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

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 blinking quantum dots. • sub-Diffusion of molecules in the live cell. Eli Barkai, Bar-Ilan Univ.

Ergodicity Ergodicity: time averages = ensemble averages. � t 0 x ( τ ) dτ . x = lim t →∞ t � ∞ −∞ xP eq ( x ) dx . � x � = Eli Barkai, Bar-Ilan Univ.

Ergodicity out of equilibrium � t − ∆ [ x ( t ′ + ∆) − x ( t ′ )] 2 dt ′ δ 2 (∆ , t ) = 0 → 2 D ∆ t − ∆ Eli Barkai, Bar-Ilan Univ.

Quantum Jumps: Atoms Stefani, Hoogenboom, Barkai Physics Today 62, 34 (2009). Eli Barkai, Bar-Ilan Univ.

Quantum Dots Stefani, Hoogenboom, Barkai Physics Today 62, 34 (2009). Eli Barkai, Bar-Ilan Univ.

Blinking Nano Crystals (coated CdSe) 100 0 0 100 200 300 400 500 100 0 600 700 800 900 1000 1100 100 Intensity 0 1200 1300 1400 1500 1600 1700 100 0 1800 1900 2000 2100 2200 2300 100 0 2400 2500 2600 2700 2800 2900 100 0 3000 3100 3200 3300 3400 3500 time (sec) Eli Barkai, Bar-Ilan Univ.

Non-ergodic Intensity Correlation Functions 1 Intensity correlation function 0.8 0.6 0.4 0.2 0 −4 −2 0 10 10 10 t / T Experiment Brokmann, Dahan et al Phys. Rev. Lett. (2003). Theory: Margolin, EB Phys. Rev. Lett. 90, 104101 (2005). Eli Barkai, Bar-Ilan Univ.

Power Law Distribution of on and Off times τ on τ off 1 10 0.039 τ −1.51 0 10 PDF −1 10 −2 10 −3 10 −4 10 −1 0 1 10 10 10 τ Power law waiting time ψ ( τ ) ∼ τ − ( 1+ α off ) . Averaged time in States On and Off is infinite � τ � = ∞ . Eli Barkai, Bar-Ilan Univ.

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.

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.

Average Waiting Time is ∞ . Diffusion is anomalous � r 2 � ∝ t α . Eli Barkai, Bar-Ilan Univ.

mRNA diffusing in a cell Golding and Cox Eli Barkai, Bar-Ilan Univ.

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.

CTRW • Renewal type of Random walk on lattice. • Jumps to nearest neighbors only. • q x (1 − q x ) 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.

Time Averages • Occupation fraction p x = t x t . • Time average: � O = O x p x x = − L,L • For example L � X = xp x . x = − L Eli Barkai, Bar-Ilan Univ.

Trajectory Unbiased RW q = 1 / 2 α = 1 / 2 4 2 Cell Number 0 -2 -4 0 20000 40000 80000 60000 1e+05 time Eli Barkai, Bar-Ilan Univ.

Ergodic vs Non-ergodic Phases α=2 α=0.5 1 1 Fraction Occupation time Fraction Occupation time 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -4 -2 0 2 4 -4 -2 0 2 4 Cell Number Cell Number Eli Barkai, Bar-Ilan Univ.

Population Dynamics in Step Number P x ( n + 1) = q x +1 P x +1 ( n ) + (1 − q x − 1 ) P x − 1 ( n ) . When n → ∞ , an equilibrium is obtained P eq x ( n + 1) = P eq x ( n ) Eli Barkai, Bar-Ilan Univ.

Levy Statistics • n x number of times particle visits site x . • When n → ∞ n x /n = P eq x • t x total time spent in state x . Sum i.i.d r.v. whose mean is infinite. • Apply Lévy’s limit theorem f ( t x ) = l α,AαP eq x n ( t x ) . • Use t x p x = � L x = − L t x Eli Barkai, Bar-Ilan Univ.

The PDF of TIME AVERAGES Using O = � x p x O x � α − 1 � L x =1 P eq � O − O x + iǫ = − 1 ǫ → 0 Im x � � f α O π lim � α . � L x =1 P eq � O − O x + iǫ x Ergodicity if α → 1 � � � � f α =1 O = δ O − �O� . Localization when α → 0 L � P eq � � � � α → 0 f α lim O = x δ O − O x . x =1 Rebenshtok, Barkai PRL 99 210601 (2007) Eli Barkai, Bar-Ilan Univ.

PDF of X UNBIASED CTRW α =0 6 α =0.2 α =0.5 α =0.8 α =1 4 f( X / L ) 2 0 −0.5 −0.25 0 0.25 0.5 X / L Eli Barkai, Bar-Ilan Univ.

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.

