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Reaction-Diffusion Processes On regular and random graphs Angelo Vulpiani Dep. Physics Universit Sapienza Roma jeudi 18 juillet 2013 Thanks to Federico Bianco Universit Pierre et Marie Curie, Paris Raffaella Burioni Universit di

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Reaction-Diffusion Processes On regular and random graphs

Angelo Vulpiani

  • Dep. Physics

Università “Sapienza” Roma

jeudi 18 juillet 2013

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SLIDE 2
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Burioni et al. Reaction spreading on graphs Physical Review E 86, 055101(R) (2012) Bianco et al. Reaction spreading on percolating clusters Physical Review E 87, 062811 (2013)

Thanks to

Federico Bianco Université Pierre et Marie Curie, Paris Raffaella Burioni Università di Parma Sergio Chibbaro Université Pierre et Marie Curie, Paris Davide Vergni IAC-CNR, Roma

jeudi 18 juillet 2013

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SLIDE 3
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Motivations

Progression of an epidemic process Vespignani Nature Phys 2011 Brain, social network, internet Chemical kinetic Benichou et al Nature Chem 2010 Chemical fronts in porous media Atis, Saha, Auradou, Salin, Talon PRL 2013

jeudi 18 juillet 2013

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  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

General framework

microscopic point of view, molecules: diffusion (jumps) advection (In presence of stirring) reaction for ex. ( ) A + B → 2A

At macro-hydrodynamic level ADR equation

∂tθ = ˆ Lθ + 1 τ f(θ) ˆ L General advection-diffusion operator

jeudi 18 juillet 2013

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SLIDE 5
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

ADR eq.

ˆ L = −u · ∇ + D∆

Advection by a fluid flow and Diffusion

f(θ)/τ

Non-linear local reaction

τ

reaction time-rate

ˆ L = 1 rd−1 ∂ ∂r

  • k(r)rd−1 ∂

∂r

  • (Richardson, Procaccia

O'Shaughnessy) Effective diffusion

jeudi 18 juillet 2013

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SLIDE 6
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Probabilistic interpretation

dx/dt = u + √ 2Dη

ˆ L = −u · ∇ + D∆

θ(x, t) =

  • θ(x, 0) exp

1 τ t f(θ(x(s; t), s)) θ(x(s; t), s) ds

  • advection-reaction=Fokker-Planck

transport + reaction Freidlin formula

Complex geometry

jeudi 18 juillet 2013

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SLIDE 7
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Time Discretisation

Limit case δ − impulse Lagrangian and reaction maps discrete-time ARD Even for non-gaussian diffusion

jeudi 18 juillet 2013

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SLIDE 8
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Space Discretisation

θ(x, t + ∆t) = +∞

−∞

dwPα,∆t(w)θ(x − w, t + 0+) = +∞

−∞

dwPα,∆t(w)G(θ(x − w, t))

θn(t + ∆t) =

  • j

P (∆t)

j→nθj(t)

P (∆t)

n→n = 1 − 2W∆t

P (∆t)

n→n−1 = P (∆t) n→n+1 = W∆t

Discrete process

n

jeudi 18 juillet 2013

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SLIDE 9
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Aij = 1 if (i, j) ∈ E if (i, j) ∈ E dθi dt = w

  • j

∆ijθj + 1 τ f(θi) θn(t + ∆t) =

  • j

P (∆t)

j→nθj(t)

θn(t + ∆t) = G∆t  

j

P (∆t)

j→nθj(t)

 

Discretisation: master eq.

G(θ) = θ + ∆t τ θ(1 − θ)

FKPP

W

P (∆t)

i→j

= WAij∆t if i = j P (∆t)

i→i

= 1 − kiW∆t if i = j

jeudi 18 juillet 2013

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SLIDE 10
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Some relevant quantities

df #(r) ∼ rdf ds = lim

t→∞ −2ln Pii(t)

ln t

Topology and geometry of the graphs

#(l) ∼ ldl Connectivity dimension Spectral dimension fractal dimension dl M(t) = 1 N

  • i∈V

θi(t) total quantity of the reaction product

jeudi 18 juillet 2013

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SLIDE 11
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Spreading on a T-fractal where the front is in red. The percentage of quantity

  • f product M(t)τ vs t. Numerical results for Equation with w = 0.5 are com-

pared to prediction tdl. For this graph dl = ln 3/ ln 2 ≃ 1.585, dl = 2 ln 3/ ln 5 ≃ 1.365.

