Weak Ergodicity Breaking on the Nano-Scale Eli Barkai Bar-Ilan - - PowerPoint PPT Presentation

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Weak Ergodicity Breaking on the Nano-Scale Eli Barkai Bar-Ilan University Bel, Burov, Margolin, Metzler, Rebenshtok Kyoto 2015 Eli Barkai, Bar-Ilan Univ. Outline Single molecule experiments exhibit weak ergodicity breaking. Power law

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SLIDE 1

Weak Ergodicity Breaking on the Nano-Scale

Eli Barkai Bar-Ilan University Bel, Burov, Margolin, Metzler, Rebenshtok

Kyoto 2015

Eli Barkai, Bar-Ilan Univ.

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SLIDE 2

Outline

  • Single molecule experiments exhibit weak ergodicity breaking.
  • Power law blinking quantum dots.
  • sub-Diffusion of molecules in the live cell.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 3

Ergodicity

Ergodicity: time averages = ensemble averages.

x = limt→∞

t

0 x(τ)dτ

t

.

x = ∞

−∞ xP eq(x)dx.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 4

Ergodicity out of equilibrium

δ2 (∆, t) = t−∆ [x(t′ + ∆) − x(t′)]2 dt′ t − ∆ → 2D∆

Eli Barkai, Bar-Ilan Univ.

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SLIDE 5

Quantum Jumps: Atoms

Stefani, Hoogenboom, Barkai Physics Today 62, 34 (2009).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 6

Quantum Dots

Stefani, Hoogenboom, Barkai Physics Today 62, 34 (2009).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 7

Blinking Nano Crystals (coated CdSe)

100 200 300 400 500 100 600 700 800 900 1000 1100 100 1200 1300 1400 1500 1600 1700 100 1800 1900 2000 2100 2200 2300 100

Intensity

2400 2500 2600 2700 2800 2900 100 3000 3100 3200 3300 3400 3500 100

time (sec)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 8

Non-ergodic Intensity Correlation Functions

10

−4

10

−2

10 0.2 0.4 0.6 0.8 1

t / T Intensity correlation function

Experiment Brokmann, Dahan et al Phys. Rev. Lett. (2003). Theory: Margolin, EB Phys. Rev. Lett. 90, 104101 (2005).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 9

Power Law Distribution of on and Off times

10

−1

10 10

1

10

−4

10

−3

10

−2

10

−1

10 10

1

τ PDF

τon τoff 0.039τ −1.51

Power law waiting time ψ(τ) ∼ τ −(1+αoff). Averaged time in States On and Off is infinite τ = ∞.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 10

Weak and strong Ergodicity Breaking

System is decomposed → strong ergodicity breaking. System’s space explored → weak ergodicity breaking.

  • J. Bouchaud J. Phys. I France (1992).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 11

Continuous Time Random Walk (CTRW)

Dispersive Transport in Amorphous Material Scher-Montroll (1975). Bead Diffusing in Polymer Network Weitz (2004).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 12

Average Waiting Time is ∞. Diffusion is anomalous r2 ∝ tα.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 13

mRNA diffusing in a cell Golding and Cox

Eli Barkai, Bar-Ilan Univ.

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SLIDE 14

Golding and Cox PRL (2006) He Burov Metzler EB PRL (2008). Lubelski, Sokolov Klafter (ibid). Kepten, .... EB, Garini PRL (2009).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 15

CTRW

  • Renewal type of Random walk on lattice.
  • Jumps to nearest neighbors only.
  • qx (1 − qx) Prob. of jumping from x to x − 1 (x + 1).
  • waiting times are i.i.d r.v with pdf

ψ(t) ∝ t−1−α 0 < α < 1

Eli Barkai, Bar-Ilan Univ.

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SLIDE 16

Time Averages

  • Occupation fraction

px = tx t .

  • Time average:

O =

  • x=−L,L

Oxpx

  • For example

X =

L

  • x=−L

xpx.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 17

Trajectory Unbiased RW q = 1/2 α = 1/2

20000 40000 60000 80000 1e+05

time

  • 4
  • 2

2 4

Cell Number

Eli Barkai, Bar-Ilan Univ.

