Self-consistent Electrostatic Field Research Articles

From the Vlasov-Poisson-Boltzmann System to the Vlasov-Poisson-Landau System with Coulomb Potential

In the celebrated work [Y. Guo., J. Amer. Math. Soc., 25 (2012), pp. 759–812], Guo proved global well-posedness of the Vlasov–Poisson–Landau (VPL) system with Coulomb potential in algebraically weighted Sobolev spaces. Thanks to this breakthrough, there are rich developments on the VPL system and also on the Vlasov–Poisson–Boltzmann (VPB) system with cutoff or noncutoff Boltzmann kernels. In this work, we are interested in the connection between the two important physical systems where particles interact through the Coulomb force and are influenced by their self-consistent electrostatic field. The Coulomb potential yields the famous Rutherford scattering cross section which is too singular in the angular variable for the Boltzmann equation to be meaningful. With an angular cutoff and a proper scaling as in [R. Alexandre and C. Villani, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 61–95], we prove global well-posedness of VPB in algebraically weighted Sobolev spaces. To this end, we develop a new two-stage energy method and merely rely on the -norm (from the dissipation of the linearized Boltzmann operator) to control the nonlinear terms containing the electrostatic field. Consequently, the same well-posedness result also holds for the VPB system with all the other soft potentials, which removes the technical constraint in [R. Duan and S. Liu, Comm. Math. Phys., 324 (2013), pp. 1–45]. Moreover, when the cutoff parameter tends to zero, convergence of VPB to VPL is established with an explicit rate.

The Vlasov–Poisson–Landau System with the Specular-Reflection Boundary Condition

We consider the Vlasov–Poisson–Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular reflection boundary condition. We provide a new improved \(L^{2}\rightarrow L^{\infty }\) framework: \(L^{2}\) energy estimate combines only with \(S_{ }^{p}\) estimate for the ultra-parabolic equation.

Accelerated steady-state electrostatic particle-in-cell simulation of Langmuir probes

First-principles particle-in-cell (PIC) simulation is a powerful tool for understanding plasma behavior, but this power often comes at great computational expense. Artificially reducing the ion/electron mass ratio is a time-honored practice to reduce simulation costs. Usually, this is a severe approximation. However, for steady-state collisionless, electrostatic (Vlasov–Poisson) systems, the solution with reduced mass ratio can be scaled to the solution for the real mass ratio, with no approximation. This “scaled mass” method, which works with already-existing PIC codes, can reduce the computation time for a large class of electrostatic PIC simulations by the square root of the mass ratio. The particle distributions of the resulting steady state must be trivially rescaled to yield the true distributions, but the self-consistent electrostatic field is independent of the mass ratio. This method is equivalent to “numerical timestepping,” an approach that evolves electron and ion populations with different time steps. Numerical timestepping can be viewed as a special case of the speed-limited PIC (SLPIC) method, which is not restricted to steady-state phenomena. Although the scaled-mass approach is simplest, numerical timestepping and SLPIC more easily generalize to include other effects, such as collisions. The equivalence of these new approaches is demonstrated by applying them to simulate a cylindrical Langmuir probe in electron–argon plasma, speeding up simulation by two orders of magnitude. Methods such as SLPIC can therefore play an invaluable role in interpreting probe measurements by including geometric effects, collisions, secondary emission, and non-Maxwellian distributions.

Simulation of the Charge Carriers Distribution in the Active Region of the P-I-N-Diodes by the Perturbation Theory Methods

A mathematical model of the electron-hole plasma stationary distribution in the active region (i-region) of p-i-n-diodes in the dif-fusion-drift approximation is proposed. The model is represented in the form of a nonlinear singularly perturbed boundary value prob-lem for the system of equations of the electron-hole currents conti-nuity, the Poisson equation and the corresponding boundary condi-tions. The decomposition of the nonlinear boundary value problem of modeling the stationary distribution of charge carriers in the plasma of p-i-n-diodes is carried out on the basis of the solutions asymptotic representation. The model problem is reduced to a se-quence of the linear boundary value problems with a characteristic separation of the main (regular) components of the asymptotics and a boundary corrections. It was found that the formulation of the problem for finding the zero term of the asymptotics regular part coincides with the classical formulation of the p-i-n-diodes charac-teristics modeling problem, which is carried out in the approxima-tion of the ambipolar diffusion (approximation of a self-consistent electrostatic field).The proposed mathematical model and the method of its linearization make it possible to determing the main components in the diffusion-drift process and to study their role. For example, it becomes possible to study (including by analytical methods) the behavior of plasma in the p-i-, n-i-contacts zones. The results of the study are aimed at developing methods for de-signing p-i-n-diode structures, used, in particular, as active ele-ments of the signals switches of a microwave data transmission systems and the corresponding protective devices.

