Abstract
The mean field for a complex network consisting of a large but finite number of random two-state elements, \(M\), has been shown to satisfy a nonlinear Langevin equation. The noise intensity is inversely proportional to \(\sqrt{M} \). In the limiting case \(M = \infty \), the solution to the Langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point. When \(M < \infty \), the inverse power law is truncated by an exponential decay with rate \(\varGamma \), the evaluation of which is the main purpose of this article. An analytic/numeric approach is used to obtain the lowest-order eigenvalues in the spectral decomposition of the solution to the corresponding Fokker–Planck equation and its equivalent Schrödinger equation representation.
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Notes
Note that in Ref. [13] Eq. (5) was wrongly written as \(\varGamma = 1.49 (\gamma D)^{1/2}\). This means that the heuristic argument adopted in Ref. [13] yields a result very close to that of the more rigorous approach illustrated in this article.
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Acknowledgments
MB acknowledges financial support from FONDECYT Project No. 1120344. MTB, AS and PG warmly thank ARO and Welch for their support through Grants No. W911NF-11-1-0478 and No. B-1577, respectively.
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Bologna, M., Beig, M.T., Svenkeson, A. et al. Spectral Decomposition of a Fokker–Planck Equation at Criticality. J Stat Phys 160, 466–476 (2015). https://doi.org/10.1007/s10955-015-1262-5
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DOI: https://doi.org/10.1007/s10955-015-1262-5