Weak Ergodicity Breaking. Manchester 2016

Transcription

1 Weak Ergodicity Breaking Eli Barkai Bar-Ilan University Burov, Froemberg, Garini, Metzler PCCP 16 (44), (2014) Akimoto, Saito Manchester 2016

2 Outline Experiments: anomalous diffusion of single molecules in the cell. Important for regulation of cell function. Random time scale invariant diffusion and transport coefficients. Ergodic hypothesis does not work. Theory of weak ergodicity breaking.

3 Brownian Motion δ 2 (, t) = t 0 [x(t + ) x(t )] 2 dt t 2D

4 Ergodicity for Brownian particles Time averages are reproducible. Two measurements of δ 2 yield the same result. The time and ensemble averages coincide (ergodicity). δ 2 = x 2. Diffusion is normal δ 2. Measure between (0, t) and (t, 2t) yield same results (stationary). These properties are broken in single molecule experiments in cells.

5 mrna diffusing in a cell Golding and Cox

6 Golding and Cox PRL (2006). Anomalous diffusion and randomness of time averages is common.

7 Telomeres in the Nucleus of Mammalian Cell Bronstien,... Barkai, Garini PRL (2009).

8 Bronstien,... Barkai, Garini PRL (2009).

9 Questions Characterize the anomalous diffusion in live cells. Why are time averages irreproducible? Popular models: CTRW, fbm, RW on Fractal, quenched trap model, are they useful? Previous work x 2. We need δ 2. Biological significance of anomalous diffusion? Physical consequence: time averaged transport and diffusion constants in disordered medium are random.

10 Continuous Time Random Walk (CTRW) Dispersive Transport in Amorphous Material Scher-Montroll (1975). Bead Diffusing in Polymer Network Weitz (2004).

11 Average Waiting Time is. Diffusion is anomalous r 2 t α.

12 CTRW: power law waiting times ψ(t) t (α+1)

13 8 6 α = x tim e Random walk on lattice with jumps to nearest neighbours only. ψ(t) t 1 α with 0 < α < 1 gives r 2 t α

14 CTRW and ergodicity breaking Einstein D = δx2 2 τ. Boltzmann-Gibbs: If measurement time t >> τ expect ergodicity. Scher-Montroll: If τ, D 0 and the process is sub-diffusive. Bouchaud: If τ expect weak ergodicity breaking.

15 Random Time-Scale Invariant Diffusion Constant δ 2 ( ) δ 2 (, t) = t 0 [x(t + ) x(t )] 2 dt t He Burov Metzler Barkai PRL (2008), Lubelski, Sokolov, Klafter (ibid).

16 Anomalous Seems Normal δ 2 2D α Γ(1 + α) t 1 α Normal diffusion δ 2 = 2D. For anomalous diffusion D(t) d x2 dt t α 1. Aging effect δ 2 decreases when measurement time increases.

17 Fluctuations of the time average δ 2 [x (t+ )-x (t)] 2 4 = x (t) α = T IM E δ 2 s t number of jumps in (0, t) is s. Most of measurement time [x(t + ) x(t )] 2 = 0 due to long trapping.

18 Distribution of δ 2 ξ = s s = δ2 δ 2, lim φ α (ξ) = Γ1/α (1 + α) t αξ 1+1/α l α [ Γ 1/α ] (1 + α). ξ 1/α α = 0.5 φ α ( ξ ) α = ξ

19 Finite size effect is important: Anomalous again δ 2 = t 0 [ x 2 (t + ) + x 2 (t ) 2 x(t + )x(t ) ] dt. t We consider the fractional Fokker-Planck dynamics in a binding field V (x) If x B = 0 namely V (x) = V ( x). x(t 1 )x(t 2 ) x 2 B(t 1 /t 2, α, 1 α) B. Γ(α)Γ(1 α) δ 2 x 2 B 2 sin(απ) (1 α)απ ( t ) 1 α Neusius,Sokolov, Smith (PRE) Burov, Metzler, Barkai (PNAS) 2010.

20 Aging effect (Diego Krpaf s experiment) Time averaged mean squared displacement ms 222 ms 333 ms 444 ms Measurement time t [sec] The older you get the slower you are Channel protein molecules on a membrane. Weigel Krapf PNAS (2011).

21 Waiting time distribution (Krapf) 10 4 R TH 2 = 500 nm R TH 2 =1000 nm R TH 2 =2000 nm Free channels 10 3 ψ(τ) Clustered channels τ ψ(τ) Free channels τ [sec] 10 0 Power law waiting times leads to aging and weak ergodicity breaking. Barkai, Garini, Metzler Physics Today Aug. (2012).