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SLIDE 18

Ergodic vs Non-ergodic Phases

  • 4
  • 2

2 4

Cell Number

0.2 0.4 0.6 0.8 1

Fraction Occupation time

α=2

  • 4
  • 2

2 4

Cell Number

0.2 0.4 0.6 0.8 1

Fraction Occupation time

α=0.5

Eli Barkai, Bar-Ilan Univ.

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SLIDE 19

Population Dynamics in Step Number

Px (n + 1) = qx+1Px+1 (n) + (1 − qx−1) Px−1 (n) . When n → ∞, an equilibrium is obtained P eq

x (n + 1) = P eq x (n)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 20

Levy Statistics

  • nx number of times particle visits site x.
  • When n → ∞ nx/n = P eq

x

  • tx total time spent in state x. Sum i.i.d r.v. whose mean is infinite.
  • Apply Lévy’s limit theorem

f (tx) = lα,AαP eq

x n (tx) .

  • Use

px = tx L

x=−L tx Eli Barkai, Bar-Ilan Univ.

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SLIDE 21

The PDF of TIME AVERAGES

Using O =

x pxOx

fα

  • O
  • = −1

π lim

ǫ→0 Im

L

x=1 P eq x

  • O − Ox + iǫ

α−1 L

x=1 P eq x

  • O − Ox + iǫ

α .

Ergodicity if α → 1

fα=1

  • O
  • = δ
  • O − O
  • .

Localization when α → 0

lim

α→0 fα

  • O
  • =

L

  • x=1

P eq

x δ

  • O − Ox
  • .

Rebenshtok, Barkai PRL 99 210601 (2007)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 22

PDF of X UNBIASED CTRW

−0.5 −0.25 0.25 0.5 2 4 6 X / L f( X / L ) α=0 α=0.2 α=0.5 α=0.8 α=1

Eli Barkai, Bar-Ilan Univ.

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SLIDE 23

Directions

Blinking QDs Margolin, Kuno.

1/f noise Kantz, Niemann, Krapf, Leibovich.

Deterministic models, relation with weak chaos Bel, Korabel, Akimoto. Disordered systems Burov. Distribution of Diffusion and Transport Coefficients Burov, Metzler. fBM Deng Lévy walks CTRW Bel, Rebenshtok. Fractional Feynman-Kac functionals Turgeman, Carmi. Aging correlation functions Margolin, Leibovich. Infinite Ergodic theory Korabel, Akimoto, Hanggi.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 24

100 200 300 400 500 100 600 700 800 900 1000 1100 100 1200 1300 1400 1500 1600 1700 100 1800 1900 2000 2100 2200 2300 100

Intensity

2400 2500 2600 2700 2800 2900 100 3000 3100 3200 3300 3400 3500 100

time (sec)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 25

Group Material Nu. Radii T αon αoff Dahan CdSe-ZnS 215 1.8nm 300 K 0.58(0.17) 0.48(0.15) Orrit CdS 2.85 1.2 EXP 0.65(0.2) Bawendi CdTe.... 200 1.5 10 − 300 0.5(0.1) 0.5(0.1) Kuno CdSe-ZnS 300 2.7 300 0.8 − 1.0 0.5 Cichos Si 300 0.8 − 1.0 0.5 Ha CdSe(coat) 300 Exp? 1

Eli Barkai, Bar-Ilan Univ.

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SLIDE 26

Uncapped NC

  • n (short)
  • ff (long)

Capped NC

  • n (short)
  • n (long)
  • ff (long)

Efros, Orrit, Onsager, Hong-Noolandi rOns = e2 kbTǫ ≃ 7nm (1)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 27

Distribution of time averaged intensity I

0.2 0.4 0.6 0.8 1 10 20 30 Experiment T’ = 36s I P(I) 0.2 0.4 0.6 0.8 1 10 20 30 Experiment T’ = 360s I P(I) 0.2 0.4 0.6 0.8 1 10 20 30 Experiment T’ = 3600s I P(I) 0.2 0.4 0.6 0.8 1 10 20 30 Simulations T’ = 36s I P(I) 0.2 0.4 0.6 0.8 1 10 20 30 Simulations T’ = 360s I P(I) 0.2 0.4 0.6 0.8 1 10 20 30 Simulations T’ = 3600s I P(I)