Results: fractals

ds

jeudi 18 juillet 2013

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SLIDE 12
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Results: fractals

Sierpinski carpet

M(t) ∼ tdl

Main result

jeudi 18 juillet 2013

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SLIDE 13
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Results: fractals

reaction spreading <=> short-time n random walkers

C0j(t) = 1 −

t

  • τ=0

F0j(τ).

Sn(t) =

N

  • j=0

1 − C0j(t)n

Probability first passage at time t

F0j(t)

Probability no passage in j at time t Independent n walkers

P(no j)n = C0j(t)n P(j)n = 1 − C0j(t)n

Number of sites visited by n walkers

n → ∞ → Sn(t) ∼ tdc

Validity regime

Pm =< k >−t nPm ≫ 1 t < ¯ t ∼ log n

jeudi 18 juillet 2013

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SLIDE 14
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Erdos-Renyi random graph

< k >= p(N − 1)

dl = ∞

#(t) ∼ et

p > log(N) N

the graph is globally connected

if

jeudi 18 juillet 2013

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SLIDE 15
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Erdos-Renyi random graph

M(t) ∼ eαt Scaling

Mean-field eq.

τ = 0.1

Fast reaction

α(k, τ) ≃ Cτ β log < k >

Slow reaction

α = 1/τ ∂tρ(t) = τ β log(< k >)ρ(t)(1 − ρ(t))

jeudi 18 juillet 2013

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SLIDE 16
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Percolation

Percolation in a square lattice

df ≃ 1.896 dl ≃ 1.67 ds ≃ 1.36

critical point p ≈ 0.595

jeudi 18 juillet 2013

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SLIDE 17
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

10

−2

10 10

2

10

4

10

−2

10 10

2

10

4

10

6

t M(t)/! !=0.3 !=1 !=3 !=10 !=30 " td

l

Percolation: reaction spreading

M(t) ≃ αtdl p = pc

jeudi 18 juillet 2013

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SLIDE 18
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Porous media: experiments with glass spheres

Chemical fronts in porous media Atis, Saha, Auradou, Salin, Talon PRL 2013

Percolation: travelling front

0.2 0.4 0.6 0.8 1 200 400 600 800 1000 !i i

Numerical simulations Ly

jeudi 18 juillet 2013

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SLIDE 19
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Percolation: travelling front

v = lim

t→∞

M(t) Npt

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 vf(p)/vf(1) p Ly= 50 Ly=100 Ly=200 Ly=400 0.4 0.5 50 100 200 400 vf(pc)/vf(1) Ly

Velocity M(t) ≃ Npvt

p ≈ pc Np ∼ Ldf

y

Ly = Ldf −1

y

p ≫ pc Np ≃ pLy

u(p) = P(p)vf(p) vf(1) , ,

jeudi 18 juillet 2013

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SLIDE 20
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Percolation: travelling front

v = lim

t→∞

M(t) Npt Front speed u(p) = P(p)vf(p) vf(1)

u(p) = P(p)vf(p) vf(1) ∼ P(p) p − pc 1 − pc γ

u(p) = P(p)vf(p) vf(1) ∼ P(p)Np Ly

α

small

α

large

jeudi 18 juillet 2013

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SLIDE 21
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Front width

Percolation: travelling front

jeudi 18 juillet 2013

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SLIDE 22
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

Conclusions and perspectives

Advection-Reaction-Diffusion fundamental framework Complex heterogeneous geometry Finite-size effects prevalence of fluctuations

  • Flow-chemistry interaction
  • Analysis of experiments in porous media
  • Realistic simulations for epidemics networks
  • Chemistry Role

jeudi 18 juillet 2013

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SLIDE 23
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

jeudi 18 juillet 2013

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SLIDE 24
  • Dep. Physics Univ. Sapienza Rome

angelo.vulpiani@roma1.infn.it

m(t) ∼ tdl. m(t) ∼ r(t)df . Therefore r(t) ∼ tdl/df , and v = dr

dt ∼ tdl/df −1 ∼

r1−dmin, where dmin = df

dl . Furthermore, if the linear size of the region is r < ξ,

where ξ is the correlation length the cluster is self-similar and then v ∼ ξ1−dmin. Moreover, analysis of the percolation phase transition gives ξ ∼ |p−pc|−ν, with ν = 4/3 for d = 2 [?], which gives the final scaling v ∼ (p − pc)γ, where γ = −ν(1 − dmin). For the average velocity, the scaling is: u(p) = P(p)vf(p) vf(1) ∼ P(p) p − pc 1 − pc γ .df (1)

jeudi 18 juillet 2013