We present a new approach to distinguish between non-ergodic and ergodic behavior. Performing ensemble averaging in a subpopulation of individual molecules leads to a mean value that can be similar to the mean value obtained in an ergodic system. The averaging is carried out by minimizing the variation between the sum of the temporal averaged mean square deviation of the simulated data with respect to the logarithmic scaling behavior of the subpopulation. For this reason, we first introduce a kind of Continuous Time Random Walks (CTRW), which we call Limited Continuous Time Random Walks (LCTRW) on fractal support. The random waiting time distributions are sampled at points which fulfill the condition N < 1, where N is the Poisson probability of finding a single molecule in the femtoliter-sized observation volume ΔV at the single-molecule level. Given a subpopulation of different single molecules of the same kind, the ratio T/ Tm between the measurement time T and the meaningful time Tm, which is the time for observing just one and the same single molecule, is the experimentally accessible quantity that allows to compare different molecule numbers in the subpopulation. In addition, the mean square displacement traveled by the molecule during the time t is determined by an upper limit of the geometric dimension of the living cell or its nucleus.
Keywords: Anomalous motion, broken ergodicity, continuous time random walks (CTRW), continuous time random walks (CTRW) on fractal supports, Limited Continuous Time Random Walks (LCTRW) on fractal supports, molecular crowding, ergodicity, FCS, FCCS, fluorescence fluctuation microscopy, heterogeneity, living cells, complex body fluids like blood and its components, interpretation of subdiffusive measurements, meaningful time for studying just one single molecule, physical model of crowding, physical model of temporal heterogeneity, random walks on fractal supports, resolution limits of measured diffusion times for two components, temporal autocorrelation, temporal two-color crosscorrelation, fluorescence imaging, time dependence of apparent diffusion coefficients
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