Simulations of microturbulence in magnetised plasmas using a delta-f gyrokinetic approach with an evolving background Maxwellian

The gyrokinetic delta-f particle-in-cell (PIC) approach is known to be successful for simulating turbulence in the core of magnetic fusion plasmas, where fluctuations are relatively small and therefore the unperturbed particle distribution function, usually represented by a stationary Maxwellian f 0, remains a good choice of a control variate for reducing statistical sampling noise. However, towards the plasma edge, characterized by low density and temperature and strong gradients, relative deviation amplitudes typically become large, so that the essential assumption of |δf/f 0| << 1 underlying the delta-f PIC approach will not be valid, where δf is the fluctuating part of distribution. This motivates the study of the limits of the delta-f approach in a simplified system mimicking the plasma edge. To this end, simulations are run using GK-engine, which is a delta-f PIC code that solves the nonlinear gyrokinetic equation in a sheared slab geometry, using B-spline finite elements to represent the self-consistent electrostatic field. Initial radial density and ion temperature profiles exhibiting high logarithmic gradients representing plasma edge conditions are used. In order to avoid practical problems of particles exiting the simulation domain as the ion temperature profile relaxes, all profiles are mirrored at domain-centre and periodic boundary conditions are imposed. The validity of the delta-f approach is measured by statistical noise estimates, while monitoring relative deviation levels of temperature via the kinetic energy. In particular, the effect of background profile gradients on these measures is investigated. As a first step towards reducing the amplitude of the deviation δf, an adaptive Maxwellian f 0 is implemented, whose time dependent temperature profiles are obtained by locally relaxing kinetic energy accumulating in δf into f 0.

Mass-limited plasmas heated by laser-driven fast electrons as a powerful source of neutron and hard x-ray radiation

To create a plasma with extreme thermodynamic parameters, we propose to heat with a laser-accelerated fast electron beam a target of a size less than the mean free path of the heating particles. The effect of capture of fast electrons generated in an electrically neutral target due to the action of a self-consistent electrostatic field at its boundaries allows us to volumetrically heat a target over multiple flights of fast electrons through it. Using such a heating mode enables control of the target mass to be significantly less than the mass stopping range of the heating particles. Heating a mass-limited target by laser-driven relativistic electrons can produce a plasma with a temperature of ∼10’s keV and a density close to its initial solid-state density. Such plasma objects are expected to serve as powerful sources of neutron and hard x-ray radiation.

Afterlive: A performant code for Vlasov-Hybrid simulations

A parallelized implementation of the Vlasov-Hybrid method (Nunn, 1993) is presented. This method is a hybrid between a gridded Eulerian description and Lagrangian meta-particles. Unlike the Particle-in-Cell method (Dawson, 1983) which simply adds up the contribution of meta-particles, this method does a reconstruction of the distribution function f in every time step for each species. This interpolation method combines meta-particles with different weights in such a way that particles with large weight do not drown out particles that represent small contributions to the phase space density. These core properties allow the use of a much larger range of macro factors and can thus represent a much larger dynamic range in phase space density.The reconstructed phase space density f is used to calculate momenta of the distribution function such as the charge density ρ. The charge density ρ is also used as input into a spectral solver that calculates the self-consistent electrostatic field which is used to update the particles for the next time-step.Afterlive (AF ourier-based T ool in the E lectrostatic limit for the R apid L ow-noise I ntegration of the V lasov E quation) is fully parallelized using MPI and writes output using parallel HDF5. The input to the simulation is read from a JSON description that sets the initial particle distributions as well as domain size and discretization constraints. The implementation presented here is intentionally limited to one spatial dimension and resolves one or three dimensions in velocity space. Additional spatial dimensions can be added in a straight forward way, but make runs computationally even more costly. Program summaryProgram Title: Afterlive (AF ourier-based T ool in the E lectrostatic limit for the R apid L ow-noise I ntegration of the V lasov E quation)Program Files doi:http://dx.doi.org/10.17632/nt79b5nfsc.1Licensing provisions: GNU Public Licence v3Programming language: C++External routines/libraries: C++ compiler (tested with g++ 4.8, 4.9, 5.3 and 6.3), MPI 1.1 (tested with OpenMPI 1.6.5, 1.8.1, and 1.10.2 and MPICH 3.1), HDF5 with support for parallel I/O(tested with version 1.8.13, 1.8.15, 1.8.16 and 1.10.0), FFTW3 (tested with version 3.3.3, 3.3.4 and 3.3.5), Blitz++ (tested with version 0.10), Jansson (tested with version 2.7.1, 2.7.3, 2.7.5 and 2.9.1) and pkg-configNature of problem: Kinetic simulations of collisionless plasma are often done with particle-in-cell codes which also mix Eulerian (fields on a grid) and Lagrangian (freely flowing meta-particles to track particle densities) description. In these codes however computational meta-particles are added up in the deposition scheme. Thus particles are usually equal weights and the noise level is considerable. Purely Eulerian codes must go to great length to yield a stable scheme without excessive numerical diffusion.Solution method: In this method an Eulerian description of the electrostatic field is also combined with a Lagrangian description of particles, but the distribution function f is reconstructed over the full phase space in a way that avoids drowning out particles with a low phase space weight. This allows for the use of a wide range of macro factors for different tracer particles and an excellent dynamic range in densities.Restrictions: The current implementation handles one dimension in space and one or three dimensions in velocity only and is applicable to electrostatic scenarios. Using the two extra components of the velocity is computationally costly.Unusual features: The configuration is stored in JSON files.

