Weak Ergodicity Breaking Eli Barkai Bar-Ilan University Bel, Burov, Korabel, Margolin, Rebenshtok WCHAOS 211
Outline Single molecule experiments exhibit weak ergodicity breaking. Blinking quantum dots, sub-diffusion of mrna in the cell. Distribution of time averages. Disordered system.
Ergodicity Ergodicity: time averages = ensemble averages. x = lim t!1 R t x()d t. hxi = R 11 xp eq (x)dx.
Main Result: DISTRIBUTION of TIME AVERAGES Let O be a physical observable, and O its time average, f O = P L 1 1 lim! Im x=1 P eq x O Ox + i P : L x=1 P eq x O Ox + i Weak Ergodicity Breaking is related to Anomalous Diffusion hx 2 i t < < 1: Ergodic case = 1 f1 O = O hoi : Rebenshtok, Barkai PRL 99 2161 (27)
Ergodicity out of equilibrium 2 ( ; t) = R t [x(t + ) x(t )] 2 dt t! 2D
Quantum Jumps: Atoms Stefani, Hoogenboom, Barkai Physics Today 62, 34 (29).
Quantum Dots Stefani, Hoogenboom, Barkai Physics Today 62, 34 (29).
Blinking Nano Crystals (coated CdSe) Intensity 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 21 22 23 24 25 26 27 28 29 3 31 32 33 34 35 time (sec)
Non-ergodic Intensity Correlation Functions Intensity correlation function 1.8.6.4.2 1 4 1 2 1 t / T Experiment Brokmann, Dahan et al Phys. Rev. Lett. (23). Theory: Margolin, EB Phys. Rev. Lett. 9, 1411 (25).
Power Law Distribution of on and Off times τ on 1 1 τ off.39τ 1.51 1 PDF 1 1 1 2 1 3 1 4 1 1 1 1 1 τ Power law waiting time ( ) ( 1+ of f ). Averaged time in States On and Off is infinite h i = 1.
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).
Continuous Time Random Walk (CTRW) Dispersive Transport in Amorphous Material Scher-Montroll (1975). Bead Diffusing in Polymer Network Weitz (24).
Average Waiting Time is 1. Diffusion is anomalous hr 2 i / t.
mrna diffusing in a cell Golding and Cox
Golding and Cox PRL (26) He Burov Metzler EB PRL (28). Lubelski, Sokolov Klafter (ibid).... EB, Garini PRL (29).
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 (t) / t 1 < < 1
Time Averages Occupation fraction p x = t x t : Time average: O = X x= L;L O x p x For example X = LX x= L xp x :
Trajectory Unbiased RW q = 1=2 = 1=2 4 Cell Number 2-2 -4 2 4 6 8 1e+5 time
Ergodic vs Non-ergodic Phases α=2 α=.5 1 1 Fraction Occupation time.8.6.4.2 Fraction Occupation time.8.6.4.2-4 -2 2 4 Cell Number -4-2 2 4 Cell Number
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! 1, an equilibrium is obtained P eq x (n + 1) = P eq x (n)
Levy Statistics n x number of times particle visits site x. When n! 1 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 ;A P eq x n (t x) : Use p x = t x P L x= L t x
The PDF of TIME AVERAGES Using O = P x p xo x f O = Ergodicity if! 1 1 lim P L Im x=1 P eq 1 x O O x + i P! L x=1 P eq f =1 O = O hoi : x O O x + i : Localization when! lim f O =! LX x=1 P eq x O O x : Rebenshtok, Barkai PRL 99 2161 (27)
PDF of X UNBIASED CTRW f( X / L ) 6 4 α= α=.2 α=.5 α=.8 α=1 2.5.25.25.5 X / L
Directions Blinking QDs Margolin, Kuno. Stochastic Models: CTRW model Bel, Rebenshtok. Deterministic models, relation with weak chaos Bel, Korabel. Disordered systems Burov. Distribution of Diffusion and Transport Coefficients Burov, Metzler. Transition from single particle to the ensemble Burov. Fractional Feynman-Kac functionals Turgeman, Carmi.
