Overdamped Brownian motion in periodic symmetric potentials
Bibliographic record
Abstract
The dynamics of an overdamped Brownian particle in the field of a one-dimensional symmetric periodic potential U(x;α) have been studied by numerical solution of the Smoluchowski diffusion equation and the Langevin equation using the Brownian Dynamics method. The parameter α controls the shape and height of the potential barrier, which ranges from a sinusoidal spatial dependence for low barrier heights (α small) to a near delta-function appearance for barrier heights tending to infinity (α very large). Both the mean square displacement (MSD) dα(t), and the probability density n(x,t|x0), where x0 denotes the initial position, have been calculated. The MSD over a wide time domain has been obtained for a number of values of α. The exact asymptotic (t→∞) form of the diffusion coefficient has been exploited to obtain an accurate representation for dα(t) at long times. The function, dα(t) changes its form in the range α=8–10, with the appearance of a “plateau” which signals a transition in the particle’s Brownian dynamics from a weakly hindered (but continuous) mechanism to essentially jump diffusion. In the limit α→∞, each well of U(x;α) becomes similar to the classical square well (SW), which we have revisited as it provides a valuable limiting case for dα(t) at α≫1. An effective “attraction” of the probability density towards the SW walls is observed for off-center initial starting positions, and it is suggested that this could explain an observed change in the analytic form of the SW MSD, dsw(t), at long times. Two approximate analytic forms for dsw(t) at short times have been derived. The relaxation of the Brownian particle distribution n(x,t|x0) in the initial-well of U(x;α) has been studied.
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How this classification was reachedexpand
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".