Multiparticle collision simulations of two-dimensional one-component plasmas: Anomalous transport and dimensional crossovers.

By means of hybrid multiparticle collsion-particle-in-cell (MPC-PIC) simulations we study the dynamical scaling of energy and density correlations at equilibrium in moderately coupled two-dimensional (2D) and quasi-one-dimensional (1D) plasmas. We find that the predictions of nonlinear fluctuating hydrodynamics for the structure factors of density and energy fluctuations in 1D systems with three global conservation laws hold true also for 2D systems that are more extended along one of the two spatial dimensions. Moreover, from the analysis of the equilibrium energy correlators and density structure factors of both 1D and 2D neutral plasmas, we find that neglecting the contribution of the fluctuations of the vanishing self-consistent electrostatic fields overestimates the interval of frequencies over which the anomalous transport is observed. Such violations of the expected scaling in the currents correlation are found in different regimes, hindering the observation of the asymptotic scaling predicted by the theory.

Towards the optimization of a gyrokinetic Particle-In-Cell (PIC) code on large-scale hybrid architectures

With the aim of enabling state-of-the-art gyrokinetic PIC codes to benefit from the performance of recent multithreaded devices, we developed an application from a platform called the “PIC-engine” [1, 2, 3] embedding simplified basic features of the PIC method. The application solves the gyrokinetic equations in a sheared plasma slab using B-spline finite elements up to fourth order to represent the self-consistent electrostatic field. Preliminary studies of the so-called Particle-In-Fourier (PIF) approach, which uses Fourier modes as basis functions in the periodic dimensions of the system instead of the real-space grid, show that this method can be faster than PIC for simulations with a small number of Fourier modes. Similarly to the PIC-engine, multiple levels of parallelism have been implemented using MPI+OpenMP [2] and MPI+OpenACC [1], the latter exploiting the computational power of GPUs without requiring complete code rewriting. It is shown that sorting particles [3] can lead to performance improvement by increasing data locality and vectorizing grid memory access. Weak scalability tests have been successfully run on the GPU-equipped Cray XC30 Piz Daint (at CSCS) up to 4,096 nodes. The reduced time-to-solution will enable more realistic and thus more computationally intensive simulations of turbulent transport in magnetic fusion devices.

The Vlasov–Poisson–Boltzmann system for the whole range of cutoff soft potentials

The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov–Poisson–Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. For cutoff intermolecular interactions, although there is some progress on the construction of global smooth solutions to its Cauchy problem near Maxwellians recently, the problem for the case of very soft potentials remains unsolved. By introducing a new time–velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables, it is shown in this manuscript that the Cauchy problem of the one-species Vlasov–Poisson–Boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition imposed in [13] for the case of cutoff moderately soft potentials but is assumed to be small in certain weighted Sobolev spaces. Our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well-posedness theory to the one-species Vlasov–Poisson–Boltzmann system near Maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework.

