This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov-Poisson formulation), ours is based on a nonlinearly converged Vlasov-Ampère (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant-Friedrichs-Lewy) and spatial (aliasing) instabilities. It is charge-and energy-conserving to numerical round-off for arbitrary implicit time steps (unlike the earlier "energy-conserving" explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton-Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible.

Proteins, the workhorses of living systems, are constructed from chains of amino acids, which are synthesized in the cell based on the instructions of the genetic code and then folded into working proteins. The time for folding varies from microseconds to hours. What controls the folding rate is hotly debated. We postulate here that folding has the same temperature dependence as the ␣-fluctuations in the bulk solvent but is much slower. We call this behavior slaving. Slaving has been observed in folded proteins: Large-scale protein motions follow the solvent fluctuations with rate coefficient k␣ but can be slower by a large factor. Slowing occurs because large-scale motions proceed in many small steps, each determined by k␣. If conformational motions of folded proteins are slaved, so a fortiori must be the motions during folding. The unfolded protein makes a Brownian walk in the conformational space to the folded structure, with each step controlled by k␣. Because the number of conformational substates in the unfolded protein is extremely large, the folding rate coefficient, kf, is much smaller than k␣. The slaving model implies that the activation enthalpy of folding is dominated by the solvent, whereas the number of steps nf ‫؍‬ k␣͞kf is controlled by the number of accessible substates in the unfolded protein and the solvent. Proteins, however, undergo not only ␣-but also ␤-fluctuations. These additional fluctuations are local protein motions that are essentially independent of the bulk solvent fluctuations and may be relevant at late stages of folding.folding energy landscape ͉ fractional viscosity dependence ͉ internal viscosity ͉ Maxwell relation ͉ protein-solvent interaction P roteins in cells fold and unfold continuously. Consequently, an understanding of folding rates is key. The distribution and strength of contacts in the native state is one ingredient that influences the rate of folding (1). A second ingredient is the effect of the solvent, because protein motions are intimately linked to the motions of the environment. Our slaving model quantifies the linkage. The model is based on three concepts: Proteins assume a large number of different conformations or substates (2), their organization is described by a hierarchic energy landscape (3), and large-scale protein fluctuations follow the ␣-relaxation (Debye or dielectric relaxation) in the bulk solvent (4-6). Here we propose that these concepts also are valid for folding and that they lead to the model pictured in Fig. 1. Fig. 1a is a cartoon of folding and a 1D cross-section through the high-dimensional energy landscape. Fig. 1b is a 2D cross-section.

For decades, the Vlasov-Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. Here, we explore a fully implicit PIC algorithm for the Vlasov-Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space-time-centered, employing particle sub-cycling and orbit-averaging. The algorithm conserves total energy, local charge, canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large time steps and cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. We demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D-3V.

Helicon discharges produce plasmas with a density gradient across the confining magnetic field. Such plasmas can create a radial potential well for nonaxisymmetric whistlers, allowing radially localized helicon ͑RLH͒ waves. This work presents new evidence that RLH waves play a significant role in helicon plasma sources. An experimentally measured plasma density profile in an argon helicon discharge is used to calculate the rf field structure. The calculations are performed using a two-dimensional field solver under the assumption that the density profile is axisymmetric. It is found that RLH waves with an azimuthal wave number m = 1 form a standing wave structure in the axial direction and that the frequency of the RLH eigenmode is close to the driving frequency of the rf antenna. The calculated resonant power absorption, associated with the RLH eigenmode, accounts for most of the rf power deposited into the plasma in the experiment.

A multiple magnetic mirror array is formed at the Large Plasma Device (LAPD) [W. Gekelman, H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. Maggs, Rev. Sci. Instrum. 62, 2875 (1991)] to study axial periodicity-influenced Alfvén spectra. Shear Alfvén waves (SAW) are launched by antennas inserted in the LAPD plasma and diagnosed by B-dot probes at many axial locations. Alfvén wave spectral gaps and continua are formed similar to wave propagation in other periodic media due to the Bragg effect. The measured width of the propagation gap increases with the modulation amplitude as predicted by the solutions to Mathieu’s equation. A two-dimensional finite-difference code modeling SAW in a mirror array configuration shows similar spectral features. Machine end-reflection conditions and damping mechanisms including electron-ion Coulomb collision and electron Landau damping are important for simulation.