Statistical model of fluorescence blinking

Blinking of single molecules and nanocrystals is modeled as a two-state renewal process with on (fluorescent) and off (non-fluorescent) states. The on and off-times may have power-law or exponential distributions. A fractional generalization of the exponential function is used to develop a unified treatment of the blinking statistics for both types of distributions. In the framework of the two-state model, an equation for the probability density p(t (on)|t) of the total on-time is derived. As applied to power-law blinking, the equation contains derivatives of fractional orders alpha and beta equal to the exponents of the on and off-time power-law distributions, respectively. In the limit case of alpha = beta = 1, the distributions become exponential and the fractional differential equation reduces to an integer order differential equation. Solutions to these equations are expressed in terms of fractional stable distributions. The Poisson transform of p(t (on)|t) is the photon number distribution that determines the photon counting statistics. It is shown that the long-time asymptotic behavior of Mandel's Q parameter follows a power law: M(t) ae t (gamma) . The function gamma(alpha, beta) is defined on the (alpha, beta) plane. An analysis of the relative variance of the total on-time shows that it decays only when alpha = beta = 1 or alpha < beta. Otherwise, relative fluctuations either exhibit asymptotic power-law growth or approach a constant level. Analytical calculations are in good agreement with the results of Monte Carlo simulations.