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Register a new userSystems with Correlations in the Variance: Generating Power-Law Tails in Probability Distributions
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We study how the presence of correlations in physical variables contributes to the form of probability distributions. We investigate a process with correlations in the variance generated by (i) a Gaussian or (ii) a truncated L{e}vy distribution. For both (i) and (ii), we find that due to the correlations in the variance, the process ``dynamically generates power-law tails in the distributions, whose exponents can be controlled through the way the correlations in the variance are introduced. For (ii), we find that the process can extend a truncated distribution {it beyond the truncation cutoff}, which leads to a crossover between a L{e}vy stable power law and the present ``dynamically-generated power law. We show that the process can explain the crossover behavior recently observed in the $S&P500$ stock index.
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In a quenched mesoscopic fluid, modelling transport processes at high densities, we perform computer simulations of the single particle energy autocorrelation function C_e(t), which is essentially a return probability. This is done to test the predictions for power law tails, obtained from mode coupling theory. We study both off and on-lattice systems in one- and two-dimensions. The predicted long time tail ~ t^{-d/2} is in excellent agreement with the results of computer simulations. We also account for finite size effects, such that smaller systems are fully covered by the present theory as well.
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