Don S. Lemons

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)]

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.

Simple criteria for the absence of the beam‐Weibel instability

Rigorously sufficient and approximately necessary conditions for the absence of the beam‐Weibel instability are derived. These conditions include previously known stability criteria and resolve the seeming contradiction that these modes can be stabilized by beam temperature when the plasma is cold, but they cannot be stabilized by beam temperature when the plasma has infinitesimally small temperature.

Two-dimensional wake potentials in sub- and supersonic dusty plasmas

Hot electrons and sub- and supersonic flows of cold ions around a charged dust particle create steady state wake and Debye screening fields. These linear, electrostatic fields are studied in two-dimensional planar or cylindrical geometry. An asymptotic analysis in the limit of large (compared to Debye length) downstream coordinate z yields analytic wakefields that are in good agreement with numerical integrations of the linear, steady state response function.

Ion kinetic effects on the wake potential behind a dust grain in a flowing plasma

The structure of the wake potential downstream of a stationary dust grain in a flowing plasma is studied on ion time scales using particle-in-cell simulation methods. The scaling of the wake is investigated as a function of Mach number and other parameters as well as the dimensionality of the system. The results are compared and discussed in relation to various theoretical expressions for the wake. Consistent with theory, in one dimension the wake wavelength scales as MλDe(1−M2)−1/2 for M 1. In two dimensions, a wake is formed for both M 1, while the wake wavelength scales as MλDe in both regimes. The amplitude of the wake peaks at M≈1 in both the one- and two-dimensional simulations.

Unstable oscillatory Pierce modes of neutralized electron beams

Oscillatory modes of the Pierce system have been calculated. These modes are found to have growth rates comparable to the previously investigated purely growing modes. When these modes are included, it is found that the Pierce system is unstable for most values of ωp L/V0≳π.

Nonlinear theory of the Weibel instability

A canonical distribution function is proposed to describe the instantaneous state of a single nonlinear wave–plasma system as it evolves quasi-statically in time. This function is based on two single particle constants of motion for a charged particle in a zero-frequency transverse magnetic wave and determines a wavenumber condition and two system energy constants. In the case of a onecomponent bi-Maxwellian plasma with T ⊥ /T ‖ >1, these relations are particularly simple and yield expressions for the energy in the magnetic wave field, the wavenumber, the temperatures, and the entropy of the system in terms of one unknown parameter, chosen to be the instantaneous temperature ratio, T ⊥ /T ‖ The maximum value of the field energy is expressed in terms of only the initial temperature anisotropy, and is shown to be always less than of the systems total energy. The results are in good agreement with computer simulations of the electron Weibel instability.

Expanding plasma structure and its evolution toward long wavelengths

The expansion of a plasma slab across an initially uniform magnetic field is simulated by the use of a two‐dimensional electromagnetic hybrid (particle ions, fluid electrons of nonzero mass) computer code. The expanding plasma develops magnetic‐field‐aligned structure on time scales faster than an ion gyroperiod. Through the full duration of the mi/me =100 simulation, the structure wavelength is well predicted by the wavelength at maximum growth rate from the linear Vlasov theory of the lower hybrid drift instability modified by deceleration. At mi/me =400, the late time structure wavelength is about 1.5 times the early time value. At mi/me =1836, the structure wavelength at early times is close to that corresponding to the maximum growth rate of linear theory, while at later times the structure wavelength becomes about twice as long as its early time value. These three results suggest that the ratio of the late time wavelength to the early time value gradually increases with mi/me. Extrapolation of this ...

Molecular dynamics simulations of plasma crystal formation including wake effects

Molecular dynamics (MD) simulations are used to study dusty plasma crystal formation in three dimensions. The grain interaction model includes a spherically symmetric Debye-Huckel potential, an asymmetric wake potential, and a unidirectional external potential representing gravity and the sheath potential. We use a new form for the wake with ion-neutral collisions that reduce the interaction length of the wake. For the parameters considered, we obtain quasi-ordered structures in which the grains align into well-formed strings in the vertical direction and a more amorphous alignment of the strings themselves. Changes in the vertical alignment as a function of the wake parameters are analyzed.

On the numbers of things and the distribution of first digits

I answer the question ‘‘Why are there more small things in the world than large things?’’ in terms of a probabilisitc model of partitioning a conserved quantity. Benford’s empirical rule for lists of numbers, that the proportion of numbers with first digit m is log10(m+1)−log10 m, is an exact consequence of the model.

Quiet direct simulation of plasmas

A new approach to particle simulation, called “quiet direct simulation Monte Carlo” (QDSMC), is described that can be applied to many problems of interest, including hydrodynamics, magnetohydrodynamics (MHD), and the modeling of collision plasmas. The essence of QDSMC is the use of carefully chosen weights for the particles (e.g., Gauss–Hermite, for Maxwellian distributions), which are destroyed each time step after the particle information is deposited onto the grid and reconstructed at the beginning of the next time step. The method overcomes the limited dynamical range and statistical noise typically found in particle simulations. In this article QDSMC is applied to hydrodynamics and MHD test problems, and its suitability for modeling semi-collisional plasma dynamics is considered.