One-Dimensional Plasma Model at Thermodynamic Equilibrium

Eldridge, O. C.; Feix, Marc

doi:10.1063/1.1724476

Cited by 59 publications

(29 citation statements)

References 9 publications

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“…This is why a particle-in-cell simulation with N D ∼ 100 may be a tolerable representation of a real plasma with N D ∼ 10 6 or more. This does not, of course, mean that spurious relaxation is never important, and kinetic properties of collisionless particle-in-cell simulations have been extensively studied [19][20][21][24][25][26][27]. In the context of one-dimensional simulations, the most salient result of these enquiries is that the rate of relaxation of the velocity distribution is proportional to 1/N 2 D , that is, there is no relaxation at first order in the plasma parameter.…”

Section: Limitations Of the Particle-in-cell Methodsmentioning

confidence: 90%

“…The second kind of error is spurious relaxation of the distribution function toward a Maxwellian, an effect sometimes called "collisions," although to minimize confusion we will generally avoid this term in the present discussion. This also has been discussed in considerable detail [18][19][20][21][22][23]. However, the all these works discuss simulation of collisionless plasmas.…”

Section: Limitations Of the Particle-in-cell Methodsmentioning

confidence: 99%

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Kinetic properties of particle-in-cell simulations compromised by Monte Carlo collisions

Turner

2006

Physics of Plasmas

104

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The particle-in-cell method with Monte Carlo collisions is frequently employed when a detailed kinetic simulation of a weakly collisional plasma is required. In such cases, one usually desires, inter alia, an accurate calculation of the particle distribution functions in velocity space. However, velocity space diffusion affects most, perhaps all, kinetic simulations to some degree, leading to numerical thermalization, and consequently distortion of the velocity distribution functions, among other undesirable effects. The rate of such thermalization can be considered a figure of merit for kinetic simulations. This paper shows that, contrary to previous assumption, the addition of Monte Carlo collisions to a one-dimensional particle-in-cell simulation seriously degrades certain properties of the simulation. In particular, the thermalization time can be reduced by as much as three orders of magnitude. This effect makes obtaining a strictly converged simulation results difficult in many cases of practical interest. * Electronic address: miles.turner@dcu.ie

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Section: Limitations Of the Particle-in-cell Methodsmentioning

confidence: 90%

Section: Limitations Of the Particle-in-cell Methodsmentioning

confidence: 99%

Kinetic properties of particle-in-cell simulations compromised by Monte Carlo collisions

Turner

2006

Physics of Plasmas

104

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“…He simulated an electrostatic plasma of 512 ions and electrons in one dimension, and showed that particle codes could be used to study the linear, nonlinear, and saturation phases of instabilities. At the time, the relevance of simulations with so few particles per Debye sphere was not clear and in 1962Dawson (1962 and Eldridge & Feix (1962) made an important contribution by showing that correct thermal behavior was produced. All these algorithms used particle-particle interactions, and the first particle-mesh codes to introduce a grid appeared only later (Burger 1965;Hockney 1966;Yu et al 1965).…”

Section: Introductionmentioning

confidence: 99%

Apar-T: code, validation, and physical interpretation of particle-in-cell results

Melzani

Winisdoerffer

Walder

et al. 2013

A&A

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We present the parallel particle-in-cell (PIC) code Apar-T and, more importantly, address the fundamental question of the relations between the PIC model, the Vlasov-Maxwell theory, and real plasmas. First, we present four validation tests: spectra from simulations of thermal plasmas, linear growth rates of the relativistic tearing instability and of the filamentation instability, and nonlinear filamentation merging phase. For the filamentation instability we show that the effective growth rates measured on the total energy can differ by more than 50% from the linear cold predictions and from the fastest modes of the simulation. We link these discrepancies to the superparticle number per cell and to the level of field fluctuations. Second, we detail a new method for initial loading of Maxwell-Jüttner particle distributions with relativistic bulk velocity and relativistic temperature, and explain why the traditional method with individual particle boosting fails. The formulation of the relativistic Harris equilibrium is generalized to arbitrary temperature and mass ratios. Both are required for the tearing instability setup. Third, we turn to the key point of this paper and scrutinize the question of what description of (weakly coupled) physical plasmas is obtained by PIC models. These models rely on two building blocks: coarse-graining, i.e., grouping of the order of p ∼ 10 10 real particles into a single computer superparticle, and field storage on a grid with its subsequent finite superparticle size. We introduce the notion of coarse-graining dependent quantities, i.e., quantities depending on p. They derive from the PIC plasma parameter Λ PIC , which we show to behave as Λ PIC ∝ 1/p. We explore two important implications. One is that PIC collision-and fluctuation-induced thermalization times are expected to scale with the number of superparticles per grid cell, and thus to be a factor p ∼ 10 10 smaller than in real plasmas, a fact that we confirm with simulations. The other is that the level of electric field fluctuations scales as 1/Λ PIC ∝ p. We provide a corresponding exact expression, taking into account the finite superparticle size. We confirm both expectations with simulations. Fourth, we compare the Vlasov-Maxwell theory, often used for code benchmarking, to the PIC model. The former describes a phase-space fluid with Λ = +∞ and no correlations, while the PIC plasma features a small Λ and a high level of correlations when compared to a real plasma. These differences have to be kept in mind when interpreting and validating PIC results against the Vlasov-Maxwell theory and when modeling real physical plasmas.

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“…(n LD Eldridge and Feix (1962) have shawn from a linearized theory that the thermodynamic fluctuations are given by…”

Section: Numerical Results For Stable Initial Conditionsmentioning

confidence: 99%

Comparison of various numerical solutions to the nonlinear Vlasov equation

Emery¹

1972

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One-Dimensional Plasma Model at Thermodynamic Equilibrium

Cited by 59 publications

References 9 publications

Kinetic properties of particle-in-cell simulations compromised by Monte Carlo collisions

Kinetic properties of particle-in-cell simulations compromised by Monte Carlo collisions

Apar-T: code, validation, and physical interpretation of particle-in-cell results

Comparison of various numerical solutions to the nonlinear Vlasov equation

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