Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?

P. Paradisi Institute of Atmospheric Sciences and Climate (CNR), Lecce UnitS. BiancoCenter for Nonlinear Sciences, University of North TexasP. Grigolini, Institute of Chemical and Physical Processes (CNR), PisaCenter for Nonlinear Sciences, University of North TexasDepartment of Physics, Pisa University

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 ?

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

eventsD.R. Cox, Renewal Theory, Chapman and Hall, London (1962)

• Poisson processes:

exponential distribution of WTs

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

Example: Manneville Map

Model for Turbulence Intermittency:alternance of Laminar Regions and Chaotic Bursts

1;0;)1(mod)(1 zyyy znnn

Laminar Regions with long Residence (Exit) Times Waiting TimesShort and Intense Bursts have the effect of erasingmemory random critical event

P. Manneville, J. Physique 41, 1235 (1980)

Renewal AgingManneville-type stochastic model (z > 1)

10 yyrdt

dy z

Critical event: Exit from y=1 WT= Exit TimeRandom back injection, uniform in [0,1]

Pareto distribution of WTs

1;0;)(

)1()(1

TT

T

P. Allegrini et al., Phys. Rev. E 68, 056123 (2003)

Liouville equation for the time evolutionof the probability distribution:

),1(),(

),(tptypy

yr

t

typ z

After a critical event, the system restarts from a newrandom initial condition (uniform distribution).

P. Allegrini et al., Phys. Rev. E 68, 056123 (2003)

Aging in Renewal Processes is related to the time evolution of p(y,t)

Starting observation at time ta implies observing an aged WT statistics

Possibility of using this property as an indicator of “Renewal Aging”

ddytypata )(),(

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

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:

dWTd Pr)(P. Allegrini et al., Phys. Rev. E 73, 046136 (2006)

S. Bianco et al., J. Chem. Phys. 123, 174704 (2005)

)(1

)(0

ydyK

a

a

a

t

t

rent

Renewal Aged PDF (approximated expression)

Experimental Aged PDF

WTs of hystogram)(exp TRUNCATEDat

Survival Probability

0

')'(1')'()( dd

G. Aquino et al., Phys. Rev. E 70, 036105 (2004)PP et al., AIP Proceedings 800 (1), 92-97 (2005)

Aging Intensity Function (AIF)

)()(

)()()(

exp

rent

ta

a

aI

RenewalAging

1)()()(exp arentt Iaa

No Aging0)()()(exp at Ia

Modulation Processes

0

)()|()( drrprvpvp eqMB

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)

Slow Modulation of a Poisson process

0

)|(;)()|()( rPeqP errdrrpr

ondistributi Pareto)(ondistributi)( rpeq

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): τn

j , j=1,Nm

• Slow Modulation Limit: Nm → ∞

P. Allegrini et al., Phys. Rev. E 73, 046136 (2006)

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

Asymptotic value of AIF → Aging Indicator (AI) independent from τ

S. Bianco et al., J. Chem. Phys. 123, 174704 (2005)PP et al., AIP Proceedings 800 (1), 92-97 (2005)

7.0;)(

mNAI

Poisson pseudo-events and critical events

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)

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

ta AI σ

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 σ

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

OFF State ON State

S. Bianco et al., J. Chem. Phys. 123, 174704 (2005)

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)

)( renta