Diffusion with resetting in a logarithmic potential
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Bibliographic record
Abstract
We study the effect of resetting on diffusion in a logarithmic potential. In this model, a particle diffusing in a potential U(x) = U0 log |x| is reset, i.e., taken back to its initial position, with a constant rate r. We show that this analytically tractable model system exhibits a series of transitions as a function of a single parameter, βU0, the ratio of the strength of the potential to the thermal energy. For βU0 < −1, the potential is strongly repulsive, preventing the particle from reaching the origin. Resetting then generates a non-equilibrium steady state, which is exactly characterized and thoroughly analyzed. In contrast, for βU0 > −1, the potential is either weakly repulsive or attractive, and the diffusing particle eventually reaches the origin. In this case, we provide a closed-form expression for the subsequent first-passage time distribution and show that a resetting transition occurs at βU0 = 5. Namely, we find that resetting can expedite arrival to the origin when −1 < βU0 < 5, but not when βU0 > 5. The results presented herein generalize the results for simple diffusion with resetting—a widely applicable model that is obtained from ours by setting U0 = 0. Extending to general potential strengths, our work opens the door to theoretical and experimental investigation of a plethora of problems that bring together resetting and diffusion in logarithmic potential.
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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