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Statistical Physics of Transportation Networks: Insights and Models for Optimal Channel Dynamics

This paper explores the statistical physics underlying transportation networks, emphasizing optimal channel dynamics through various modeling approaches such as the Scheidegger model, spanning trees, and random directed trees. We discuss concepts of finite size scaling, universality classes, and the interaction between topology and dynamics in fluvial systems. The research encompasses theoretical frameworks supported by observational data, illustrating how local minima in optimal networks reflect erosion dynamics. Key findings highlight the reparametrization invariance of erosion processes and implications for landscape evolution.

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Statistical Physics of Transportation Networks: Insights and Models for Optimal Channel Dynamics

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  1. Statistical physics of transportation networks Cieplak, Colaiori, Damuth, Flammini, Giacometti, Marsili, Rodriguez-Iturbe, Swift Amos Maritan, Andrea Rinaldo 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)

  2. 1 1 1 1 1 4 3 1 1 3 1 2 1 1 12 1 1 2 1 16 3 6 1 2 25 Digital elevation map  Spanning Tree

  3. Upstream length

  4. Scheidegger model – equal weight for all directed networks Huber, Swift, Takayasu .....

  5. Peano Basin Random spanning trees (all trees have equal weight) Coniglio, Dhar, Duplantier, Majumdar, Manna, Sire …..

  6. Dynamics of optimal channel networkexcellent accord with data Only able to access local minima Rinaldo & Rodriguez-Iturbe

  7. Topology of optimal network 2: Electrical network 1: Random directed trees ½: River networks 0: Random trees

  8. Finite size scaling – verified in observational data Maritan, Meakin, Rothman …..

  9. Finite size scaling (contd.) Scheidegger model: H=1/2; Mean field: H=1; Random trees: dl = 5/4; Peano Basin: H = dl = 1

  10. Universality classes of optimal channel networks in D = 2 3 universality classes none of which agrees with observational data

  11. Disorder is irrelevant

  12. 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

  13. 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

  14. Data (& More Recent Data) on Kleiber’s law Brown & West, Physics Today, 2004 B M

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