CdSe-ZnS NC. Margolin, Kuno, Barkai (2006)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 28

Random Time-Scale Invariant Diffusion Constant

10

4

10

5

10

−1

10 10

1

10

2

10

3

∆ δ2 ( ∆ ) 2 × 103

δ2 (∆, t) = t−∆ [x(t′ + ∆) − x(t′)]2 dt′ t − ∆

He Burov Metzler EB PRL (2008)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 29

Anomalous Seems Normal

δ2 ∼ 2Dα Γ(1 + α) ∆ t1−α

  • A taste for this: if α = 1 δ2 = 2D∆.

For anomalous diffusion D(t) ∼ dx2/dt ∼ tα−1.

  • We see aging effect δ2 decreases when measurement time increases.
  • Anomalous diffusion seems normal δ2 ∼ ∆.
  • For closed system different behavior δ2 ∼ ∆1−α where ∆ < t.
  • Burov, metzler, Barkai PNAS (2010)

Eli Barkai, Bar-Ilan Univ.

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SLIDE 30
  • δ2 ∼ N. [Hint [x(t′ + ∆) − x(t′)]2 = 0 when particle is trapped].
  • ξ = δ2/δ2

1 2 3 4 5 0.5 1 1 2 3 4 5 0.5 1 ξ φα ( ξ ) α = 0.5 α = 0.75

lim

t→∞ φα (ξ) = Γ1/α (1 + α)

αξ1+1/α lα Γ1/α (1 + α) ξ1/α

  • .

Eli Barkai, Bar-Ilan Univ.

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SLIDE 31

Aging effect (Diego Krapf’s experiment)

5000 10000 20000 30000 1 10 100

Time averaged mean squared displacement Measurement time t [sec]

111 ms 222 ms 333 ms 444 ms

  • The older you get the slower you are.
  • Channel protein molecules on a membrane.
  • Weigel · · · Krapf PNAS 2011

Eli Barkai, Bar-Ilan Univ.

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SLIDE 32

Waiting time distribution (Krapf)

101 102 103 104 0.1 1 10 100 101 102 103

!(") Clustered channels !(") Free channels " [sec] "-1.9

RTH

2= 500 nm

RTH

2=1000 nm

RTH

2=2000 nm

Free channels

Power law waiting times lead to aging and weak ergodicity breaking Barkai, Garini and Metzler Physics Today Aug. (2012).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 33

Boltzmann--Gibbs WEB

normal diffusion anomalous diffusion r2 ∼ tα Gaussian Lévy f1

  • O
  • = δ
  • O − O
  • fα
  • O
  • = − 1

π limǫ→0 Im L

x=1 P eq x (O−Ox+iǫ) α−1

L

x=1 P eq x (O−Ox+iǫ) α .

Chaos λ = 0, Infinite Invariant Density δ2 = x2 Transport Coefficients Random

Eli Barkai, Bar-Ilan Univ.

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SLIDE 34

Reviews

  • Stefani, Hoogenboom, and Barkai Beyond Quantum Jumps: Blinking Nano-scale

Light Emitters Physics Today 62 nu. 2, p. 34 (February 2009).

  • E. Barkai, Y. Garini and R. Metzler Strange Kinetics of Single Molecules in the

Cell Physics Today 65(8), 29 (2012).

  • R. Metzler, J, H. Jeon, A. G. Cherstvy, and E. Barkai Anomalous diffusion

models and their properties: non-stationarity, non-ergodicity and ageing at the centenary of single particle tracking Phys. Chem. Chem. Phys. 16 (44), 24128 - 24164 (2014).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 35

Quenched Trap Model (Burov EB)

x

U(x)/kT

∆x ∆x ∆x ∆x ∆x

ρ(E) = 1 Tg exp(− E Tg ). Ux = U det

x

− Ex.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 36

Dynamics and Occupation Fraction

The dynamics are described by the master equation

d dtPx(t) = − 1 τx Px(t) + 1 2τx+1 Px+1(t) + 1 2τx−1 Px−1(t) τi = exp(Ex T ).

