Brownian motion (and more) in disordered media John Lapeyre - PowerPoint PPT Presentation

Brownian motion (and more) in disordered media John Lapeyre IDAEA/CSIC, Barcelona July 7, 2015 Barcelona MHetScale

What are the possible sources of observed anomalous (sub)diffusion? Need a more precise question. What does subdiffusive mean ? � x 2 ( t ) � ∼ t α , 0 < α < 1 (or logarithms etc.) • Stochastic process: correlated increments. non-stationary increments. • Fractional Brownian motion (homogeneous medium). Diffusion on scale-free disorder: percolation clusters, fractals. • Continuous time random walk with anomalously long waiting times between steps. Traps. • Aggregating particles. What is diffusion on (scale-free) disordered media ? • Disordered diffusivity. • Disordered confinement (reflecting barriers or confining potentials)

Continuous time random walk x(t) P ( x, t ) � ∞ ψ ( t, x ) = ψ ( t ) λ ( x ) Ψ( t ) = ψ ( t ′ ) d t ′ t Ψ( s ) P ( k, s ) = 1 − ψ ( s, k ) � ∞ � ∞ � x 2 � step = x 2 λ ( x ) d x � t � step = tψ ( t ) d t finite ? finite ? 0 0 � x 2 ( t ) � ∼ Dt � x 2 ( t ) � ∼ K α t α 0 < α < 1

Manzo, Torreno-Pina, Massignan, Lapeyre, Lewenstein, Garc´ ıa-Parajo, PRX 5 011021 (2015)

P D,r ( D, r ) = P D ( D ) P r ( r | D ) P D ( D ) ∼ D σ − 1 with σ > 0 , for small D (e.g. Γ dist.) P r ( r | D ) has mean E [ r | D ] = D (1 − γ ) / 2 , −∞ < γ < ∞ or P τ ( τ | D ) has mean E [ τ | D ] = D − γ � − x 2 � 1 , ψ ( τ ) ∼ τ − σ/γ − 1 λ ( x | τ, t ) = � exp 2 D ( τ ) t 2 πD ( τ ) t � x 2 ( t ) � ∼ tα (0) (I) (II) σ < γ < σ + 1 σ + 1 < γ γ < σ Annealed 1 σ / γ 1 - 1/ γ Quenched 1d 1 2 σ /( σ + γ ) ? Massignan, Manzo, Torreno-Pina, Garc´ ıa-Parajo, Lewenstein, Lapeyre, PRL 112 150603 (2014)

Continuous time Random Walk (CTRW) Waiting time distribution ψ ( t ) ∼ t − α − 1 0 < α < 1 Ensemble averaged MSD � x 2 ( t ) � ∼ t α x (0) x ( T ) t Time-ensemble Avg. MSD � t � 1 − α t α Time ensemble averaged MSD � x 2 ( t ) � T ∼ T α − 1 t = T t ≪ T He, Burov,Metzler,Barkai PRL (2008) Lubelski, Sokolov, Klafter, PRL (2008) Subordination. CTRW is “subordinator” of another process. 2nd process. MSD � x 2 ( t ) � ∼ t β 0 < β < 1 Combined processes. MSD � x 2 ( t ) � ∼ t αβ � t � 1 − α t αβ Time-ensemble MSD � x 2 ( t ) � T ∼ T α − 1 t 1 − α + αβ = T Meroz, Sokolov, Klafter PRE (rc) (2010) Weigel, Simon, Tamkun, Krapf, PNAS (2011) � � 1 − α � x 2 ( t ) � t � x 2 ( t ) � T = t ≪ T T

Disordered Confinement P ( r ) ∼ r − 1 − c , c > 0 Nat.Phys. (2013) Sheinman, Sharma, Alvarado, Koenderink, MacKintosh PRL (2015) Pr( |C| = s ) ∼ s − 0 . 82 (!) τ ≈ 1 . 82 Fischer exponent P ∗ ( r ) ∼ r − 1 − c , c = 1 . 64

