aging in blinking quantum dots: renewal or slow modulation ?
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/1.jpg)
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
![Page 2: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/2.jpg)
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 ?
![Page 3: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/3.jpg)
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)
![Page 4: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/4.jpg)
• Poisson processes:
exponential distribution of WTs
• Interesting case: power-law tail in the distribution of WTs (Non-Poisson renewal processes)
![Page 5: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/5.jpg)
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)
![Page 6: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/6.jpg)
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)
![Page 7: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/7.jpg)
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)
![Page 8: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/8.jpg)
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 )(),(
![Page 9: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/9.jpg)
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
![Page 10: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/10.jpg)
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)
![Page 11: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/11.jpg)
)(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)
![Page 12: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/12.jpg)
Aging Intensity Function (AIF)
)()(
)()()(
exp
rent
ta
a
aI
RenewalAging
1)()()(exp arentt Iaa
No Aging0)()()(exp at Ia
![Page 13: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/13.jpg)
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)
![Page 14: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/14.jpg)
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)
![Page 15: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/15.jpg)
Pareto distribution with T=1 and μ=1.8ta = 0, 20, 60
![Page 16: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/16.jpg)
![Page 17: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/17.jpg)
![Page 18: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/18.jpg)
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)
![Page 19: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/19.jpg)
7.0;)(
mNAI
Poisson pseudo-events and critical events
![Page 20: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/20.jpg)
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)
![Page 21: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/21.jpg)
Example of BQD Emission Intensity Sequence(typical jumps between ON and OFF state)
![Page 22: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/22.jpg)
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)
![Page 23: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/23.jpg)
![Page 24: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/24.jpg)
![Page 25: Aging in Blinking Quantum Dots: Renewal or Slow Modulation ?](https://reader035.vdocument.in/reader035/viewer/2022062409/56815152550346895dbf7697/html5/thumbnails/25.jpg)
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