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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)
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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:
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.Electrodeposition
More examples with different voltages:
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Voltage: a:2V, b:3V, c:4V, d:6V, e:10V, f:
12V, g:16V
Voltage: a:2V, b:3V, c:4V, d:6V, e:10V, f:
12V, g:16V
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Formation of conducting regions inside isolating matter submitted to high electric potential.
Lightnings:
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Bacteria colonies with various quantities of nutriments:
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D) Croissance des mégapoles
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These pictures indicate the need of a unique model with parameter
The model must consist of:
1) A probability law for the choice of the boundary point.
2) An object to attach.
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Dielectric breakdown models
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A) Eden ’s model.
•Model used in biology:
•Growth of bacteria colonies with abundance of nutriments
•Growth of tumors.
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DLA Model (Diffusion-limited aggregation)
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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.
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The HL(0) process
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DETERMINISTIC MODELS
We consider growth models for which the size of the added objects is infinitesimally small with appropriate time change.
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Loewner processes
Conformal mapping
The fact that the process is increasing translates into
Which implies the existence of measures (µt ) such that
We get Loewner equation:
And every (reasonnable) family (µt ) of positive measures can be obtained in this way .
Re(A(t,z))=
C(t) is the capacity of Kt
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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.
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REGULARIZATION
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Proof:
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