Particle-in-cell (PIC) methods have a long history in the study of laser-plasma interactions. Early electromagnetic codes used the Yee staggered grid for field variables combined with a leapfrog EM-field update and the Boris algorithm for particle pushing. The general properties of such schemes are well documented. Modern PIC codes tend to add to these high-order shape functions for particles, Poisson preserving field updates, collisions, ionisation, a hybrid scheme for solid density and high-field QED effects. In addition to these physics packages, the increase in computing power now allows simulations with real mass ratios, full 3D dynamics and multi-speckle interaction. This paper presents a review of the core algorithms used in current laser-plasma specific PIC codes. Also reported are estimates of self-heating rates, convergence of collisional routines and test of ionisation models which are not readily available elsewhere. Having reviewed the status of PIC algorithms we present a summary of recent applications of such codes in laser-plasma physics, concentrating on SRS, short-pulse laser-solid interactions, fast-electron transport, and QED effects.

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Contemporary particle-in-cell approach to laser-plasma modelling

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

https://iopscience.iop.org/article/10.1088/0034-4885/76/4/046602/pdf

Anomalous transport in the crowded world of biological cells.

Many-particle charged-particle plasma simulations using spatial meshes for the electromagnetic field solutions, particle-in-cell (PIC) merged with Monte Carlo collision (MCC) calculations, are coming into wide use for application to partially ionized gases. The author emphasizes the development of PIC computer experiments since the 1950s starting with one-dimensional (1-D) charged-sheet models, the addition of the mesh, and fast direct Poisson equation solvers for 2-D and 3-D. Details are provided for adding the collisions between the charged particles and neutral atoms. The result is many-particle simulations with many of the features met in low-temperature collision plasmas; for example, with applications to plasma-assisted materials processing, but also related to warmer plasmas at the edges of magnetized fusion plasmas. >

Particle-in-Cell Charged Particle Simulations, plus Monte Carlo Collisions with Neutral Atoms, PIC-MCC

Complex (dusty) plasmas: current status, open issues, perspectives

[1] The flux of energetic electrons in the Earth's outer radiation belt can vary by several orders of magnitude over time scales less than a day, in response to changes in properties of the solar wind instigated by solar activity. Variability in the radiation belts is due to an imbalance between the dominant source and loss processes, caused by a violation of one or more of the adiabatic invariants. For radiation belt electrons, non-adiabatic behavior is primarily associated with energy and momentum transfer during interactions with various magnetospheric waves. A review is presented here of recent advances in both our understanding and global modeling of such wave-particle interactions, which have led to a paradigm shift in our understanding of electron acceleration in the radiation belts; internal local acceleration, rather than radial diffusion now appears to be the dominant acceleration process during the recovery phase of magnetic storms.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2010GL044990

Radiation belt dynamics: The importance of wave-particle interactions

We present a translation of Paul Langevin’s landmark paper. In it Langevin successfully applied Newtonian dynamics to a Brownian particle and so invented an analytical approach to random processes which has remained useful to this day.

Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530–533 (1908)]

Contents: Preface and Acknowledgments Chapter 1: Random Variables Chapter 2: Expected Values Chapter 3: Random Steps Chapter 4: Continuous Random Variables Chapter 5: Normal Variable Theorems Chapter 6: Einstein's Brownian Motion Chapter 7: Ornstein-Uhlenbeck Processes Chapter 8: Langevin's Brownian Motion Chapter 9: Other Physical Processes Chapter 10: Fluctuations without Dissipation Appendix A: "On the Theory of Brownian Motion," by Paul Langevin, translated by Anthony Gythiel Appendix B: Kinetic Equations Answers to Problems References Index

An Introduction to Stochastic Processes in Physics

A Grid-Based Coulomb Collision Model for PIC Codes

The paper deals with electron variations across simple high-speed streams. Comprehensive scans of the shapes of electron distributions measured at the highest bulk speeds confirm the results of Rosenbauer et al. (1976, 1977) and show that the electron velocity distributions can be broken down into a low-energy or core component and a high-energy strongly beamed component. The low-energy component displays many characteristics expected from a fluid: the internal particle coupling necessary to maintain this state must result from both binary Coulomb collisions and wave-particle interactions. The high-energy or halo component displays many characteristics expected to develop in the absence of collisions beyond a certain base radius. These electrons appear to evolve under the primary influence of static interplanetary magnetic and electric fields and, therefore, develop very anisotropic velocity distributions.

Characteristic electron variations across simple high-speed solar wind streams

The linear theory of fully electromagnetic plasma instabilities driven by currents flowing across a magnetic field is investigated, with applications to the laminar perpendicular bow shock. The theory utilizes the local approximation, assumes unmagnetized ions and magnetized electrons, and includes the effects of gradients in the magnetic field magnitude and density. Oblique propagation is considered, and wave numbers may be of the order of or larger than the reciprocal electron Larmor radius. An increase in the electron beta does not significantly affect the ion acoustic instability but does reduce the growth rates of the Buneman and the modified two-stream instabilities. If Te » Ti within the shock, results indicate that the ion acoustic instability is the most important growing mode, for any reasonable beta. If Te < Tι throughout the shock, further work is necessary to determine which instability is most important.

Don S. Lemons

Papers

Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530–533 (1908)]

An Introduction to Stochastic Processes in Physics

A Grid-Based Coulomb Collision Model for PIC Codes

Characteristic electron variations across simple high-speed solar wind streams

Current-driven instabilities in a laminar perpendicular shock