Boltzmann electron PIC simulation of the E-sail effect

Abstract. The solar wind electric sail (E-sail) is a planned in-space propulsion device that uses the natural solar wind momentum flux for spacecraft propulsion with the help of long, charged, centrifugally stretched tethers. The problem of accurately predicting the E-sail thrust is still somewhat open, however, due to a possible electron population trapped by the tether. Here we develop a new type of particle-in-cell (PIC) simulation for predicting E-sail thrust. In the new simulation, electrons are modelled as a fluid, hence resembling hybrid simulation, but in contrast to normal hybrid simulation, the Poisson equation is used as in normal PIC to calculate the self-consistent electrostatic field. For electron-repulsive parts of the potential, the Boltzmann relation is used. For electron-attractive parts of the potential we employ a power law which contains a parameter that can be used to control the number of trapped electrons. We perform a set of runs varying the parameter and select the one with the smallest number of trapped electrons which still behaves in a physically meaningful way in the sense of producing not more than one solar wind ion deflection shock upstream of the tether. By this prescription we obtain thrust per tether length values that are in line with earlier estimates, although somewhat smaller. We conclude that the Boltzmann PIC simulation is a new tool for simulating the E-sail thrust. This tool enables us to calculate solutions rapidly and allows to easily study different scenarios for trapped electrons.

Modeling of Deuterium Ionization and Extraction From an Ion Source Driven by Heated Cathode

A new simple predictor–corrector method for stationary self-consistent electrostatic field modeling is developed on the basis of Comsol Multiphysics 3.5 capabilities. The method is implemented in the script written in Comsol Script 1.3. The basic idea of the method is iterative computation of space charge density from a set of virtually traced trajectories of charged particles. The script is tailored for modeling of charged particle movement in a cylindrically symmetric self-consistent stationary electric field of a gas-filled DD-neutron tube with an ion source driven by a heated cathode. The proposed approach is useful for determining the mode of operation of a tube (whether plasma is formed in ion source or the mode of operation is plasmaless) and for verification of design modifications for plasmaless neutron tubes. The computed charge fields are shown to be self-consistent with a space charge adjustment error of 2.5% at worst.

Field theory and weak Euler-Lagrange equation for classical particle-field systems.

It is commonly believed as a fundamental principle that energy-momentum conservation of a physical system is the result of space-time symmetry. However, for classical particle-field systems, e.g., charged particles interacting through self-consistent electromagnetic or electrostatic fields, such a connection has only been cautiously suggested. It has not been formally established. The difficulty is due to the fact that the dynamics of particles and the electromagnetic fields reside on different manifolds. We show how to overcome this difficulty and establish the connection by generalizing the Euler-Lagrange equation, the central component of a field theory, to a so-called weak form. The weak Euler-Lagrange equation induces a new type of flux, called the weak Euler-Lagrange current, which enters conservation laws. Using field theory together with the weak Euler-Lagrange equation developed here, energy-momentum conservation laws that are difficult to find otherwise can be systematically derived from the underlying space-time symmetry.

Conformational Sampling Techniques

The potential energy hyper-surface of a protein relates the potential energy of the protein to its conformational space. This surface is useful in determining the native conformation of a protein or in examining a statistical-mechanical ensemble of structures (canonical ensemble). In determining the potential energy hyper-surface of a protein three aspects must be considered; reducing the degrees of freedom, a method to determine the energy of each conformation and a method to sample the conformational space. For reducing the degrees of freedom the choice of solvent, coarse graining, constraining degrees of freedom and periodic boundary conditions are discussed. The use of quantum mechanics versus molecular mechanics and the choice of force fields are also discussed, as well as the sampling of the conformational space through deterministic and heuristic approaches. Deterministic methods include knowledge-based statistical methods, rotamer libraries, homology modeling, the build-up method, self-consistent electrostatic field, deformation methods, tree-based elimination and eigenvector following routines. The heuristic methods include Monte Carlo chain growing, energy minimizations, metropolis monte carlo and molecular dynamics. In addition, various methods to enhance the conformational search including the deformation or smoothing of the surface, scaling of system parameters, and multi copy searching are also discussed.

One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space

The classical one-species Vlasov-Poisson-Landau system describes dynamics of electrons interacting with its self-consistent electrostatic field as well as its grazing collisions modeled by the famous Landau (Fokker-Planck) collision kernel. We show in this manuscript that the Cauchy problem for the one-species Vlasov-Poisson-Landau system which includes the Coulomb potential admits a unique global solution near a given global Maxwellian in the whole space $\mathbb{R}^3_x$ provided that the initial perturbation satisfies certain regularity and smallness conditions. Compared with that of [12], which, to the best of our knowledge, is the only result concerning the one-species Vlasov-Poisson-Landau system available up to now, we do not ask the initial perturbation to satisfy the neutral condition and the minimal regularity assumption we imposed on the initial perturbation is also weaker.

