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Dynamical origin of memory and renewal.

R Cakir¹, P Grigolini, A A Krokhin

¹Center for Nonlinear Science, University of North Texas, PO Box 311427, Denton, Texas 76203, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 10, 2006

Summary

The dynamic approach to fractional Brownian motion (FBM) links memoryless renewal processes to correlated dynamics. This study generalizes FBM, modeling blinking quantum dots with a Lévy process.

Area of Science:

Physics
Statistical Mechanics
Complex Systems

Background:

Fractional Brownian motion (FBM) describes processes with memory.
Renewal processes are characterized by independent time intervals between events.
Lévy processes are a generalization of Brownian motion with jumps.

Purpose of the Study:

To establish a link between renewal processes and correlated dynamics using FBM.
To generalize the understanding of Lévy processes generated by trajectory recrossings.
To develop a model for blinking quantum dot fluorescence signals.

Main Methods:

Theoretical analysis of fractional Brownian motion.
Numerical simulations to verify theoretical predictions.
Application of the generalized FBM model to experimental data.

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Main Results:

The dynamic FBM approach connects non-Poisson renewal processes to correlated dynamics.
Recrossings in dynamic FBM generate Lévy processes with an index 0 < theta < 1.
This framework successfully models the fluorescence signal of blinking quantum dots.

Conclusions:

The dynamic FBM provides a unified framework for processes with and without memory.
The generalized Lévy process offers a more comprehensive description of stochastic phenomena.
This work has implications for understanding and modeling complex systems like quantum dots.