Phase Transitions in a Relaxation Model of Mixed Type with Periodic Boundary Condition
Bibliographic record
Abstract
We study the asymptotic behavior of solutions for a 2 × 2 relaxation model of mixed type with periodic initial and boundary conditions. We prove that the asymptotic behavior of the solutions and their phase transitions are dependent on the location of the initial data and the size of the viscosity. If the average of the initial data is in the hyperbolic region and the initial data does not deviate too much from its average, we prove that there exists a unique global solution and that it converges time-asymptotically to the average in the same hyperbolic region. No phase transition occurs after initial oscillations. If the average of the initial data is in the elliptic region and the initial data does not deviate too much from its average, and in addition if the viscosity is big, then the solution converges to the average in the same elliptic region, and does not exhibit phase transitions after initial oscillations. If, however, the viscosity is small, numerical evidence indicates that the solution oscillates across the hyperbolic and elliptic regions for all time, exhibiting phase transitions. In this case, we conjecture that the solution converges to an oscillatory standing wave (steady-state solution).
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| 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".