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Interface growth in two dimensions: a Loewner-equation approach.

Miguel A Durán1, Giovani L Vasconcelos

  • 1Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary

This study explores Laplacian growth in 2D using Loewner equations, presenting an exact solution for three-finger growth and a general model for interface dynamics. The research introduces new growth models and discusses generalizations for complex interface behaviors.

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Area of Science:

  • Physics
  • Mathematics
  • Complex Systems

Background:

  • Laplacian growth describes phenomena like diffusion-limited aggregation.
  • The Loewner equation offers a powerful tool for analyzing 2D interface growth.
  • Understanding complex interface dynamics is crucial in various scientific fields.

Purpose of the Study:

  • To investigate Laplacian growth in two dimensions using the Loewner equation framework.
  • To derive a general Loewner equation for interface growth in the upper half-plane.
  • To explore various configurations of growing interfaces, including single and multiple tips.

Main Methods:

  • Utilizing the Loewner equation to model interface evolution.
  • Developing an exact solution for a specific three-finger growth configuration.
  • Deriving a general class of growth models and their corresponding Loewner equations.

Main Results:

  • An exact solution for a three-finger Laplacian growth configuration was obtained.
  • A general framework for interface growth in the upper half-plane was established.
  • Examples of interfaces with multiple tips and multiple growing interfaces were presented.

Conclusions:

  • The Loewner equation framework is effective for studying 2D Laplacian growth.
  • The developed general model accommodates diverse interface growth scenarios.
  • Further generalizations, such as "Loewner domains," offer potential for modeling more intricate growth rules.