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Laplacian networks: Growth, local symmetry, and shape optimization
O Devauchelle1, P Szymczak2, M Pecelerowicz2
1Institut de Physique du Globe de Paris, 1 rue Jussieu, 75238 Paris, France.
Physical Review. E
|April 19, 2017
Summary
Laplacian growth models, using the Loewner equation, reveal three rules governing network expansion. Mathematical equivalence is found between different direction rules, though the resulting network may differ from static flux-optimized configurations.
Area of Science:
- Complex Systems
- Mathematical Physics
- Pattern Formation
Background:
- Laplacian growth phenomena, such as river networks, exhibit complex patterns.
- The Loewner equation provides a framework for studying growth processes.
- Previous work established rules for network expansion: velocity, direction, and nucleation.
Purpose of the Study:
- To investigate the growth of finger-like networks in a diffusion field using the Loewner equation.
- To explore the mathematical equivalence of different direction rules for network growth.
- To compare dynamic growth outcomes with static configurations optimizing flux.
Main Methods:
- Utilized the Loewner equation to model network finger growth.
- Reviewed and analyzed three distinct formulations of the direction rule: geodesic, local symmetry, and maximal flux.
- Established mathematical equivalences between these direction rule formulations.
Main Results:
- Demonstrated that the Loewner equation framework simplifies network growth to three fundamental rules.
- Proved the mathematical equivalence of geodesic growth, local symmetry maintenance, and flux maximization for direction.
- Observed that the dynamic growth process, governed by these rules, can yield a network structure distinct from a static flux-optimized configuration.
Conclusions:
- The Loewner equation effectively models complex network formation driven by diffusion.
- Multiple growth strategies (geodesic, symmetric, flux-maximizing) are mathematically equivalent under this formalism.
- Dynamic growth pathways can lead to emergent network structures that differ from static optima.
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