limit c → ∞ its solutions converge in L1 to solutions of the non-relativistic Fokker-Planck equation, uniformly in compact intervals of time. Moreover in the case of spatially homogeneous solutions, and provided the temperature of the thermal bath is sufficiently small, exponential trend to equilibrium in L1 is established. The dependence of the rate of convergence on the speed of light is estimated. Finally, it is proved that exponential convergence to equilibrium for all temperatures holds in a weighted L2 norm.

Bremsstrahlung of relativistic electrons accelerated in solar flares: The intensity and degree of polarization

The generation of bremsstrahlung by relativistic electrons accelerated in solar flares is considered. The electron distribution function is calculated numerically using the relativistic Fokker-Planck equation. The intensity and degree of polarization of hard X rays are calculated as functions of the energy, observation angle, and electron distributions in pitch angle and energy. The degree of polarization does not exceed 40% and depends most strongly on the observation angle and hardness of the electron energy spectra. The results of model computation are compared with data for solar flares observed July 23, 2002 and October 28, 2003.

Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations

A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the velocity field, the energy function and the diffusion matrix, it is shown that global regular solutions converge in time to equilibrium with exponential rate. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by Villani. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by Bakry and Emery. The result applies to the relativistic Fokker-Planck equation in the low temperature regime, for which exponential trend to equilibrium was previously unknown.

Leading relativistic corrections to the Kompaneets equation

We calculate the first relativistic corrections to the Kompaneets equation for the evolution of the photon frequency distribution brought about by Compton scattering. The Lorentz invariant Boltzmann equation for electron–photon scattering is first specialized to isotropic electron and photon distributions, the squared scattering amplitude and the energy–momentum conserving delta function are each expanded to order v4/c4, averages over the directions of the electron and photon momenta are then carried out, and finally an integration over the photon energy yields our Fokker–Planck equation. The Kompaneets equation, which involves only first- and second-order derivatives with respect to the photon energy, results from the order v2/c2 terms, while the first relativistic corrections of order v4/c4 introduce third- and fourth-order derivatives. We emphasize that our result holds when neither the electrons nor the photons are in thermal equilibrium; two effective temperatures characterize a general, non-thermal electron distribution. When the electrons are in thermal equilibrium our relativistic Fokker–Planck equation is in complete agreement with the most recent published results, but we both disagree with older work.

On a relativistic Fokker-Planck equation in kinetic theory

A relativistic kinetic Fokker-Planck equation that has been recently proposed in the physical literature is studied. It is shown that, in contrast to other existing relativistic models, the one considered in this paper is invariant under Lorentz transformations in the absence of friction. A similar property (invariance by Galilean transformations in the absence of friction) is verified in the non-relativistic case. In the first part of the paper some fundamental mathematical properties of the relativistic Fokker-Planck equation are established. In particular, it is proved that the model is compatible with the finite propagation speed of particles in relativity. In the second part of the paper, two non-linear relativistic mean-field models are introduced. One is obtained by coupling the relativistic Fokker-Planck equation to the Maxwell equations of electrodynamics, and is therefore of interest in plasma physics. The other mean-field model couples the Fokker-Planck dynamics to a relativistic scalar theory of gravity (the Nordström theory) and is therefore of interest in gravitational physics. In both cases the existence of steady states for all possible prescribed values of the mass is established. In the gravitational case this result is better than for the corresponding non-relativistic model, the Vlasov-Poisson-Fokker-Planck system, for which existence of steady states is known only for small mass.

Fast electron energy deposition in a magnetized plasma: Kinetic theory and particle-in-cell simulation

The collisional dynamics of a relativistic electron jet in a magnetized plasma are investigated within the framework of kinetic theory. The relativistic Fokker–Planck equation describing slowing down, pitch angle scattering, and cyclotron rotation is derived and solved. Based on the solution of this Fokker–Planck equation, an analytical formula for the root mean square spot size transverse to the magnetic field is derived and this result predicts a reduction in radial transport. Some comparisons with particle-in-cell simulation are made and confirm striking agreement between the theory and the simulation. For fast electron with 1 MeV typical kinetic energy interacting with a solid density hydrogen plasma, the energy deposition density in the transverse direction increases by a factor 2 for magnetic field of the order of 1 T. Along the magnetic field, the energy deposition profile is unaltered compared with the field-free case.

The propagation and deposition of fast muons in dense DT mixture

The propagation of the fast muon population mainly due to collisional effect in a dense deuterium–tritium (DT for short) mixture is investigated and analysed within the framework of the relativistic Fokker–Planck equation. Without the approximation that the muons propagate straightly in the DT mixture, the muon penetration length, the straggling length, and the mean transverse dispersion radius are calculated for different initial energies, and especially for different densities of the densely compressed DT mixture in our suggested muon-driven fast ignition (FI). Unlike laser-driven FI requiring super-high temperature, muons can catalyze DT fusion at lower temperatures and may generate an ignition sparkle before the self-heating fusion follows. Our calculation is important for the feasibility and the experimental study of muon-driven FI.

Anomalous diffusion in relativistic heavy-ion collisions

On the basis of non-ideal plasma effects, due to strongly coupled plasma in the early stage of relativistic heavy-ion collisions, we show that the broad rapidity shape measured at RHIC can be very well reproduced in the framework of a non-linear relativistic Fokker-Planck equation which incorporates anomalous diffusion effects.

Signals of non-extensive statistical mechanics in high energy nuclear collisions

We investigate, from a phenomenological point of view, the relevance of non-conventional statistical mechanics effects on the rapidity spectra of net proton yield at AGS, SPS and RHIC. We show that the broad rapidity shape measured at RHIC can be very well reproduced in the framework of a non-linear relativistic Fokker–Planck equation which incorporates non-extensive statistics and anomalous diffusion.