Analytical solution of the space-dependent cosmic ray Fokker-Planck equation using Airy functions
Bibliographic record
Abstract
Abstract The motion of energetic particles such as cosmic rays is complicated due to their interaction with turbulent magnetic fields. In particular, an analytical description of such interactions is difficult to achieve due to their stochastic and nonlinear nature. Therefore, transport equations are used to describe the motion of energetic particles. In phase-space, where the particle distribution function depends on position, time, and velocity, one uses a Fokker-Planck partial differential equation. While the general case of such a transport equation is too complicated to solve analytically, we consider the special case of a constant pitch-angle scattering coefficient. For this special case we explore analytical solutions of the Fokker-Planck equation by using Airy functions. We also develop an N-dimensional subspace method for this case corresponding to a semi-analytical approach. We compare the different methods, obtained results, and computational times needed with each other. We also determine different expectation values which are relevant for applications in particle transport theory.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".