Barrier crossing driven by Lévy noise: Universality and the role of noise intensity
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Bibliographic record
Abstract
We study the barrier crossing of a particle driven by white symmetric L\'evy noise of index $\ensuremath{\alpha}$ and intensity $D$ for three different generic types of potentials: (a) a bistable potential, (b) a metastable potential, and (c) a truncated harmonic potential. For the low noise intensity regime we recover the previously proposed algebraic dependence on $D$ of the characteristic escape time, ${T}_{\mathrm{esc}}\ensuremath{\simeq}C(\ensuremath{\alpha})∕{D}^{\ensuremath{\mu}(\ensuremath{\alpha})}$, where $C(\ensuremath{\alpha})$ is a coefficient. It is shown that the exponent $\ensuremath{\mu}(\ensuremath{\alpha})$ remains approximately constant, $\ensuremath{\mu}\ensuremath{\approx}1$ for $0<\ensuremath{\alpha}<2$; at $\ensuremath{\alpha}=2$ the power-law form of ${T}_{\mathrm{esc}}$ changes into the known exponential dependence on $1∕D$; it exhibits a divergencelike behavior as $\ensuremath{\alpha}$ approaches 2. In this regime we observe a monotonous increase of the escape time ${T}_{\mathrm{esc}}$ with increasing $\ensuremath{\alpha}$ (keeping the noise intensity $D$ constant). The probability density of the escape time decays exponentially. In addition, for low noise intensities the escape times correspond to barrier crossing by multiple L\'evy steps. For high noise intensities, the escape time curves collapse for all values of $\ensuremath{\alpha}$. At intermediate noise intensities, the escape time exhibits nonmonotonic dependence on the index $\ensuremath{\alpha}$, while still retaining the exponential form of the escape time density.
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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) | 0.000 | 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) | 0.000 | 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