Since Ei are exponentially distributed

ψ(τ) = T Tg τ

−1− T

Tg

When T/Tg < 1 the model exhibits anomalous diffusion.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 37

Occupation Fraction for the Quenched Trap Model

The occupation fraction in a domain x1 < x < x2

p = tx t ∼ ZObs Z = x2

x=x1 exp

  • −Udet

x

−Ex T

  • Z

where Z is the normalizing partition function. For a single realization of disorder, and for a finite system, the

  • ccupation fraction is given by Boltzmann statistics.

The occupation fraction is a random variable since {Ex} are random variables.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 38

Tg < T is the effective temperature of the system.

Our main result for T/Tg < 1

f (p) ∼ δT/Tg [Rx (Tg) , p] Rx(Tg) = PB(Tg) 1 − PB(Tg) PB(Tg) = x2

x=x1 exp

  • −Udet

Tg

  • Z

.

The temperature Tg yield the statistical properties of the occupation fraction. For T > Tg standard Boltzmann Gibbs statistics is valid, even after averaging over disorder

f (p) ∼ δ(p − PB).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 39

PDF of occupation fraction α = T/Tg = 0.3

0.2 0.4 0.6 0.8 1

tx/t

0.5 1 1.5 2 2.5

PDF

Theory Simulation

Eli Barkai, Bar-Ilan Univ.

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SLIDE 40

PDF of occupation fraction α = T/Tg = 0.7

0.2 0.4 0.6 0.8 1

tx/t

1 2 3 4

PDF

Theory Simulation

U(x) = x, Tg = 1, observation domain 0 < x < 1.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 41

PDF of occupation fraction α = T/Tg = 3

0.2 0.4 0.6 0.8 1

tx/t

50 100 150 200 250 300

PDF

U(x) = x, Tg = 1, observation domain 0 < x < 1.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 42

Generality of Result for Quenched Disorder

Quenched disorder: p = tx

t ∼ ZObs Z .

Weak Ergodicity Breaking: p = tx

t =

  • i τi(x)

t

. If Z is Lévy distributed behavior similar to weak ergodicity is found. Models of anomalous diffusion in disorder systems: Z Lévy distributed.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 43

The Geisel Map

  • 0.5

0.5 1 1.5 2 2.5

xt

  • 1

1 2 3

xt+1

mapping rule path

In a unit cell

xt+1 = xt + axz, 0 < x < 0.5

Eli Barkai, Bar-Ilan Univ.

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SLIDE 44

CTRW Dynamics

In vicinity of fixed point

dx dt = axz

Smooth injection of trajectories

ψ(t) ∝ t−(1+α), α = 1 z − 1.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 45

Random Occupation Times

2000 4000 6000 8000 10000 time (a)

  • 4
  • 2

2 4 Cell Number(t)

  • 4
  • 2

2 4 Cell Number (b) 0.2 0.4 0.6 0.8 1 Fraction of Occupation Time

  • 4
  • 2

2 4 Cell Number (c) 0.2 0.4 0.6 0.8 1 Fraction of Occupation Time

  • 4
  • 2

2 4 Cell Number (d) 0.2 0.4 0.6 0.8 1 Fraction of Occupation Time Eli Barkai, Bar-Ilan Univ.

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SLIDE 46

Occupation Time Statistics

0 0.2 0.4 0.6 0.8 1

(a)

0.1 0.2 0.3 0.4 0.5

f(tm/t)

0 0.2 0.4 0.6 0.8 1

tm/t (b)

0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1

(c)

0.0005 0.001 0.0015 0.002 0.0025 0.003 Theoretical Simulation

Bel, Barkai Europhysics Letters 74 15 (2006).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 47

Visitation Fraction

  • 4
  • 2

2 4

(a) Cell Number (b)

0.05 0.1

Visitation Fraction

  • 4
  • 2

2 4 0.05 0.1

Visitation fraction is uniform, in and out of the ergodic phase, hence weak ergodicity breaking.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 48

Intermittency, Zero Lyapunov Exponent

Pesin identity λ = hks. Intermittent dynamics: zero Lyapunov exponent λ = 0. Stretched exponential separation of nearby trajectories:

δx = δx0 exp(λαtα).

