Critical role of friction for a single particle falling through a funnel
Bibliographic record
Abstract
We investigate a single frictional, inelastic, spherical particle falling under gravity through a symmetric funnel. A recent study showed that, for a frictionless particle in such a system, several anomalous phenomena occur: The particle can stay longer, lose more energy, and exert more impulsive force in a funnel with steeper walls. For frictionless particles, such phenomena exist for many small ranges of funnel angles and are a consequence of the many possible repeated patterns in particle trajectories. However, in reality, friction always exists and it is a natural question whether the anomalous phenomena still exist for frictional particles in such systems. We show that, surprisingly, the inclusion of friction in the dynamics actually dramatically enhances the anomalous phenomena. For frictional particles, the anomalous phenomena exist for all funnel angles steeper than 45^{°} and are thus more robust than the frictionless case. Furthermore, instead of many possible complicated repeated patterns in particle trajectories, there is a unique repeated pattern for frictional particles. Moreover, this is the simplest possible repeated pattern. We derive an analytical expression for this unique repeated pattern and provide a theoretical explanation for the anomalous phenomena observed in frictional particle systems. We further show that the friction, no matter how small, plays a critical role in the dynamics, that is, the dynamics of the frictionless particle system is singular.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".