Brownian motion on a stochastic harmonic oscillator chain: limitations of the Langevin equation
Bibliographic record
Abstract
Abstract Diffusion of a Brownian particle along a linear chain of coupled stochastic harmonic oscillators is investigated using molecular dynamics (MD) and stochastic modeling. The latter technique is based on the Langevin equation (LE) derived by applying linear response theory to the chain degrees of freedom. When the coupling strength between the particle and the chain oscillators is comparable to or exceeds the chain coupling strength, the LE becomes inaccurate in its predictions of the diffusion coefficient value; however, it does reproduce qualitatively correctly the non-monotonic dependence of the diffusion coefficient on the particle-chain coupling strength, also found in MD simulations. The diffusion coefficient versus temperature curves determined from MD and Langevin simulations agree very well with each other at low temperatures. At high temperatures, the diffusion coefficient obtained from Langevin simulations is proportional to temperature, as predicted by Einstein’s relation. In contrast, the diffusion coefficient from MD is a non-linear function of temperature and is significantly greater than in Langevin simulations. Next, it is shown that Langevin description breaks down when the coupling between the chain oscillators is too strong. Finally, when an external constant force is applied to the particle, the Langevin description becomes qualitatively wrong. Namely, MD shows that the particle quickly detaches itself from the chain and moves at a constant acceleration due to the external force, whereas Langevin simulations predict constant terminal velocity of the particle.
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| 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".