1 / 30

Deterministic and random Growth Models. (Some remarks on Laplacian growth).

This article discusses deterministic and random growth models, focusing on Laplacian growth. It explores physical phenomena modeled by random growth processes, such as electrodeposition, lightning formation, and bacteria colonies. The need for a unique model with specific parameters is emphasized, and various growth models such as Eden's model and DLA model are examined. The study of growth processes involves comparing the diameter and length of clusters over time. The article also introduces deterministic models, Loewner processes, and regularization techniques.

kathleenryan
Download Presentation

Deterministic and random Growth Models. (Some remarks on Laplacian growth).

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Deterministic and random Growth Models. (Some remarks on Laplacian growth). S.Rohde (University of Washington) M.Zinsmeister (MAPMO,Université d’Orléans et PMC, Ecole Polytechnique)

  2. Some physical phenomena are modelized by random growth processes: cluster at time n+1 is obtained by choosing at random a point on the boundary of the cluster at time n and adding at this point some object Here are some examples:

  3. Electrodeposition . More examples with different voltages:

  4. Voltage: a:2V, b:3V, c:4V, d:6V, e:10V, f: 12V, g:16V

  5. Lightnings: Formation of conducting regions inside isolating matter submitted to high electric potential.

  6. Bacteria colonies with various quantities of nutriments:

  7. D) Croissance des mégapoles

  8. These pictures indicate the need of a unique model with parameter • The model must consist of: • A probability law for the choice of the boundary point. • 2) An object to attach.

  9. Dielectric breakdown models

  10. A) Eden ’s model. • Model used in biology: • Growth of bacteria colonies with abundance of nutriments • Growth of tumors.

  11. DLA Model (Diffusion-limited aggregation)

  12. The study of the growth process consists in comparing the diameter Dn of the cluster at time n and its length Ln. An important remark is that in the case of HL(0) Cn=Cn for some C>1.

  13. The HL(0) process

  14. DETERMINISTIC MODELS We consider growth models for which the size of the added objects is infinitesimally small with appropriate time change.

  15. Loewner processes Conformal mapping C(t) is the capacity of Kt The fact that the process is increasing translates into We get Loewner equation: Re(A(t,z))= Which implies the existence of measures (µt ) such that And every (reasonnable) family (µt ) of positive measures can be obtained in this way .

  16. Case alpha=2; Hele-Shaw flows, supposedly modelising introduction of a non-viscous fluid into a viscous one. Picture= experience with coloured water into oil.

  17. REGULARIZATION

  18. Proof:

More Related