Statistical physics of transportation networks

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Statistical physics of transportation networks

• Amos Maritan, Andrea Rinaldo

Cieplak, Colaiori, Damuth, Flammini, Giacometti,
Marsili, Rodriguez-Iturbe, Swift
Science 272, 984 (1996) PRL 77, 5288 (1996), 78,
4522 (1997), 79, 3278 (1997), 84, 4745 (2000)
Rev. Mod. Phys. 68, 963 (1996) PRE 55, 1298
(1997) Nature 399, 130 (1999) J. Stat. Phys.
104, 1 (2001) Geophys. Res. Lett. 29, 1508
(2002) PNAS 99, 10506 (2002) Physica A340, 749
(2004) Water Res. Res. 42, W06D07 (2006)
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Digital elevation map ? Spanning Tree
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Upstream length
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Scheidegger model equal weight for all directed
networks
Huber, Swift, Takayasu .....
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Peano Basin
Random spanning trees (all trees have equal
weight)
Coniglio, Dhar, Duplantier, Majumdar, Manna, Sire
..
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Dynamics of optimal channel networkexcellent
accord with data
Only able to access local minima
Rinaldo Rodriguez-Iturbe
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Topology of optimal network
2 Electrical network 1 Random directed trees
½ River networks 0 Random trees
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Finite size scaling verified in observational
data
Maritan, Meakin, Rothman ..
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Finite size scaling (contd.)
Scheidegger model H1/2 Mean field H1 Random
trees dl 5/4 Peano Basin H dl 1
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Universality classes of optimal channel networks
in D 2

• 3 universality classes none of which agrees with
  observational data

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Disorder is irrelevant
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Sculpting of a fractal river basin

• Landscape evolution equation
• erosion ? to local flow A(x,t) (no flow - no
  erosion)
• reparametrization invariance
• small gradient expansion

Somfai Sander, Ball Sinclair
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Non-local, non-linear equation amenable to
exact solution in one dimension

• Consequences in two dimensions
• Slope discharge relationship
• Quantitative accord with observational data
• Local minima of optimal channel networks are
  stationary solutions of erosion equation
• Two disparate time scales connectivity of the
  spanning tree established early, soil height
  acquires stable profile much later

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Data ( More Recent Data) on Kleibers law
Brown West, Physics Today, 2004
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