Our aim: Generalize Pesin Identity. Take Away: Intermittency is related to Weak Ergodicity Breaking.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 49

Pomeau Manneville Map

0,0 0,5 1,0 0,0 0,5 1,0

M(x) x

ξ x0

xt+1 = M(xt) M(xt) ∼ xt + a(xt)z xt → 0 λα = t−1

t=0 ln M ′(xt)

tα

Eli Barkai, Bar-Ilan Univ.

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SLIDE 50

Trajectory

2000 4000 0,0 0,5 1,0 1,5 1 ln(|M'(x t )|) t S

t−1

  • t=0

ln M ′(xt) ∝ t/τ ∝ t t tt−1−αdt ∝ tα Distribution of number of renewals in (0, t) yields distribution of λα.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 51

Distribution of generalized Lyapunov Exp.

ζ = λα/λα Renewal Theory: distribution of λα is Mittag-Leffler. Korabel Barkai Phys. Rev. Lett. 102, 050601 (2009).

Eli Barkai, Bar-Ilan Univ.

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SLIDE 52

Infinite Invariant Measure (Aaronson, Thaler, · · ·)

10
  • 2
10
  • 1
10 10 10 1 10 2 10 3 10 4 x c (t 3 ) x c (t 4 ) (x) x x c (t 2 ) ~x
  • 1/

¯ ρ(x) = ρ(x, t) tα−1 ¯ ρ(x) ∝ x−1/α Non Normalizable.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 53

Generalized Lyapunov Exp.

1 2 3 4 5 1 2 3 4 5 < > z

λα =

  • ln |M ′(x)|¯

ρ(x)dx Even though ¯ ρ(x) non normalizable, it yields the average.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 54

Pesin Type of Identity

Krengel Entropy hα is the Kolomogorov Sinai entropy of the first return map (Zweimüller, Thaler).

st = 0 left branch. st = 1 right branch. S = 00011110101 · · · = (0)(00)(1)(11)(10)(101) · · ·. n(t) number of words (n(t) = 6). hα = n log2 n

tα

  • hα = αλα.

A link between separation of trajectories and entropy.

Eli Barkai, Bar-Ilan Univ.

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SLIDE 55

Boltzmann--Gibbs WEB

normal diffusion anomalous diffusion r2 ∼ tα Gaussian Lévy -- Lamperti f1

  • O
  • = δ
  • O − O
  • fα
  • O
  • = − 1

π limǫ→0 Im L

x=1 P eq x (O−Ox+iǫ) α−1

L

x=1 P eq x (O−Ox+iǫ) α .

Chaos λ = 0, Infinite Invariant Density δ2 = x2 Transport Coefficients Random

Eli Barkai, Bar-Ilan Univ.

slide-56
SLIDE 56
  • Refs. and THANKS
  • Turgeman, Carmi, Barkai Phys. Rev. Lett. Fractional Feynman-Kac Equation

for non Brownian Functionals Phys. Rev. Lett. 103, 190201 (2009).

  • Stefani, Hoogenboom, and Barkai Beyond Quantum Jumps: Blinking Nano-scale

Light Emitters Physics Today 62 nu. 2, p. 34 (February 2009).

  • Korabel, Barkai Pesin-Type Identity for Intermittent Dynamics with a Zero

Lyapunov Exponent Phys. Rev. Lett. 102, 050601 (2009).

  • Rebenshtok, Barkai Weakly non-Ergodic Statistical Physics Journal of Statistical

Mechanics 133 565 (2008).

  • He,

Burov, Metzler, Barkai Random Time-Scale Invariant Diffusion and Transport Coefficients Phys. Rev. Lett. (2008).

  • Burov, Barkai, Occupation Time Statistics in the Quenched Trap Model. Phys.
  • Rev. Lett. 98 250601 (2007).
  • Bel, Barkai Weak Ergodicity Breaking in the Continuous-Time Random Walk
  • Phys. Rev. Lett. 94 240602 (2005).
  • Margolin Barkai Non-ergodicity of Blinking Nano Crystals and Other Lévy Walk

Processes Phys. Rev. Letters 94 080601 (2005).

Eli Barkai, Bar-Ilan Univ.