Intensity 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 21 22 23 24 25 26 27 28 29 3 31 32 33 34 35 time (sec)
Group Material Nu. Radii T on of f Dahan CdSe-ZnS 215 1:8nm 3 K :58(:17) :48(:15) Orrit CdS 2:85 1:2 EXP :65(:2) Bawendi CdTe... 2 1:5 1 3 :5(:1) :5(:1) Kuno CdSe-ZnS 3 2:7 3 :8 1: :5 Cichos Si 3 :8 1: :5 Ha CdSe(coat) 3 Exp? 1
Uncapped NC on (short) off (long) Capped NC on (short) on (long) off (long) Efros, Orrit, Onsager, Hong-Noolandi r Ons = e2 k b T ' 7nm (1)
Distribution of time averaged intensity I 3 2 P (I) Experiment T = 36s 3 2 P (I) Simulations T = 36s 1 1 2 P (I).2.4.6.8 1 I 3 1 2 P (I) Experiment T = 36s.2.4.6.8 1 I 3 1 Experiment T = 36s.2.4.6.8 1 I 2 P (I).2.4.6.8 1 I 3 1 2 P (I) Simulations T = 36s.2.4.6.8 1 I 3 1 Simulations T = 36s.2.4.6.8 1 I CdSe-ZnS NC. Margolin, Kuno, Barkai (26)
Quenched Trap Model U(x)/kT x x x x x x (E) = 1 T g exp( U x = U det x E x : E T g ):
Dynamics and Occupation Fraction The dynamics are described by the master equation d dt P x(t) = 1 P x (t) + 1 P x+1 (t) + 1 P x 1 (t) x 2 x+1 2 x 1 i = exp( E x T ): Since E i are exponentially distributed ( ) = T T g 1 T Tg When T=T g < 1 the model exhibits anomalous diffusion.
Occupation Fraction for the Quenched Trap Model The occupation fraction in a domain x 1 < x < x 2 p t x = t ZObs Z = P x2 x=x 1 exp U det where Z is the normalizing partition function. Z x E x T For a single realization of disorder, and for a finite system, the occupation fraction is given by Boltzmann statistics. The occupation fraction is a random variable since fe x g are random variables.
T g < T is the effective temperature of the system. Our main result for T=T g < 1 f (p) T =Tg [R x (T g ) ; p] The temperature T g fraction. R x (T P B(T g ) g ) = 1 P B (T g ) P B (T g ) = P x2 x=x 1 exp Z U det T g yield the statistical properties of the occupation For T > T g standard Boltzmann Gibbs statistics is valid, even after averaging over disorder f (p) (p P B ): :
PDF of occupation fraction = T =T g = :3 2.5 2 Theory Simulation 1.5 PDF 1.5.2.4.6.8 1 t x /t
PDF of occupation fraction = T =T g = :7 4 Theory Simulation 3 PDF 2 1.2.4.6.8 1 t x /t U (x) = x, T g = 1, observation domain < x < 1.
PDF of occupation fraction = T =T g = 3 3 25 2 PDF 15 1 5.2.4.6.8 1 t x /t U (x) = x, T g = 1, observation domain < x < 1.
The Averaged Occupation Fraction U det (x) = x.8 Simulation Theory.6 <t x /t>.4.2 1 2 3 4 5 T/Tg
Generality of Result for Quenched Disorder Quenched disorder: p = t x t ZObs Z. Weak Ergodicity Breaking: p = t x t = P i i(x) t. If Z is Lévy distributed behavior similar to weak ergodicity is found. Models of anomalous diffusion in disorder systems: Z Lévy distributed.
Boltzmann--Gibbs normal diffusion WEB anomalous diffusion hr 2 i t Gaussian Lévy -- Lamperti P L 1 f 1 O = O hoi f O = lim! Im x=1 P eq x (O O x +i) 1 P L : x=1 P x eq (O O x +i) Chaos =, Infinite Invariant Density 2 = hx 2 i Transport Coefficients Random
Refs. and THANKS Stefani, Hoogenboom, and Barkai Beyond Quantum Jumps: Blinking Nano-scale Light Emitters Physics Today 62 nu. 2, p. 34 (February 29). Rebenshtok, Barkai Weakly non-ergodic Statistical Physics Journal of Statistical Mechanics 133 565 (28). Burov, Barkai, Occupation Time Statistics in the Quenched Trap Model. Phys. Rev. Lett. 98 2561 (27). Bel, Barkai Weak Ergodicity Breaking in the Continuous-Time Random Walk Phys. Rev. Lett. 94 2462 (25). Margolin Barkai Non-ergodicity of Blinking Nano Crystals and Other Lévy Walk Processes Phys. Rev. Letters 94 861 (25).
