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 gt 1)

Critical event Exit from y1 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 ??
• Theres 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 G
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 G distribution, n1,2,
• For each r(n), draw Nm WTs from exponential
• PDF with rate r(n) tnj , j1,Nm
• Slow Modulation Limit Nm ? 8

P. Allegrini et al., Phys. Rev. E 73, 046136
(2006)
15
Pareto distribution with T1 and µ1.8 ta 0,
20, 60
16
(No Transcript)
17
(No Transcript)
18
Asymptotic value of AIF ? Aging Indicator (AI)

independent from t
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 ton and toff)

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
ta AI s
20 1.04 0.02
60 0.99 0.01
100 1.03 0.01
140 1.01 0.01
180 0.95 0.01
220 0.96 0.02
ta AI s
20 0.68 0.03
60 0.69 0.02
100 0.81 0.02
140 0.80 0.02
180 0.74 0.02
220 0.87 0.02
S. Bianco et al., J. Chem. Phys. 123, 174704
(2005)
23
(No Transcript)
24
(No Transcript)
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)