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 ? x2(t) ∼ tα, 0 < α < 1 (or logarithms etc.)

• Stochastic process: correlated increments. non-stationary

increments.

• Fractional Brownian motion (homogeneous medium). Diffusion
• n 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)

P(x, t)

Continuous time random walk x(t)

ψ(t, x) = ψ(t)λ(x) Ψ(t) = ∞

t

ψ(t′) dt′ P(k, s) = Ψ(s) 1 − ψ(s, k)

tstep = ∞ tψ(t) dt

finite ?

x2step = ∞ x2λ(x) dx

finite ?

x2(t) ∼ Dt x2(t) ∼ Kαtα 0 < α < 1

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

PD,r(D, r) = PD(D)Pr(r|D) PD(D) ∼ Dσ−1 with σ > 0, for small D (e.g. Γ dist.) Pr(r|D) has mean E [r|D] = D(1−γ)/2, −∞ < γ < ∞

• r Pτ(τ|D) has mean E [τ|D] = D−γ

λ(x|τ, t) = 1

• 2πD(τ)t

exp −x2 2D(τ)t

• , ψ(τ) ∼ τ −σ/γ−1

(0) (I) (II) γ < σ σ < γ < σ + 1 σ + 1 < γ Annealed 1 σ/γ 1 - 1/γ Quenched 1d 1 2σ/(σ + γ) ?

x2(t) ∼ tα

Massignan, Manzo, Torreno-Pina, Garc´ ıa-Parajo, Lewenstein, Lapeyre, PRL 112 150603 (2014)

• Subordination. CTRW is “subordinator” of another process.

2nd process. MSD x2(t) ∼ tβ Combined processes. MSD x2(t) ∼ tαβ Time-ensemble MSD x2(t)T ∼ T α−1t1−α+αβ =

t T 1−αtαβ

Weigel, Simon, Tamkun, Krapf, PNAS (2011) Meroz, Sokolov, Klafter PRE (rc) (2010)

x2(t)T =

• t

T

1−αx2(t)

Continuous time Random Walk (CTRW) Waiting time distribution ψ(t) ∼ t−α−1 0 < α < 1 Ensemble averaged MSD x2(t) ∼ tα

He, Burov,Metzler,Barkai PRL (2008) Lubelski, Sokolov, Klafter, PRL (2008)

Time ensemble averaged MSD x2(t)T ∼ T α−1t =

t T 1−αtα Time-ensemble Avg. MSD

x(0) x(T)

t

t ≪ T 0 < β < 1

t ≪ T

Disordered Confinement

P(r) ∼ r−1−c, c > 0

τ ≈ 1.82 Pr(|C| = s) ∼ s−0.82 (!) P ∗(r) ∼ r−1−c, c = 1.64

Fischer exponent

Sheinman, Sharma, Alvarado, Koenderink, MacKintosh PRL (2015) Nat.Phys. (2013)

• x2(t) ∼

∞

r∗ rc 0 r−c−1Da ta f

Da ta r2

• dr

f(z) ∼ z−1 Converges for 0 < c < 2

• x2(t) ∼ rc

0D

2−c 2

a

t

a(2−c) 2

z∗ z

c−2 2 f(z) dz

∞ Change variable. Get dimensionless integral.. . . Convergence ?

Displacement of every particle bounded. Ensemble MSD unbounded

• x2(t) ∼ rc

0 D 1− c

2

a

ta(1− c

2), 0 < c < 2

x2(t)T ∼ rc

0 D 1− c

2

a

t T 1−α ta(1− c

2)

x2(t)T = t

T

1−αx2(t)

a → a

• 1 − c

2

• Probability density of “radii”

P(r) ∼ rc

0 r−c−1,

0 < c

• x2(t) =

∞ P(r)x2(t)r dr

Average MSD over random radii Lapeyre arXiv:1504.07158

0.001 0.01 0.1 1 10 100 1000 10000 101 102 103 104 105 106 107 108 t (lag time) α = . 8 α = 0.7 α = 0.9

x2(t)T ∼ T 1−α t1−α+αβ(1− c

2)

β = 0.7, 0.8, 0.9, 1.0 c = 0.8, 1.0

T ≈ const.

All finite clusters Only infinite cluster

k = 2ν − β 2ν − β + µ. k′ = 2ν 2ν − β + µ

Gefen, Aharony, Alexander, PRL (1983)

Walk

x2(t) ∼ tk Percolation: “Natural” scale-free confined disorder.

Walk on percolation p = pc, x2(t) ∼ tk

k′ = 2ν 2ν − β + µ Free diffusion on “incipient” infinite cluster Subdiffusion due purely to walk on random fractal

ta → ta(1− c

2)

Scale-free confinement → Ratio of exponents = 1 − c

2 r ∼ sσν

Pr(|C| = s) ∼ s1−τ 2 − τ = −σβ From known exponents, one easily finds c = 3β/ν for percolation.

c = β ν

Does it agree? 1−c 2 = 1− β 2ν = 2ν − β 2ν = k k′ k k′ = 2ν − β 2ν Ratio of exponents, with and without confinement No conductivity exponent µ.

k = 2ν − β 2ν − β + µ Walk only on all finite clusters of occupied sites. Subdiffusion has two sources: 1) Walk on random fractal 2) Scale free confinement.

Thanks!

• F. H¨
• fling 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
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• 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

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anggi, and A. Gadomski Phys. Rev. E 51, 57625769 (1995) Aggregation

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090602 (2011) disorder on periodic potential

• J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A 35, 3081 (1987) CTRW