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Exponents in Lifetime and Power Spectral Density Forms in Self-Organized Critical Systems--A Mathematica Application

L. Meisel
P. Cote
Journal / Anthology

Computers in Physics
Year: 1993
Volume: 7
Issue: 6
Page range: 710-713

Bak, Tang, and Wiesenfeld (BTW) [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 36 (1988)] established that power law frequency dependencies in the power spectral density (PSD) and size effect modified power law distributions of lifetimes are the fingerprints of self-organized critical systems. Jensen, Christensen, and Fogeby (JCF) [Phys. Rev B 40, 7425 (1989)] clarified the ideas introduced by BTW and established the connection between the distribution of lifetimes and the PSD for the case of exponentially cutoff ("size effect" modified) distributions of lifetimes. Here the JCF connection between the PSD and the PSD may be expressed in terms of generalized hypergeometric functions in this case. A detailed discussion of the JCF connections is presented for a subset of values of the lifetime distribution exponent, for which the generalized hypergeometric functions reduce to Fresnel integrals and sine and cosine integrals, which were the subject of a recent "Numerical Recipes" column [Press and Teukolsky, Comput. Phys. 6, 670-672 (1992)]. All calculations were performed in Mathematica.

*Mathematics > Calculus and Analysis > Dynamical Systems