Fokker-Planck Equation for an Inverse-Square Force

Rosenbluth, M. N.; MacDonald, William M.; Judd, D.L.

doi:10.1103/physrev.107.1

Cited by 1,140 publications

(834 citation statements)

References 4 publications

Supporting

Mentioning

809

Contrasting

Unclassified

Order By: Relevance

“…This is of the form advocated by Rosenbluth et al [15], a form also described by Trubnikov [16]. These authors work with a completely arbitrary background plasma, a plasma which has no aspects of thermal equilibrium.…”

Section: Velocity Fluctuationsmentioning

confidence: 99%

“…Such coefficients are sometimes described as "Rosenbluth potentials" which were introduced in Ref. [15] and discussed in several places, a good reference being Ref. [16].…”

Section: Methodsmentioning

confidence: 99%

“…the dispersion form (3.9) gives 15) with the second equality just a repetition of the reflection property (3.13). These results show that…”

Section: A Classical Regimementioning

confidence: 99%

“…Here we shall obtain a precise evaluation of the tensor functions C kl b that appear in the Fokker-Planck equation. This we shall do to the accuracy of the stopping power dE/dx that 15 In this regard, we should note that although a "diffusion approximation" to the Boltzmann equation is outlined in Lifshitz and Pitaevskii [26], that discussion is in the context of dilute heavy particles moving in a gas of light particles. This is a kinematical restriction that is quite different than the dynamical case of sharply peaked forward scattering that we examine in Appendix C. To make this distinction clear, we also review the work of Lifshitz and Pitaevskii in Appendix C, where we show by a simple example that their result is not internally consistent unless further restrictions are imposed.…”

Section: General Formulationmentioning

confidence: 99%

See 3 more Smart Citations

Charged particle motion in a highly ionized plasma

2005

View full text Add to dashboard Cite

A recently introduced method utilizing dimensional continuation is employed to compute the energy loss rate for a non-relativistic particle moving through a highly ionized plasma. No restriction is made on the charge, mass, or speed of this particle. It is, however, assumed that the plasma is not strongly coupled in the sense that the dimensionless plasma coupling parameter g = e 2 κ D /4πT is small, where κ D is the Debye wave number of the plasma. To leading and next-to-leading order in this coupling, dE/dx is of the generic form g 2 ln[Cg 2 ]. The precise numerical coefficient out in front of the logarithm is well known. We compute the constant C under the logarithm exactly for arbitrary particle speeds. Our exact results differ from approximations given in the literature. The differences are in the range of 20% for cases relevant to inertial confinement fusion experiments. The same method is also employed to compute the rate of momentum loss for a projectile moving in a plasma, and the rate at which two plasmas at different temperatures come into thermal equilibrium. Again these calculations are done precisely to the order given above. The loss rates of energy and momentum uniquely define a Fokker-Planck equation that describes particle motion in the plasma. The coefficients determined in this way are thus well-defined, contain no arbitrary parameters or cutoffs, and are accurate to the order described. This FokkerPlanck equation describes the straggling -the spreading in the longitudinal position of a group of particles with a common initial velocity and positionand the transverse diffusion of a beam of particles. It should be emphasized that our work does not involve a model, but rather it is a precisely defined evaluation of the leading terms in a well-defined perturbation theory.

show abstract

Section: Velocity Fluctuationsmentioning

confidence: 99%

“…Such coefficients are sometimes described as "Rosenbluth potentials" which were introduced in Ref. [15] and discussed in several places, a good reference being Ref. [16].…”

Section: Methodsmentioning

confidence: 99%

“…the dispersion form (3.9) gives 15) with the second equality just a repetition of the reflection property (3.13). These results show that…”

Section: A Classical Regimementioning

confidence: 99%

Section: General Formulationmentioning

confidence: 99%

See 2 more Smart Citations

Charged particle motion in a highly ionized plasma

2005

View full text Add to dashboard Cite

show abstract

“…Landau derived it for an electronic plasma from the Boltzmann equation in a weak deflexion limit, using a linear trajectory approximation [34]. It can also be obtained from the Fokker-Planck equation by calculating the first and second moments of the velocity increments induced by a succession of two-body encounters [35]. The Landau equation can be further simplified by explicitly evaluating the tensor (41).…”

Section: F the Landau Equationmentioning

confidence: 99%

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

204

View full text Add to dashboard Cite

We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the FokkerPlanck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N → +∞, a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1/N , the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard-Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N → +∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ → +∞, or for large times t ≫ ξ −1 , it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.

show abstract

ISR Spectra Simulations With Electron‐Ion Coulomb Collisions

et al. 2018

View full text Add to dashboard Cite

Incoherent scatter radars (ISR) rely on Thomson scattering of very high frequency or ultrahigh frequency radio waves off electrons in the ionosphere and measure the backscattered power spectra in order to estimate altitude profiles of plasma density, electron temperature, ion temperature, and ion drift speed. These spectra result from the collective behavior of coupled ion and electron dynamics, and, for most cases, existing theories predict these well. However, when the radar points nearly perpendicular to the Earth's magnetic field, the motion of the plasma across the field lines becomes complex and Coulomb collisions between electrons and ions become important in interpreting ISR measurements. This paper presents the first fully kinetic, self‐consistent, particle‐in‐cell simulations of ISR spectra with electron‐ion Coulomb collisions. We implement a grid‐based Coulomb collision algorithm in the Electrostatic Parallel Particle‐in‐Cell simulator and obtain ISR spectra from simulations both with and without collisions. For radar directions greater than 5° away from perpendicular to the magnetic field, both sets of simulations match collisionless ISR theory well. For angles between 3° and 5°, the collisional simulation is well described by a simplified Brownian motion collision process. At angles less than 3° away from perpendicular the Brownian motion model fails, and the collisional simulation qualitatively agrees with previous single particle simulations. For radar directions exactly perpendicular to the magnetic field the simulated collisional spectra match those from the Brownian motion collision theory, in agreement with previous single particle simulations.

show abstract

Fokker-Planck Equation for an Inverse-Square Force

Cited by 1,140 publications

References 4 publications

Charged particle motion in a highly ionized plasma

Charged particle motion in a highly ionized plasma

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

ISR Spectra Simulations With Electron‐Ion Coulomb Collisions

Contact Info

Product

Resources

About