Propagation of a laser-driven relativistic electron beam inside a solid dielectric

Laser probe diagnostics: shadowgraphy, interferometry, and polarimetry were used for a comprehensive characterization of ionization wave dynamics inside a glass target induced by a laser-driven, relativistic electron beam. Experiments were done using the 50-TW Leopard laser at the University of Nevada, Reno. We show that for a laser flux of ∼2 × 10(18) W/cm2 a hemispherical ionization wave propagates at c/3 for 10 ps and has a smooth electron-density distribution. The maximum free-electron density inside the glass target is ∼2 × 10(19) cm-3, which corresponds to an ionization level of ∼0.1%. Magnetic fields and electric fields do not exceed ∼15 kG and ∼1 MV/cm, respectively. The electron temperature has a hot, ringlike structure with a maximum of ∼0.7 eV. The topology of the interference phase shift shows the signature of the "fountain effect", a narrow electron beam that fans out from the propagation axis and heads back to the target surface. Two-dimensional particle-in-cell (PIC) computer simulations demonstrate radial spreading of fast electrons by self-consistent electrostatic fields driven by laser. The very low ionization observed after the laser heating pulse suggests a fast recombination on the sub-ps time scale.

An arbitrary curvilinear-coordinate method for particle-in-cell modeling

A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in- Cell (PIC) simulation method, is explored. In the two dimensional (2d) electrostatic version of our method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out in 2d on a uniform grid on the unit square logical domain, and mapped to a nonuniform boundary-fitted grid on the physical domain. As the resulting logical grid equations of motion are not separable, we have developed an extension of the semi-implicit Modified Leapfrog (ML) integration technique to preserve the symplectic nature of the logical grid particle mover. A generalized, curvilinear coordinate formulation of Poisson's equations to solve for the electrostatic fields on the uniform logical grid is also developed. By our formulation, we compute the plasma charge density on the logical grid based on the particles' positions on the logical domain. That is, the plasma particles are weighted to the uniform logical grid and the self-consistent mean electrostatic fields obtained from the solution of the logical grid Poisson equation are interpolated to the particle positions on the logical grid. This process eliminates the complexity associated with the weighting and interpolation processes on the nonuniform physical grid and allows us to run the PIC method on arbitrary boundary-fitted meshes.

On the cometary flow equations with force fields

The Cauchy problem of a nonlinear kinetic equation modeling the time evolution of a cometary flow interacting with a force field is discussed, two kinds of existence results for weak solutions are established for initial data having finite mass and finite kinetic energy. The first one concerns a given force field which is assumed to be divergence free with respect to the velocity variable, it is shown that there exists a nonnegative weak solution to the Cauchy problem when the initial datum and the force field have reasonable integrability. As a special case, we also consider a Lorentz field and give another type of existence result. The second one deals with self-consistent electrostatic field, we show that when the initial datum has an L2 integrability the system has a global nonnegative solution which extends a previous result obtained by one of the authors.

Fountain effect of laser-driven relativistic electrons inside a solid dielectric

Ultrafast interferometry with sub-ps resolution has been applied for the direct measurement of an electron density induced by a laser-driven relativistic electron beam inside a solid dielectric. The topology of the interference phase shift shows the signature of the “fountain effect,” a narrow electron beam that fans out from the propagation axis and heads back to the target surface. Two-dimensional particle-in-cell (PIC) computer simulations demonstrate radial spreading of fast electrons by self-consistent electrostatic fields. The very low ionization, ∼0.1%, observed after the heating pulse suggests a fast recombination at the sub-ps time scale.

Self-similar nonlinear dynamical solutions for one-component nonneutral plasma in a time-dependent linear focusing field

In a linear trap confining a one-component nonneutral plasma, the external focusing force is a linear function of the configuration coordinates and/or the velocity coordinates. Linear traps include the classical Paul trap and the Penning trap, as well as the newly proposed rotating-radio-frequency traps and the Mobius accelerator. This paper describes a class of self-similar nonlinear solutions of nonneutral plasma in general time-dependent linear focusing devices, with self-consistent electrostatic field. This class of nonlinear solutions includes many known solutions as special cases.