22 !! "#$! Granule subdiffusion: finite size effect #!" #!" #!" #!"!!"!"!"# %&!"#!" $%&'()*+! Jeon... Barkai, Oddershede, Metzler PRL 106, (2011). $%!"!"#$!!

23 Three more experiments showing ageing MSD Receptor Motion in Living Cells Manzo... Garcia Parajo PRX (2015). Insulin granule in pancreatic cell Tabei... Scherer PNAS (2013). Myosin motors in filament system Burov... Dinner PNAS (2013).

24 Quenched trap model τ i = exp(e x /k B T ) and E x IID RV. Exponential density of states ρ(e) = exp( E/T g )/T g Sojourn time PDF ψ(τ) τ (1+T/T g)

25 Ergodic properties of the quenched trap model For a finite system of size L the largest waiting time is finite. It is determined by the deepest trap. Quenched trap model is ergodic when we take t before L. Quenched model exhibits non self averaging.

26 Nonergodicity mimics inhomogeneity (SOKOLOV) How to quantify the fluctuations of the quenched model? Annealed versus quenched models. Where are fluctuations larger? Is Mittag-Leffler statistics universal? Role of initial conditions? state. equilibrium versus non equilibrium initial Role of dimension d = 2 is critical.

27 Distribution of diffusion constant The size of the system is crucial Γ ( α 1) δ 2 dis = 2 αγ (1 α) 1/α L 1/α 1. Starting from thermal initial conditions, periodic boundary conditions, the ensemble average MSD is r 2 eq = t i τ i/l. The SA parameter SA = O2 dis O 2 dis. O 2 dis The EB parameter EB = O2 O 2 O 2.

28 SA versus EB parameters SA α

29 Distribution of diffusivities α = 2/3, D = P(D) D

30 Ergodicity in equilibrium O = O(x)µ(dx) O = lim t 1 t t 0 O(t )dt For example x = x.

31 Weak Ergodicity Breaking in CTRW O = x p x O x p x = t x /t the occupation fraction O x value of the observable in state x. ergodicity p x Px eq. ( ) Lx=1 f α O = 1 π lim Px ɛ 0 Im eq (O O x +iɛ) α 1 Lx=1 Px eq (O O x +iɛ) α. Ergodicity if α 1 f α=1 ( O ) = δ ( O O ). Rebenshtok, Barkai PRL (2007)

32 PDF of X UNBIASED CTRW f( X / L ) 6 4 α=0 α=0.2 α=0.5 α=0.8 α= X / L

33 Summary Single molecule tracking in the cell is important for qualitative modelling of dynamics relevant for life. The time averaged mean square displacement is a random variable, similar to the CTRW and QTM model predictions. In equilibrium, for closed system, fluctuations of quenched system far exceed the annealed case. Aging effect is observed, δ 2 with lag time. decreases with measurement time t and increases Weak ergodicity breaking describes statistics of irreproducible time averages of several other systems and models: blinking quantum dots and deterministic dynamics with weak chaos.

34 Refs. and THANKS Theory: Barkai, Garini and Metzler Strange Kinetics of Single Molecules in the Cell Physics Today 65(8), 29 (2012). Metzler, Jeon, Cherstvy, and Barkai Anomalous diffusion models and their properties: non-stationarity, non-ergodicity and ageing at the centenary of single particle tracking Physical Chemistry Chemical Physics 16 (44), (2014). Akimoto, Saito EB (unpublished). Experiments: Bronstein, Israel, Kepten, Mai, Shav-Tal, Barkai, Garini Transient Anomalous Diffusion of Telomeres in the Nucleus of Mammalian Cells Phys. Rev. Lett. 103, (2009). Jeon, Tejedor, Burov, Barkai, Selhuber-Unkel, Berg-Sorensen, Oddershede, and Metzler In vivo anomalous diffusion and weak ergodicity breaking of lipid granules Phys. Rev. Lett. 106, (2011).

35 Diffusion maps (a) y (b) y x x 0

36 Fractional Fokker Planck vs Fractional Brownian Motio Fractional Fokker Planck Equation (non-ergodic, non stationary) α t αp (x, t) = L fpp (x, t) Physical picture: trap model, CTRW. Fractional Langevin equation (ergodic, stationary) α 1 mẍ + mγ a t α 1ẋ(t) + U (x) = F noise (t) Physical picture: single file diffusion, certain polymer models. Deng, Barkai PRE (2009). Metzler, Barkai, Klafter PRL 82, 3563 (1999).

37 And what about active super diffusion? Certain active processes in cell exhibit super diffusion δ 2 ξ and ξ > 1. Lévy walks and fractional Brownian motions are models of such behaviour. The time average MSD remains random when average sojourn time diverges. Froemberg EB PRE (R) (2013).