Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?

1. Aging in Blinking Quantum Dots: Renewal or Slow Modulation ? P. Paradisi Institute of Atmospheric Sciences and Climate (CNR), Lecce Unit S. Bianco Center for Nonlinear Sciences, University of North Texas P. Grigolini, Institute of Chemical and Physical Processes (CNR), Pisa Center for Nonlinear Sciences, University of North Texas Department of Physics, Pisa University

2. Outline • Renewal processes • an example: the Manneville Map • Renewal Aging • How can we evaluate the amount of Renewal Aging in a time series ? • Renewal Aging in Modulation processes • Application to Blinking Quantum Dots: renewal or slow modulation ?

3. Renewal Processes • Stochastic process with: - recurrent (critical) events associated with some pattern of the system variables - Waiting Times (WTs) are mutually independent random variables - WT = time interval between two critical events D.R. Cox, Renewal Theory, Chapman and Hall, London (1962)

4. Poisson processes: exponential distribution of WTs • Interesting case: power-law tail in the distribution of WTs (Non-Poisson renewal processes)

5. Example: Manneville Map Model for Turbulence Intermittency: alternance of Laminar Regions and Chaotic Bursts Laminar Regions with long Residence (Exit) Times Waiting Times Short and Intense Bursts have the effect of erasing memory random critical event P. Manneville, J. Physique 41, 1235 (1980)

6. Renewal Aging Manneville-type stochastic model (z > 1) Critical event: Exit from y=1 WT= Exit Time Random back injection, uniform in [0,1] Pareto distribution of WTs P. Allegrini et al., Phys. Rev. E 68, 056123 (2003)

7. Liouville equation for the time evolution of the probability distribution: After a critical event, the system restarts from a new random initial condition (uniform distribution). P. Allegrini et al., Phys. Rev. E 68, 056123 (2003)

8. Aging in Renewal Processes is related to the time evolution of p(y,t) Starting observation at time ta implies observing an aged WT statisticsPossibility of using this property as an indicator of “Renewal Aging”

9. Important Facts • Poisson processes have zero renewal aging • Non-zero Renewal aging for Non-Poisson renewal processes • Dependence on the distribution of WTs • Approximate analytical results available for Pareto (power-law) distribution of WTs

10. Description of the method • Definition of critical events in the time series • WTs sequence • WTs are correlated ? YES no renewal theory NO ?? There’s some chance of having a (Non-Poisson) renewal process • Compute hystogram of WTs: P. Allegrini et al., Phys. Rev. E 73, 046136 (2006) S. Bianco et al., J. Chem. Phys. 123, 174704 (2005)

11. Renewal Aged PDF (approximated expression) Experimental Aged PDF Survival Probability G. Aquino et al., Phys. Rev. E 70, 036105 (2004) PP et al., AIP Proceedings 800 (1), 92-97 (2005)

12. Aging Intensity Function (AIF) Renewal Aging No Aging

13. Modulation Processes Slow modulation of relaxation rate (friction) in an Orstein-Ulenbeck process (Ordinary Brownian Motion, Maxwell-Boltzmann equilibrium distribution): Equilibrium probability peq(r) given by a Γ distribution p(v) in agreement with “Tsallis” energy distribution C. Beck, Phys. Rev. Lett. 87, 180601 (2001)

14. Slow Modulation of a Poisson process • Numerical simulations: • Draw r(n) from Γ distribution, n=1,2,… • For each r(n), draw Nm WTs from exponential • PDF with rate r(n): τnj , j=1,Nm • Slow Modulation Limit: Nm → ∞ P. Allegrini et al., Phys. Rev. E 73, 046136 (2006)

15. Pareto distribution with T=1 and μ=1.8 ta = 0, 20, 60

18. Asymptotic value of AIF → Aging Indicator (AI) independent from τ PP et al., AIP Proceedings 800 (1), 92-97 (2005) S. Bianco et al., J. Chem. Phys. 123, 174704 (2005)

19. Poisson pseudo-events and critical events

20. Application to BQDs • Laser stimulation → ON-OFF intermittency • 100 sequences of Photon Emission Intensity • Duration of each experiment: 1h, f =10-3 Hz Data made available by Prof. M. Kuno and V.Protasenko, Dept. Of Chemistry and Biochemistry, University of Notre Dame • Distinction of ON and OFF states: iterative method for the definition of the threshold [Kuno et al., J. Chem. Phys. 115, 1028, 2001] • Wts are Residence Times in the ON or OFF state (distinction between τon and τoff) R.G. Neuhauser et al., Phys. Rev. Lett. 85, 3301 (2000)

21. Example of BQD Emission Intensity Sequence (typical jumps between ON and OFF state)

22. OFF State ON State S. Bianco et al., J. Chem. Phys. 123, 174704 (2005)

25. Conclusions and future developments • BQDs cannot be described by a slow modulation process • Other systems could be described by slow modulation (single enzyme catalysis, Strechted Exponential PDF, see Poster Session) • BQDs are reasonably described by a Non-Poisson renewal process (some Poisson pseudo-events) • Aging Analysis also applied to financial data (Mittag-Leffler Survival Probability) S. Bianco and P. Grigolini, Chaos Solitons and Fractals, accepted • Improvement of the method → exact expression of (Algorithm for the numerical inversion of Laplace transform)