Swarm equilibria in domains with boundaries
Bibliographic record
Abstract
This thesis involves the study of a well-known swarming model with interaction and external potentials in one and two dimensions. We refer to this model as the plain aggregation model and later study the model with nonlinear diffusion, so-called the diffusive model here. Typically set in free space, one of the novelties of this thesis is the study of such swarming models in the presence of a boundary. We consider a no-flux boundary condition enforced in a particle context via a ``slip'' condition. Of particular relevance to the context of this thesis, the swarming model used here can be formulated as an energy gradient flow and thusly, one might expect equilibrium states to be minima of the energy. In this work we demonstrate, through both analytical and numerical investigations, a continuum of equilibria of the plain aggregation model that are not minima of the energy. Furthermore, we show that these non-minimizing equilibria are achieved dynamically from a non-trivial set of initial conditions with a variety of interaction potentials and boundary geometries. Thus we show conclusively a deficiency with the plain aggregation model in domains with boundaries, namely that it appears to evolve into equilibria that are not minima of the energy. Following this we then propose a rectification to this deficiency in way of nonlinear diffusion. This choice of nonlinear diffusion is especially attractive because it preserves compact states of the plain aggregation model. We showcase how the diffusive model approaches, but does not equilibrate at, the non-minimizing equilibria of the plain aggregation model. Furthermore we demonstrate how minimizers of the diffusive model do approach minimizers of the plain aggregation model in the zero diffusion limit.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| 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".