The Geisel Map 3 mapping rule path 2 x t+1 1-1 -.5.5 1 1.5 2 2.5 x t In a unit cell x t+1 = x t + ax z ; < x < :5
CTRW Dynamics In vicinity of fixed point dx dt = axz Smooth injection of trajectories (t) / t (1+) ; = 1 z 1 :
Random Occupation Times Cell Number(t) Fraction of Occupation Time 4 2-2 -4 1.8.6.4.2 2 4 6 8 1 time (a) -4-2 2 4 Cell Number (c) Fraction of Occupation Time Fraction of Occupation Time 1.8.6.4.2 1.8.6.4.2-4 -2 2 4 Cell Number (b) -4-2 2 4 Cell Number (d)
Occupation Time Statistics Theoretical Simulation.5.3.5.4.25.4.3.2 f(t m /t).3.2.15.2.1.1.1.5.2.4.6.8 1 (a).2.4.6.8 1 t m /t (b).2.4.6.8 1 (c) Bel, Barkai Europhysics Letters 74 15 (26).
Visitation Fraction Visitation Fraction.1.5.1.5-4 -2 2 4-4 -2 2 4 (a) Cell Number (b) Visitation fraction is uniform, in and out of the ergodic phase, hence weak ergodicity breaking.
Intermittency, Zero Lyapunov Exponent Pesin identity = h ks. Intermittent dynamics: zero Lyapunov exponent =. Stretched exponential separation of nearby trajectories: x = x exp( t ): Our aim: Generalize Pesin Identity. Take Away: Intermittency is related to Weak Ergodicity Breaking.
Pomeau Manneville Map 1, M (x ),5 x,,,5 ξ 1, x x t+1 = M (x t ) M (x t ) x t + a(x t ) z x t! = P t 1 t= ln M (x t ) t
Trajectory 1,5 ln( M'(x t ) ) 1, S,5 1, 2 4 t Xt 1 t= ln M (x t ) / t=hi / t R t tt 1 dt / t Distribution of number of renewals in (; t) yields distribution of.
Distribution of generalized Lyapunov Exp. = =h i Renewal Theory: distribution of is Mittag-Leffler. Korabel Barkai Phys. Rev. Lett. 12, 561 (29).
Infinite Invariant Measure (Aaronson, Thaler, ) x c (t 4 ) 1 4 x c (t 3 ) (x) 1 3 1 2 x c (t 2 ) 1 1 ~x -1/ 1 1-2 1-1 1 x (x) = (x; t) t 1 (x) / x 1= Non Normalizable.
Generalized Lyapunov Exp. 5 4 < > 3 2 1 1 2 3 4 5 z h i = Z ln jm (x)j(x)dx Even though (x) non normalizable, it yields the average.
Pesin Type of Identity Krengel Entropy h is the Kolomogorov Sinai entropy of the first return map (Zweimüller, Thaler). s t = left branch. s t = 1 right branch. S = 111111 = ()()(1)(11)(1)(11). n(t) number of words (n(t) = 6). h = h n log 2 n i t h = h i: A link between separation of trajectories and entropy.
Boltzmann--Gibbs normal diffusion WEB anomalous diffusion hr 2 i t Gaussian Lévy -- Lamperti P L 1 f 1 O = O hoi f O = lim! Im x=1 P eq x (O O x +i) 1 P L : x=1 P x eq (O O x +i) Chaos =, Infinite Invariant Density 2 = hx 2 i Transport Coefficients Random
Refs. and THANKS Turgeman, Carmi, Barkai Phys. Rev. Lett. Fractional Feynman-Kac Equation for non Brownian Functionals Phys. Rev. Lett. 13, 1921 (29). Stefani, Hoogenboom, and Barkai Beyond Quantum Jumps: Blinking Nano-scale Light Emitters Physics Today 62 nu. 2, p. 34 (February 29). Korabel, Barkai Pesin-Type Identity for Intermittent Dynamics with a Zero Lyapunov Exponent Phys. Rev. Lett. 12, 561 (29). Rebenshtok, Barkai Weakly non-ergodic Statistical Physics Journal of Statistical Mechanics 133 565 (28). He, Burov, Metzler, Barkai Random Time-Scale Invariant Diffusion and Transport Coefficients Phys. Rev. Lett. (28). Burov, Barkai, Occupation Time Statistics in the Quenched Trap Model. Phys. Rev. Lett. 98 2561 (27). Bel, Barkai Weak Ergodicity Breaking in the Continuous-Time Random Walk Phys. Rev. Lett. 94 2462 (25). Margolin Barkai Non-ergodicity of Blinking Nano Crystals and Other Lévy Walk Processes Phys. Rev. Letters 94 861 (25).