No Arabic abstract
We present an unbiased and robust analysis method for power-law blinking statistics in the photoluminescence of single nano-emitters, allowing us to extract both the bright- and dark-state power-law exponents from the emitters intensity autocorrelation functions. As opposed to the widely-used threshold method, our technique therefore does not require discriminating the emission levels of bright and dark states in the experimental intensity timetraces. We rely on the simultaneous recording of 450 emission timetraces of single CdSe/CdS core/shell quantum dots at a frame rate of 250 Hz with single photon sensitivity. Under these conditions, our approach can determine ON and OFF power-law exponents with a precision of 3% from a comparison to numerical simulations, even for shot-noise-dominated emission signals with an average intensity below 1 photon per frame and per quantum dot. These capabilities pave the way for the unbiased, threshold-free determination of blinking power-law exponents at the micro-second timescale.
Many time series produced by complex systems are empirically found to follow power-law distributions with different exponents $alpha$. By permuting the independently drawn samples from a power-law distribution, we present non-trivial bounds on the memory strength (1st-order autocorrelation) as a function of $alpha$, which are markedly different from the ordinary $pm 1$ bounds for Gaussian or uniform distributions. When $1 < alpha leq 3$, as $alpha$ grows bigger, the upper bound increases from 0 to +1 while the lower bound remains 0; when $alpha > 3$, the upper bound remains +1 while the lower bound descends below 0. Theoretical bounds agree well with numerical simulations. Based on the posts on Twitter, ratings of MovieLens, calling records of the mobile operator Orange, and browsing behavior of Taobao, we find that empirical power-law distributed data produced by human activities obey such constraints. The present findings explain some observed constraints in bursty time series and scale-free networks, and challenge the validity of measures like autocorrelation and assortativity coefficient in heterogeneous systems.
The photoluminescence intermittency (blinking) of quantum dots is interesting because it is an easily-measured quantum process whose transition statistics cannot be explained by Fermis Golden Rule. Commonly, the transition statistics are power-law distributed, implying that quantum dots possess at least trivial memories. By investigating the temporal correlations in the blinking data, we demonstrate with high statistical confidence that quantum dot blinking data has non-trivial memory, which we define to be statistical complexity greater than one. We show that this memory cannot be discovered using the transition distribution. We show by simulation that this memory does not arise from standard data manipulations. Finally, we conclude that at least three physical mechanisms can explain the measured non-trivial memory: 1) Storage of state information in the chemical structure of a quantum dot; 2) The existence of more than two intensity levels in a quantum dot; and 3) The overlap in the intensity distributions of the quantum dot states, which arises from fundamental photon statistics.
We study the avalanche statistics observed in a minimal random growth model. The growth is governed by a reproduction rate obeying a probability distribution with finite mean a and variance va. These two control parameters determine if the avalanche size tends to a stationary distribution, (Finite Scale statistics with finite mean and variance or Power-Law tailed statistics with exponent in (1, 3]), or instead to a non-stationary regime with Log-Normal statistics. Numerical results and their statistical analysis are presented for a uniformly distributed growth rate, which are corroborated and generalized by analytical results. The latter show that the numerically observed avalanche regimes exist for a wide family of growth rate distributions and provide a precise definition of the boundaries between the three regimes.
Some authors have recently argued that a finite-size scaling law for the text-length dependence of word-frequency distributions cannot be conceptually valid. Here we give solid quantitative evidence for the validity of such scaling law, both using careful statistical tests and analytical arguments based on the generalized central-limit theorem applied to the moments of the distribution (and obtaining a novel derivation of Heaps law as a by-product). We also find that the picture of word-frequency distributions with power-law exponents that decrease with text length [Yan and Minnhagen, Physica A 444, 828 (2016)] does not stand with rigorous statistical analysis. Instead, we show that the distributions are perfectly described by power-law tails with stable exponents, whose values are close to 2, in agreement with the classical Zipfs law. Some misconceptions about scaling are also clarified.
This paper develops an analytical and rigorous formulation of the maximum entropy generation principle. The result is suggested as the Fourth Law of Thermodynamics.