Rate processes in a delayed, stochastically driven, and overdamped system
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Bibliographic record
Abstract
A Fokker-Planck formulation of systems described by stochastic delay differential equations has been recently proposed. A separation of time scales approximation allowing this Fokker-Planck equation to be simplified in the case of multistable systems is hereby introduced, and applied to a system consisting of a particle coupled to a delayed quartic potential. In that approximation, population numbers in each well obey a phenomenological rate law. The corresponding transition rate is expressed in terms of the noise variance and the steady-state probability density. The same type of expression is also obtained for the mean first passage time from a given point to another one. The steady-state probability density appearing in these formulas is determined both from simulations and from a small delay expansion. The results support the validity of the separation of time scales approximation. However, the results obtained using a numerically determined steady-state probability are more accurate than those obtained using the small delay expansion, thereby stressing the high sensitivity of the transition rate and mean first passage time to the shape of the steady-state probability density. Simulation results also indicate that the transition rate and the mean first passage time both follow Arrhenius' law when the noise variance is small, even if the delay is large. Finally, deterministic unbounded solutions are found to coexist with the bounded ones. In the presence of noise, the transition rate from bounded to unbounded solutions increases with the delay.
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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) | 0.001 | 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 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it