Fokker-Planck equation for Coulomb relaxation and wave-particle diffusion: Spectral solution and the stability of the Kappa distribution to Coulomb collisions
Bibliographic record
Abstract
The present paper considers the time evolution of a charged test particle of mass m in a constant temperature heat bath of a second charged particle of mass M. The time dependence of the distribution function of the test particles is given by a Fokker-Planck equation with a diffusion coefficient for Coulomb collisions as well as a diffusion coefficient for wave-particle interactions. For the mass ratio m/M→0, the steady distribution is a Kappa distribution which has been employed in space physics to fit observed particle energy spectra. The time dependence of the distribution functions with some initial value is expressed in terms of the eigenvalues and eigenfunctions of the linear Fokker-Planck operator and also interpreted with the transformation to a Schrödinger equation. We also consider the explicit time dependence of the distribution function with a discretization of the Fokker-Planck equation. We study the stability of the Kappa distribution to Coulomb collisions.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".