Reaction-Diffusion Processes On regular and random graphs Angelo - PowerPoint PPT Presentation

Reaction-Diffusion Processes On regular and random graphs Angelo Vulpiani Dep. Physics Università “Sapienza” Roma jeudi 18 juillet 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 Burioni et al. Bianco et al. Reaction spreading on graphs Reaction spreading on percolating clusters Physical Review E 86, 055101(R) (2012) Physical Review E 87, 062811 (2013) angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Motivations Brain, social network, internet Chemical kinetic Benichou et al Nature Chem 2010 Progression of an epidemic process Chemical fronts in porous media Vespignani Nature Phys 2011 Atis, Saha, Auradou, Salin, Talon PRL 2013 angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

General framework microscopic point of view, molecules: diffusion (jumps) advection (In presence of stirring) reaction for ex. ( ) A + B → 2 A At macro-hydrodynamic level ADR equation L θ + 1 ∂ t θ = ˆ ˆ L General advection-diffusion operator τ f ( θ ) angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

ADR eq. ˆ L = − u · ∇ + D ∆ Advection by a fluid flow and Diffusion f ( θ ) / τ Non-linear local reaction reaction time-rate τ � � 1 ∂ k ( r ) r d − 1 ∂ (Richardson, Procaccia ˆ Effective diffusion L = O'Shaughnessy) r d − 1 ∂ r ∂ r angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Probabilistic interpretation √ ˆ d x / d t = u + 2 D η L = − u · ∇ + D ∆ advection-reaction=Fokker-Planck � t transport + reaction � � 1 f ( θ ( x ( s ; t ) , s )) �� θ ( x , t ) = θ ( x , 0) exp ds Freidlin formula θ ( x ( s ; t ) , s ) τ 0 Complex geometry angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Time Discretisation Limit case δ − impulse Lagrangian and reaction maps discrete-time ARD Even for non-gaussian diffusion angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Space Discretisation � + ∞ d wP α , ∆ t ( w ) θ ( x − w, t + 0 + ) θ ( x, t + ∆ t ) = −∞ � + ∞ = d wP α , ∆ t ( w ) G ( θ ( x − w, t )) −∞ n Discrete process P ( ∆ t ) � θ n ( t + ∆ t ) = j → n θ j ( t ) j P ( ∆ t ) n → n − 1 = P ( ∆ t ) P ( ∆ t ) n → n = 1 − 2 W ∆ t n → n +1 = W ∆ t angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Discretisation: master eq. � 1 if ( i, j ) ∈ E A ij = 0 if ( i, j ) �∈ E ∆ ij θ j + 1 d θ i � W dt = w τ f ( θ i ) j P ( ∆ t ) WA ij ∆ t if i � = j = i → j P ( ∆ t ) � θ n ( t + ∆ t ) = j → n θ j ( t ) P ( ∆ t ) = 1 − k i W ∆ t if i � = j j i → i   G ( θ ) = θ + ∆ t P ( ∆ t ) � θ n ( t + ∆ t ) = G ∆ t j → n θ j ( t ) τ θ (1 − θ )  j FKPP angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Some relevant quantities Topology and geometry of the graphs Connectivity dimension #( l ) ∼ l d l d l t →∞ − 2ln P ii ( t ) Spectral dimension d s = lim ln t #( r ) ∼ r d f fractal dimension d f M ( t ) = 1 � total quantity of the reaction product θ i ( t ) N i ∈ V angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Results: fractals Spreading on a T-fractal where the front is in red. The percentage of quantity of product M ( t ) τ vs t . Numerical results for Equation with w = 0 . 5 are com- d s pared to prediction t d l . For this graph d l = ln 3 / ln 2 ≃ 1 . 585, d l = 2 ln 3 / ln 5 ≃ 1 . 365. angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Results: fractals Sierpinski carpet M ( t ) ∼ t d l Main result angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Results: fractals reaction spreading <=> short-time n random walkers Probability no passage in j at time t Probability first passage at time t t � C 0 j ( t ) = 1 − F 0 j ( τ ) . F 0 j ( t ) τ =0 P ( no j ) n = C 0 j ( t ) n Independent n walkers P ( j ) n = 1 − C 0 j ( t ) n N � 1 − C 0 j ( t ) n S n ( t ) = Number of sites visited by n walkers j =0 n → ∞ → S n ( t ) ∼ t d c t < ¯ P m = < k > − t t ∼ log n nP m ≫ 1 Validity regime angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Erdos-Renyi random graph #( t ) ∼ e t d l = ∞ < k > = p ( N − 1) p > log( N ) if the graph is globally connected N angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Erdos-Renyi random graph Scaling M ( t ) ∼ e α t τ = 0 . 1 α = 1 / τ Slow reaction α ( k, τ ) ≃ C τ β log < k > Fast reaction ∂ t ρ ( t ) = τ β log( < k > ) ρ ( t )(1 − ρ ( t )) Mean-field eq. angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Percolation Percolation in a square lattice critical point p ≈ 0 . 595 d f ≃ 1 . 896 d l ≃ 1 . 67 d s ≃ 1 . 36 angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Percolation: reaction spreading p = p c 6 10 4 10 M(t)/ ! M ( t ) ≃ α t d l 2 10 ! =0.3 ! =1 0 10 ! =3 ! =10 ! =30 " t d l − 2 10 − 2 0 2 4 10 10 10 10 t angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Percolation: travelling front Porous media: Numerical simulations experiments with glass spheres L y 1 0.8 0.6 ! i 0.4 0.2 0 0 200 400 600 800 1000 i Chemical fronts in porous media Atis, Saha, Auradou, Salin, Talon PRL 2013 angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Percolation: travelling front Velocity p ≈ p c N p ∼ L d f , y M ( t ) ≃ N p vt = L d f − 1 y L y , M ( t ) u ( p ) = P ( p ) v f ( p ) p ≫ p c N p ≃ pL y v = lim v f (1) N p t t →∞ Ly= 50 1 Ly=100 Ly=200 0.9 Ly=400 0.8 v f (p)/v f (1) 0.7 0.5 v f (p c )/v f (1) 0.6 0.5 0.4 0.4 50 100 200 400 Ly 0.3 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Percolation: travelling front Front speed M ( t ) v = lim u ( p ) = P ( p ) v f ( p ) N p t t →∞ v f (1) small α u ( p ) = P ( p ) v f ( p ) v f (1) ∼ P ( p ) N p L y large α � γ u ( p ) = P ( p ) v f ( p ) � p − p c v f (1) ∼ P ( p ) 1 − p c angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

Percolation: travelling front Front width angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

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 angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013

m ( t ) ∼ t d l . m ( t ) ∼ r ( t ) d f . Therefore r ( t ) ∼ t d l /d f , and v = dr dt ∼ t d l /d f − 1 ∼ r 1 − d m in , where d m in = d f d l . 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 ∼ ξ 1 − d m in . Moreover, analysis of the percolation phase transition gives ξ ∼ | p − p c | − ν , with ν = 4 / 3 for d = 2 [ ? ], which gives the final scaling v ∼ ( p − p c ) γ , where γ = − ν (1 − d m in ). For the average velocity, the scaling is: � γ u ( p ) = P ( p ) v f ( p ) � p − p c v f (1) ∼ P ( p ) .d f (1) 1 − p c angelo.vulpiani@roma1.infn.it Dep. Physics Univ. Sapienza Rome jeudi 18 juillet 2013