[Paper Review] Signatures of quantum-like chaos in spacing intervals of non-trivial Riemann zeta zeros and in turbulent fluid flows

A. M. Selvam|ArXiv.org|Feb 11, 2001

Complex Systems and Time Series Analysis34 references17 citations

TL;DR

This paper proposes that the spacing intervals between non-trivial Riemann zeta zeros exhibit quantum-like chaos, evidenced by fractal fluctuations and universal inverse power law power spectra—hallmarks of self-organized criticality. Using a cell dynamical system model originally developed for turbulent fluid flows, the authors show these spacing intervals follow the statistical normal distribution’s variance structure, linking them to quantum-like energy-level fluctuations across all scales of dynamical systems.

ABSTRACT

The spacing intervals of adjacent Riemann zeta zeros(non-trivial) exhibit fractal(irregular) fluctuations generic to dynamical systems in nature such as fluid flows, heart beat patterns, stock market price index, etc., and are associated with unpredictability or chaos. The power spectra of such fractal space-time fluctuations exhibit universal inverse power law form and signify long-range correlations, identified as self-organized criticality . A cell dynamical system model developed by the author for turbulent fluid flows provides a unique quantification for the observed power spectra in terms of the statistical normal distribution, such that the variance represents the statistical probability densities. Such a result that the additive amplitudes of eddies when squared, represent the statistical probabilities is an observed feature of the subatomic dynamics of quantum systems such as an electron or photon. Self-organized criticality is therefore a signature of quantum-like chaos in dynamical systems. The model concepts are applicable to all real world(observed) and computed(mathematical model) dynamical systems. Continuous periodogram analyses of the fractal fluctuations of Riemann zeta zero spacing intervals show that the power spectra follow the unique and universal inverse power law form of the statistical normal distribution. The Riemann zeta zeros therefore exhibit quantum-like chaos, the spacing intervals of the zeros representing the energy(variance) level spacings of quantum-like chaos inherent to dynamical systems in nature. The cell dynamical system model is a general systems theory applicable to dynamical systems of all size scales.

Motivation & Objective

• To investigate whether the spacing intervals of non-trivial Riemann zeta zeros display signatures of quantum-like chaos.
• To determine if the fractal fluctuations in these spacings exhibit long-range correlations and self-organized criticality.
• To apply a cell dynamical system model—originally for turbulent fluid flows—to quantify the power spectra of zeta zero spacings using statistical normal distribution variance.
• To establish a universal connection between mathematical structures (Riemann zeta zeros) and physical dynamical systems exhibiting quantum-like behavior.
• To demonstrate that the model applies universally across real-world and mathematical dynamical systems, from fluid flows to number theory.

Proposed method

• Continuous periodogram analysis is used to examine the fractal fluctuations in the spacing intervals of non-trivial Riemann zeta zeros.
• The power spectra of these fluctuations are compared to the universal inverse power law form derived from the statistical normal distribution.
• The cell dynamical system model is applied to map additive eddy amplitudes in fluid flows to statistical probability densities when squared, mirroring quantum systems.
• The model's framework is extended to interpret zeta zero spacing intervals as representing energy-level variances in quantum-like chaos.
• The variance of the normal distribution is interpreted as statistical probability density, linking fluid dynamics to quantum mechanics through universal power laws.
• The model is validated across observed and computed dynamical systems, confirming its general applicability.

Experimental results

Research questions

• RQ1 Do the spacing intervals between non-trivial Riemann zeta zeros exhibit fractal fluctuations indicative of chaos?
• RQ2 Do these fluctuations display universal inverse power law power spectra, signaling long-range correlations and self-organized criticality?
• RQ3 Can the cell dynamical system model used for turbulent fluid flows be applied to quantify the power spectra of Riemann zeta zero spacings?
• RQ4 Is there a quantitative link between the variance of the statistical normal distribution and the energy-level spacings in quantum-like chaos?
• RQ5 Does the model’s framework generalize across all real and mathematical dynamical systems, from fluid dynamics to number theory?

Key findings

• The power spectra of Riemann zeta zero spacing intervals follow the universal inverse power law form of the statistical normal distribution.
• The fractal fluctuations in zeta zero spacings exhibit long-range correlations, indicating self-organized criticality.
• The spacing intervals of non-trivial Riemann zeta zeros represent energy-level variances in quantum-like chaos inherent to dynamical systems.
• The additive amplitudes of eddies in the cell dynamical system model, when squared, yield statistical probabilities—mirroring quantum systems like electrons and photons.
• The cell dynamical system model provides a unified framework that quantifies chaotic fluctuations across all size scales, from fluid flows to mathematical constructs.
• The results establish that self-organized criticality is a signature of quantum-like chaos in diverse dynamical systems, including those described by the Riemann zeta function.