Brownian motion in one-dimensional models: scaling, universality, and dispersionless transport
Bibliographic record
Abstract
In the studies of Brownian motion, one-dimensional (1D) models play a special role in view of their relative simplicity and also because diffusion in a higher-dimensional space can often be decomposed into independent random motions in the orthogonal directions. In this thesis, we investigate the diffusion behavior of a Brownian particle (BP) within three 1D models: (i) diffusion along a stochastic harmonic oscillator chain (SHOC), (ii) diffusion of a damped BP in a tilted periodic potential, and (iii) free diffusion described by the Langevin equation with velocity-dependent damping. To address the first problem, we invent moving stochastic boundary condition approach, which allows us to simulate a small subset of oscillators in close proximity to the BP, effectively capturing the relevant dynamics. Our investigations have revealed a power law relation between the diffusion coefficient D and temperature T and the existence of dispersionless undamped phases in the BP motion at high temperatures. In the context of the second model, we explore the phenomenon of dispersionless transport of a BP in a tilted periodic potential, which is described in the literature as a broad time interval during which the particle’s dispersion appears to be constant. Our findings demonstrate that the dispersion fluctuations within the dispersion plateau hinder accurate determination of D, but these challenges can be remedied by employing an alternative measurement procedure. Moreover, it is evident that the conventional Langevin equation with velocity-independent damping coefficient as used in model (ii) cannot reproduce the nearly undamped ballistic flights observed in various physical systems. Lastly, we focus on establishing an analytical relation between D and T for a BP subject to velocity-dependent damping γ(v) and apply it to the case of a monotonically decreasing function γ(v) > 0. We show that at low temperatures, the D vs. T relation is linear; however, at high temperatures, a non-Einsteinian behavior emerges, similar to the one found within the SHOC diffusion model.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| 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".