Adiabatic invariant of a charged particle moving in a magnetic field with a constant gradient
Bibliographic record
Abstract
This paper presents the calculation of the adiabatic invariant for the motion of a charged particle in a two-dimensional magnetic field with a constant gradient. Magnetic field intensity is equal to zero along the neutral line for this field model. The mathematical expression for the invariant depends upon whether the particle crosses the neutral line. For trajectories that do not cross the neutral line, the adiabatic invariant reduces to the familiar expression for the magnetic moment, μ0=v2/B, for small values of the magnetic field gradient. The two expressions for the adiabatic invariant can be matched continuously across the change in the type of trajectory. When the magnetic field parameters smoothly change in time, the adiabatic invariant is conserved exponentially well as long as the type of the particle trajectory remains the same. If, however, the trajectory of a particle initially crosses the neutral line but after the magnetic field evolution stops crossing it (or vice versa), the adiabatic invariant is not conserved.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.011 | 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 teacher head, 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".