My PhD thesis on emergent geometric complexity in natural systems, using river networks as a rich example of how simple constituent interactions produce novel structures and statistical properties across multiple scales.
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Abstract
This thesis explores emergence—how novel structures arise from collective interactions between constituent entities—through the lens of river network geometry. River networks exemplify emergent complexity: simple physical processes (erosion, diffusion, pressure gradients) interact to produce geometric patterns and power-law statistics that appear fundamentally different from their microscopic origins.
The work examines emergent geometric patterns at three characteristic scales:
Valley Scale: Stream valleys form through diffusive erosion processes, creating characteristic geometries describable through free-boundary problems. The valley shape itself becomes an emergent property of the underlying diffusive dynamics.
Confluence Scale: Stream junctions exhibit preferential angles (~120°) that emerge from the interaction between subsurface pressure fields and surface topography—a geometric constraint arising from optimization principles rather than individual particle mechanics.
Basin Scale: At continental scales, river networks exhibit statistical properties (branching ratios, power-law distributions, scaling relationships) that connect to climate forcing through dimensionless parameters. These aggregate properties emerge from microscopic erosion rules but require entirely different mathematical frameworks to describe.
The thesis demonstrates how emergence manifests structurally in natural systems, producing complex geometric patterns through coupled nonlinear dynamics. Rather than a search for absolute physical laws, this work explores a theme: how macroscopic complexity arises from simple microscopic rules.
Details
PhD in Geophysics Massachusetts Institute of Technology Department of Earth, Atmospheric, and Planetary Sciences 2017
Advisor: Daniel H. Rothman
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