0 r − c − 1 , P ( r ) ∼ r c Probability density of “radii” 0 < c � ∞ � P ( r ) � x 2 ( t ) � r d r � x 2 ( t ) � = Average MSD over random radii 0 � D a t a � � ∞ � 0 r − c − 1 D a t a f r ∗ r c � x 2 ( t ) � ∼ d r r 2 Change variable. Get dimensionless integral.. . . Convergence ? ∞ � z ∗ f ( z ) ∼ z − 1 2 − c a (2 − c ) � c − 2 � x 2 ( t ) � ∼ r c 2 f ( z ) d z 2 0 D t z 2 a 0 Converges for 0 < c < 2 � � 1 − c 1 − c � � x 2 ( t ) � ∼ r c t a (1 − c 2 ) , 0 < c < 2 a → a 0 D 2 a 2 � t � 1 − α � t � 1 − α � x 2 ( t ) � � � 1 − c x 2 ( t ) � T ∼ r c t a (1 − c 2 ) � x 2 ( t ) � T = 0 D 2 a T T Displacement of every particle bounded. Ensemble MSD unbounded Lapeyre arXiv:1504.07158

10000 β = 0 . 7 , 0 . 8 , 0 . 9 , 1 . 0 1000 c = 0 . 8 , 1 . 0 100 10 1 T ≈ const. α = 0 . 9 0.1 8 . 0 = � � x 2 ( t ) � T ∼ T 1 − α t 1 − α + αβ (1 − c α = 0 . 7 α 2 ) 0.01 0.001 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 t (lag time)

Percolation: “Natural” scale-free confined disorder. All finite clusters Only infinite cluster Walk 2 ν 2 ν − β k ′ = � x 2 ( t ) � ∼ t k k = 2 ν − β + µ. 2 ν − β + µ Gefen, Aharony, Alexander, PRL (1983)

Walk on percolation p = p c , � x 2 ( t ) � ∼ t k 2 ν Free diffusion on “incipient” infinite cluster k ′ = 2 ν − β + µ Subdiffusion due purely to walk on random fractal 2 ν − β Walk only on all finite clusters of occupied sites. k = Subdiffusion has two sources: 2 ν − β + µ 1) Walk on random fractal 2) Scale free confinement. k ′ = 2 ν − β k Ratio of exponents, with and without confinement 2 ν No conductivity exponent µ . t a → t a (1 − c 2 ) Scale-free confinement → Ratio of exponents = 1 − c Pr( |C| = s ) ∼ s 1 − τ 2 r ∼ s σν 2 − τ = − σβ c = β From known exponents, one easily finds c = 3 β/ν for percolation. ν Does it agree? k ′ � 1 − c 2 = 1 − β 2 ν = 2 ν − β = k 2 ν

Thanks! • F. H¨ ofling and T. Franosch, Rep. Prog. Phys. 76, 046602 (2013) recent review • R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014) recent review • J. Klafter and I. M. Sokolov, First Steps in Random Walks (Oxford University Press, Oxford, 2011) “elementary”, but very useful • I. M. Sokolov, Physics 1 (2008), 10.1103/Physics.1.8 correlated, vs. non-stationary, aimed at experiment • S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987) classic review • J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 127 classic review (1990) • N. Destainville, A. Sauli´ ere, and L. Salom´ e, Biophys. J. 95, 3117 (2008) confinement, intermediate time anomaly • T. Neusius, I. M. Sokolov, and J. C. Smith, Phys. Rev. E 80, 011109 (2009) confinement time average • Burov, Metzler, Barkai, PNAS (2010) confinement time average • A. G. Cherstvy, A. V. Chechkin, and R. Metzler, Soft Matter 10, 1591 (2014) heterogeneous D • J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, and R. Metzler, Phys. Rev. Lett. 106, 048103 (2011) WEB in experiment • J. Luczka, P. H¨ anggi, and A. Gadomski Phys. Rev. E 51, 57625769 (1995) Aggregation • M. Khoury, A. M. Lacasta, J. M. Sancho, and K. Lindenberg, Phys. Rev. Lett. 106, 090602 (2011) disorder on periodic potential • J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A 35, 3081 (1987) CTRW