Shadowing-based Reliability Decay in Softened <i>n</i> -Body Simulations
Bibliographic record
Abstract
A shadow of a numerical solution to a chaotic system is an exact solution to the equations of motion that remains close to the numerical solution for a long time. In a collisionless n -body system, we know that particle motion is governed by the global potential rather than by interparticle interactions. As a result, the trajectory of each individual particle in the system is independently shadowable. It is thus meaningful to measure the number of particles that have shadowable trajectories as a function of time. We find that the number of shadowable particles decays exponentially with time as e -μ t and that for ∈ [~ 0.2, 1] (in units of the local mean interparticle separation ), there is an explicit relationship among the decay constant μ, the time step h of the leapfrog integrator, the softening , and the number of particles N in the simulation. Thus, given N and , it is possible to precompute the time step h necessary to achieve a desired fraction of shadowable particles after a given length of simulation time. We demonstrate that a large fraction of particles remain shadowable over ~100 crossing times even if particles travel up to about of the softening length per time step. However, a sharp decrease in the number of shadowable particles occurs if the time step increases to allow particles to travel farther than the softening length in 1 time step or if the softening is decreased below ~0.2 .
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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 | 0.003 | 0.015 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 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) | 0.001 | 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".