Weak Ergodicity Breaking in Continuous Time Random Walk

1. Weak Ergodicity Breaking in Continuous Time Random Walk June 28, 2006 Golan Bel (UCSB) Eli Barkai (BIU)

2. Ergodicity • Ensemble of non interacting particles Thermal equilibrium Partition function

3. Single particle, time measurement

4. Conditions for Ergodicity • All phase space is visited. • The fraction of occupation time is proportional to the fraction of phase space volume. • Microscopic time scale exists. • Sections of the measured signal are independent. • Independent of initial condition in the long time limit.

5. Ergodicity Breaking • Strong non-ergodicity: Phase space is apriori divided into mutually inaccessible regions. Dynamics is limited. • Weak non-ergodicity: Phase space is connected, but the fraction of occupation time is not equal to the fraction of phase space occupied. Dynamics exists over the whole phase space. J. P. Bouchaud, J. De Physique I (1992).

6. Motivation • Single molecule experiments remove the problem of ensemble average. • In many single molecule experiments the microscopic time scale diverges. • What replace Boltzmann-Gibbs statistical mechanics in this case?

7. Weitz’s Experiment I. Y. Wong et al, Phys. Rev. Lett. (2004) Trajectory

8. Power law waiting time PDF

9. Anomalous Diffusion

10. Subdiffusion in living yeast cells I. M. Tolic-Norrelikkeet al, Phys. Rev. Lett. (2004)

11. Continuous Time Random walk

13. Trajectories

14. Fraction of occupation time histogram 1 2 3 4 1,2,3, Non - Ergodic 4, Ergodic

15. Two States Process

17. First Passage Time • Relation between the Survival probability in discrete time RW and CTRW In Laplace space

18. Lamperti’s PDF For unbiased and uniformly biased CTRW an exact solution of the FPT PDF exists, allows to determine .

19. Fraction of Occupation Time PDF

20. PDF of the fraction of occupation time in unbiased CTRW

21. Consider the CTRW as function of visits number The master equation

22. Visitation Fraction The master equation describes both discrete time RW and CTRW thus the visitation fraction in both cases is equal and given by Detailed balance

23. VF in unbiased CTRW (periodic boundary conditions)

24. VF in unbiased CTRW (reflecting b.c.)

25. Visitation Fraction and Ensemble Average in Harmonic Potential

26. Derivation of Lamperti’s PDF Using the Visitation Fraction

27. Summing over n

28. Tauberian theorem Inverting the double Laplace transform This solution recovers the exact solution for the uniformly biased CTRW

29. Visitation Fraction and Ensemble Average in Harmonic Potential

30. Fraction of Occupation Time PDF on the bottom of harmonic potential

31. Fraction of occupation time PDF on the bottom of harmonic potential

34. In ergodic system microscopic time scale exists, thus the visitation fraction is equal to the fraction of occupation time, which in turn is equal to the equilibrium probability in ensemble sense. • In the case where the visitation fraction is equal to the equilibrium probability, but due to divergence of the microscopic time scale, the fraction of occupation time is not equal to the equilibrium probability, the system is said to exhibit weak ergodicity breaking.

35. If both the visitation fraction and the fraction of occupation time are not equal to the equilibrium probability, the system exhibits strong ergodicity breaking.

36. Conclusion • CTRW with power law PDF of sojourn times exhibits weak ergodicity breaking. • Weak non-ergodicity in the context of CTRW was precisely defined. • The weak non-ergodicity was quantified by the universal probability density function of the fraction of occupation time. • Generalization of Boltzmann-Gibbs statistical mechanics to weakly non-ergodic system is possible.

37. References • G. Bel, E. Barkai, PRL94, 240602 (2005). • G. Bel, E. Barkai, PRE 73, 016125 (2006). • G. Bel, E. Barkai, EPL 74, 15 (2006).