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The modeling of complex phenomena using random walks was discussed earlier. In that discussion we outlined how a simple random walk model of a normal diffusion process leads to Gaussian statistics and a mean-square displacement that increases linearly with time. We also examined how inverse power-law memory in the random fluctuations, that is, in the steps of the walker, can produce a system response that is anomalous in that the mean-square displacement is proportional to t 2H , where 0 < H < 1. We saw that such time series are random fractals with fractal dimension given by D = 2 - H. The most complex phenomena studied earlier involved the limit of fractional differences becoming fractional derivatives, so that a stochastic process with long-term memory can be generated by taking the fractional derivative of a Wiener process. We learned that such processes have Gaussian statistics, but they also have inverse power-law spectra.
KeywordsMaster Equation Fractional Derivative Langevin Equation Fractional Brownian Motion Wait Time Distribution
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