Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problems
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Bibliographic record
Abstract
We consider here two well-known variational problems associated with the phenomenon of phase separation: the isoperimetric problem and minimization of the Cahn-Hilliard energy. The two problems are related through a classical result in \Gamma -convergence and we explore the behavior of global and local minimizers for these problems in the periodic setting. More precisely, we investigate these variational problems for competitors defined on the flat 2 or 3 -torus. We view these two problems as prototypes for periodic phase separation. We give here a complete analysis of stable critical points of the 2 -d periodic isoperimetric problem and also obtain stable solutions to the 2 -d and 3 -d periodic Cahn-Hilliard problem. We also discuss some intriguing open questions regarding triply periodic constant mean curvature surfaces in 3 d and possible counterparts in the Cahn-Hilliard setting.
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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 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 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