Nonergodicity of a Time Series Obeying Lévy Statistics |
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Authors: | Gennady Margolin Eli Barkai |
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Institution: | (1) Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, IN, 46556;(2) Department of Physics, Bar Ilan University, Ramat Gan, Israel, 52900 |
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Abstract: | Time-averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power
law sojourn time distribution are studied. Such a process, called a Lévy walk, describes dynamical behaviors of many physical
systems, fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example. When
the mean sojourn time diverges the process is non-ergodic. In that case, the time average autocorrelation function is not
equal to the ensemble averaged autocorrelation function, instead it remains random even in the limit of long measurement time.
Several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges,
and favorably compared to Monte Carlo simulations. Nonergodicity of the power spectrum of the process is briefly discussed,
and a nonstationary Wiener-Khintchine theorem, relating the correlation functions and the power spectrum is presented. The
considered situation is in full contrast to the usual assumptions of ergodicity and stationarity. |
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Keywords: | Nonergodicity Time series Lévy statistics Power law Correlation function |
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