Long-time behaviour of sphalerons in $ϕ^4$ models with a false vacuum
Bibliographic record
Abstract
Abstract Evolution of sphalerons in a class of quartic (nonlinear) Klein–Gordon models are studied under a growing perturbation. Sphalerons are unstable lump-like solutions that arise from a saddle point between true and false vacua in the energy functional. Numerical simulations are presented which show the sphaleron evolving into an accelerating kink–antikink pair whose separation increases in time and asymptotically approaches the speed of light. To explain this behaviour analytically, a nonlinear collective coordinate method is developed which has three dynamical parameters and leads to an explicit asymptotic solution using a power series expansion. The solution describes the emergence of a spreading tabletop profile whose height approaches the true vacuum while its flanks steepen and accelerate outward. In addition, the energy density is shown to concentrate at the flanks, indicating the onset of a gradient blow-up at large times. These results provide a detailed description of the long-time dynamics of positively perturbed sphalerons, and reveal a universal mechanism for the formation of relativistically expanding structures in nonlinear field theories.
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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 | 0.000 | 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 | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".