100 0 0 100 200 300 400 500 100 0 600 700 800 900 1000 1100 100 Intensity 0 1200 1300 1400 1500 1600 1700 100 0 1800 1900 2000 2100 2200 2300 100 0 2400 2500 2600 2700 2800 2900 100 0 3000 3100 3200 3300 3400 3500 time (sec) Eli Barkai, Bar-Ilan Univ.

Group Material Nu. Radii T α on α off Dahan CdSe-ZnS 1 . 8 nm 300 K 215 0 . 58(0 . 17) 0 . 48(0 . 15) Orrit CdS EXP 2 . 85 1 . 2 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) Exp? 300 1 Eli Barkai, Bar-Ilan Univ.

Uncapped NC on (short) off (long) Capped NC on (short) on (long) off (long) Efros, Orrit, Onsager, Hong-Noolandi e 2 k b Tǫ ≃ 7 nm r Ons = (1) Eli Barkai, Bar-Ilan Univ.

Distribution of time averaged intensity I 30 30 Experiment Simulations T’ = 36s T’ = 36s 20 20 P ( I ) P ( I ) 10 10 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I I 30 30 Experiment Simulations T’ = 360s T’ = 360s 20 20 P ( I ) P ( I ) 10 10 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I I 30 30 Experiment Simulations T’ = 3600s T’ = 3600s 20 20 P ( I ) P ( I ) 10 10 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I I CdSe-ZnS NC. Margolin, Kuno, Barkai (2006) Eli Barkai, Bar-Ilan Univ.

Random Time-Scale Invariant Diffusion Constant 3 10 2 10 δ 2 ( ∆ ) 1 10 0 10 −1 10 2 × 10 3 4 5 10 10 ∆ � t − ∆ [ x ( t ′ + ∆) − x ( t ′ )] 2 dt ′ δ 2 (∆ , t ) = 0 t − ∆ He Burov Metzler EB PRL (2008) Eli Barkai, Bar-Ilan Univ.

Anomalous Seems Normal 2 D α ∆ � δ 2 � ∼ t 1 − α Γ(1 + α ) • A taste for this: if α = 1 � δ 2 � = 2 D ∆ . For anomalous diffusion D ( t ) ∼ d � x 2 � /dt ∼ t α − 1 . • We see aging effect � δ 2 � decreases when measurement time increases. • Anomalous diffusion seems normal � δ 2 � ∼ ∆ . • For closed system different behavior � δ 2 � ∼ ∆ 1 − α where ∆ < t . • Burov, metzler, Barkai PNAS (2010) Eli Barkai, Bar-Ilan Univ.

• δ 2 ∼ N . [Hint [ x ( t ′ + ∆) − x ( t ′ )] 2 = 0 when particle is trapped]. • ξ = δ 2 / � δ 2 � 1 α = 0.5 0.5 0 φ α ( ξ ) 0 1 2 3 4 5 1 α = 0.75 0.5 0 0 1 2 3 4 5 ξ t →∞ φ α ( ξ ) = Γ 1 /α (1 + α ) � Γ 1 /α (1 + α ) � lim l α . αξ 1+1 /α ξ 1 /α Eli Barkai, Bar-Ilan Univ.

Aging effect (Diego Krapf’s experiment) Time averaged mean squared displacement 111 ms 222 ms 333 ms 30000 444 ms 20000 10000 5000 1 10 100 Measurement time t [sec] • The older you get the slower you are. • Channel protein molecules on a membrane. • Weigel · · · Krapf PNAS 2011 Eli Barkai, Bar-Ilan Univ.

Waiting time distribution (Krapf) 2 = 500 nm R TH 10 4 10 3 2 =1000 nm R TH 2 =2000 nm R TH Free channels ! ( " ) Clustered channels ! ( " ) Free channels 10 3 10 2 " -1.9 10 2 10 1 10 1 10 0 0.1 1 10 " [sec] Power law waiting times lead to aging and weak ergodicity breaking Barkai, Garini and Metzler Physics Today Aug. (2012). Eli Barkai, Bar-Ilan Univ.

Boltzmann--Gibbs WEB normal diffusion anomalous diffusion � r 2 � ∼ t α Gaussian Lévy α − 1 � L x =1 P eq x ( O−O x + iǫ ) = − 1 π lim ǫ → 0 Im � � � � � � f 1 O = δ O − �O� f α O α . x =1 P eq � L x ( O−O x + iǫ ) Chaos λ = 0 , Infinite Invariant Density δ 2 = � x 2 � Transport Coefficients Random Eli Barkai, Bar-Ilan Univ.

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.

Quenched Trap Model (Burov EB) U(x)/kT ∆ x ∆ x ∆ x ∆ x ∆ x x ρ ( E ) = 1 exp( − E ) . T g T g U x = U det − E x . x Eli Barkai, Bar-Ilan Univ.