Aging correlation functions for blinking nanocrystals, and other on–off stochastic processes

Abstract

Following recent experiments on power law blinking behavior of single nanocrystals, we calculate two-time intensity correlation functions I(t)I(t+t') for these systems. We use a simple two state (on and off) stochastic model to describe the dynamics. We classify possible behaviors of the correlation function and show that aging, e.g., dependence of the correlation function on age of process t, is obtained for classes of the on time and off time distributions relevant to experimental situation. Analytical asymptotic scaling behaviors of the intensity correlation in the double time t and t' domain are obtained. In the scaling limit I(t)I(t+t('))-->h(x), where four classes of behaviors are found: (i) finite averaged on and off times x=t' (standard behavior); (ii) on and off times with identical power law behaviors x=t/t' (case relevant for capped nanocrystals); (iii) exponential on times and power law off times x=tt' (case relevant for uncapped nanocrystals); (iv) for defected off time distribution we also find x=t+t'. Origin of aging behavior is explained based on simple diffusion model. We argue that the diffusion controlled reaction A+B <==>AB, when followed on a single particle level exhibits aging behavior.

Scaling Green-Kubo Relation and Application to Three Aging Systems

“…As mentioned in the Introduction, Zumofen and Klafter [12] showed that the diffusivity in this model is sensitive to the initial conditions. Here, we demonstrate the applicability of our scaling Green-Kubo relation, while previous works used renewal theory approaches [7,34].…”

“…[7,38], this is no longer the case for more realistic models that take into account that there may be more than single on-state or exponential cutoff on the power-law statistics of on and/or off times [69]. Since our scaling Green-Kubo relation Eq.…”

“…periods and remain dark during others [63][64][65]. The durations of these on periods and off periods are found to be distributed according to a power law whose exponent is such that the average on and off time is infinite (i.e., of the order of the measurement time), leading to aging in the intensity autocorrelation function [7,66]. This model is the familiar Lévy walk [12,31,36,67], which is a popular stochastic framework with many applications.…”

“…For this form of the distribution, the aging intensity autocorrelation for long times and 0 < μ < 1 has been explicitly obtained in Refs. [7,34] by mapping the problem onto a Lévy walk,…”

“…Some systems may possess a stationary state, but the relaxation towards this state may be slow and thus important on experimentally relevant time scales [1][2][3][4]. Other systems may even exhibit aging on all experimentally accessible time scales [5][6][7]. The extension and generalization of known results for the stationary case to the aging one is important to better understand and characterize the behavior of aging systems [8].…”