statistical physics of transportation networks

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Statistical physics of transportation networks Amos Maritan , Andrea Rinal 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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Statistical physics of transportation networks. Cieplak, Colaiori, Damuth, Flammini, Giacometti, Marsili, Rodriguez-Iturbe, Swift. Amos Maritan , Andrea Rinaldo. - PowerPoint PPT Presentation

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Page 1: Statistical physics of transportation networks

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)

Page 2: Statistical physics of transportation networks
Page 3: Statistical physics of transportation networks

1 1 111

1

1

1

1

1

1

1

1

14 3 3

12 2

2 163 6

25 2

aap )(

1 )( aaAP

1/2agradient height local

:law discharge-Slope

Digital elevation map Spanning Tree

Page 4: Statistical physics of transportation networks

Upstream length

ha l

l l)

:Law sHack'

0.600.54 h

1.851.68

1.41.41

5

Page 5: Statistical physics of transportation networks

Huber, Swift, Takayasu .....

Scheidegger model – equal weight for all directed networks

aap )(

4/3

Page 6: Statistical physics of transportation networks

3/2

Peano Basin

Random spanning trees (all trees have equal weight)

8/11 Coniglio, Dhar, Duplantier, Majumdar, Manna, Sire …..

aap )(

Page 7: Statistical physics of transportation networks

Dynamics of optimal channel networkexcellent accord with data

Rinaldo & Rodriguez-Iturbe

i i 1/2a k E with

Only able to access local minima

Page 8: Statistical physics of transportation networks

Topology of optimal network

i i 1/2 1, 2,a k E with

2: Electrical network

1: Random directed trees

½: River networks

0: Random trees

Page 9: Statistical physics of transportation networks

L

HL

1 HH1

LL H1

with

basin ofArea

streammain ofdimension fractal

exponent Roughness

ld

H

)/(

)ldLlfl L)(l,

F(a/Lap(a,L)

Finite size scaling – verified in observational data

/ld h

1with area mean

sites thefromoutlet the todistance Average

ld dLa l

Maritan, Meakin, Rothman …..

Page 10: Statistical physics of transportation networks

h

L)dl(l,p(a,L)da

(1l

d

d

da L)p(a, aLa l

1

lll d d h d H /2,2/,2/,1

basins Fractal

Finite size scaling (contd.)

H),1/(1 h

HH) H,d

S

l ),1/((,1

networksriver affine-elf

Scheidegger model: H=1/2; Mean field: H=1; Random trees: dl = 5/4; Peano Basin: H = dl = 1

Page 11: Statistical physics of transportation networks

Universality classes of optimal channel networks in D = 2

3 universality classes none of which agrees with observational data

i ii a k E 1 1/2with

1,1

variable; random quenchedor constant either ,12/1

3/2,1 variable; random quenched ,1

2/1,1 constant; ,

Hd

k

Hdk

Hdk

l

i

l i

li

Page 12: Statistical physics of transportation networks

naggregatio minimumwith pathsExplosion

weight equal have treesspanning All

basin thedrains stream one - spiral Random

2/3 H

a ki

ii

outlet tositeeach from pathsoptimal ofUnion treeOptimal

growth linterfaciafor equation KPZ media; randomin polymer Directed

Minimize disorder; eduncorrelat,

er Scheidegg 1/2 H

aN aki

ii

walk random -outlet Path to

optimal and degenerate - networks directed All

lattice theof sites thefromoutlet the todistance Average

Minimize constant; ,

Page 13: Statistical physics of transportation networks

i

21i

i

21i

i

21i

i

ii

li

212i

L a{ Min

L~ a

L a{ Min

0 x xx

1 d H L aL a{ Min

} so and

, basin, PeanoFor

}

for

~}

12/1

Disorder is irrelevant

Page 14: Statistical physics of transportation networks

Sculpting of a fractal river basin

Landscape evolution equation: 1. erosion to local flow A(x,t) (no flow - no erosion)2. reparametrization invariance3. small gradient expansion

! 2

1 )()()(

unit timeper energy nalgravitatio Dissipated

)()(ty stationariat

uplift .......),(),(,

dropheight current

2

1

invariance reparam.

2

at x area drained

xx

xAxhxAE

xAxh

txhtxAt

txh

Somfai & Sander, Ball & Sinclair

Page 15: Statistical physics of transportation networks

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

Page 16: Statistical physics of transportation networks

M

B

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

X = Mz z

Life Span 1/4

Heart Beat Freq -1/4

Aorta Diam 3/8

Capillary Density -1/12