Optimizing the accuracy of lattice Monte Carlo algorithms for simulating diffusion
Bibliographic record
Abstract
The behavior of a lattice Monte Carlo (LMC) algorithm (if it is designed correctly) must approach that of the continuum system that it is designed to simulate as the time step and the mesh step tend to zero. However, we show for an algorithm for unbiased particle diffusion that if one of these two parameters remains fixed, the accuracy of the algorithm is optimal for a certain finite value of the other parameter. In one dimension, the optimal algorithm with moves to the two nearest neighbor sites reproduces the correct second and fourth moments (and minimizes the error for the higher moments at large times) of the particle distribution and preserves the first two moments of the first-passage time distributions. In two and three dimensions, the same level of accuracy requires simultaneous moves along two axes ("diagonal" moves). Such moves attempting to cross an impenetrable boundary should be projected along the boundary, rather than simply rejected. We also treat the case of absorbing boundaries. We discuss the relation between optimally accurate LMC algorithms and a particular case of lattice Boltzmann (LB) algorithms for simulating diffusion and compare the computational efficiency of optimal LMC and optimal LB algorithms.
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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 | 0.004 | 0.030 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 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".