Signatures of Quantum-like Chaos in Spacing Intervals of Non-trivial Riemann Zeta Zeros and in Turbulent Fluid Flows
| Scientific Paper | |
|---|---|
| Title | Signatures of Quantum-like Chaos in Spacing Intervals
of Non-trivial Riemann Zeta Zeros and in Turbulent Fluid Flows |
| Read in full | Link to paper |
| Author(s) | A Mary Selvam |
| Keywords | fractal structure of spacing intervals of Riemann zeta zeros, quantum-like chaos in Riemann zeta zeros, selforganized criticality in Riemann zeta zeros |
| Published | 2001 |
| Journal | Apeiron |
| Volume | 8 |
| Number | 4 |
| No. of pages | 31 |
| Pages | 10-40 |
Read the full paper here
Abstract
The spacing intervals of adjacent Riemann zeta zeros (nontrivial) 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 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.
Overview
A. Mary Selvam, retired from the Indian Institute of Tropical Meteorology in Pune, brings a meteorologist's tool to a number-theoretic object. Her "cell dynamical system" model was developed to describe turbulent atmospheric flows; this paper applies it to the spacing intervals between consecutive non-trivial zeros of the Riemann zeta function, and reports that those spacings show the same statistical signature as rainfall, river flows and stock market indices.
The connection is not arbitrary. Since the Montgomery–Dyson observation of 1972 it has been suspected that zeta-zero spacings follow the same statistics as energy levels of complex quantum systems that are classically chaotic — the "spectrum" interpretation of the Riemann hypothesis, developed by Berry, Gutzwiller and others, which Selvam cites through Keating, Cipra and Klarreich. Her contribution is to argue that this quantum-like behaviour is not special to zeta zeros or to quantum systems at all, but is a generic property of any system exhibiting fractal fluctuations and long-range correlations, i.e. any system displaying self-organized criticality in the sense of Bak, Tang and Wiesenfeld.
The departure from the mainstream account is the direction of explanation. Standard random-matrix theory treats the correspondence between zeta zeros and quantum spectra as a statistical universality class. Selvam instead claims a physical mechanism common to both: an eddy continuum in which large eddies form as envelopes of smaller ones, so that by the Central Limit Theorem the energy spectrum is normally distributed, the variance is the probability density, and "the additive amplitudes of eddies when squared represent probabilities" — the Born rule, obtained from turbulence rather than postulated. On this reading quantum mechanics is not a separate regime but the small-scale end of a scale-free continuum, and quantum statistics should be found "in dynamical systems of all size scales".
The argument
The cell dynamical system model
Large eddies of r.m.s. circulation speed W and radius R form as envelopes enclosing small eddies of speed w* and radius r, with
- 2πW2 = (2r/R)πw*2
Large eddies grow at unit length steps in unit time, the units being the enclosed small-eddy length and circulation time. Because the large eddy is the average of the enclosed smaller ones, the Central Limit Theorem gives an eddy energy spectrum following the statistical normal distribution — from which Selvam draws the model's central identification: the variance represents the probability density.
Four predictions follow:
- (a) Fractal fluctuations are generated by an overall logarithmic spiral trajectory whose internal structure is the quasiperiodic Penrose tiling pattern.
- (b) Continuous periodogram analysis of such a spiral reveals a continuum of periodicities with progressive increase in phase.
- (c) The broadband spectrum contains embedded dominant wavebands with peak periodicities En = Ts(2 + τ)τn, where τ = (1+√5)/2 ≈ 1.618 is the golden mean and Ts the primary perturbation length scale. For n = −1 to 6 this gives 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1 and 64.9 units.
- (d) The ratio r/R also equals the increment dθ in phase angle, so phase angle represents variance. Increments in wavelength and phase are therefore linked — a relation Selvam identifies with Berry's phase in quantum systems.
The overall spiral obeys W = (w*/k)log z, with k the steady-state fractional volume dilution of the large eddy by turbulent fluctuations. The model gives k = 1/τ2 ≈ 0.382, identified as a universal constant for deterministic chaos in fluid flows, and Selvam notes that the logarithmic wind profile is a long-established feature of the atmospheric boundary layer with von Kármán's constant measured at 0.38.
Since W is the r.m.s. eddy perturbation amplitude relative to the previous growth step, successive growth stages have standard deviations σ, 2σ, 3σ, …, corresponding to normalised standard deviations t = 0, 1, 2, 3, …. The power spectrum plotted as variance versus log frequency then becomes eddy probability density versus standard deviation, with
- t = (log L / log T50) − 1
where T50 is the period up to which cumulative contribution to total variance equals 50 %. Because t = 0 and t = 2 correspond to cumulative probabilities of 50 % and 95 % in the normal distribution, and t is the eddy growth step n, the model predicts
- T50 = Ts(2 + τ)τ0 ≈ 3.6 unit spacing intervals
- T95 = Ts(2 + τ)τ2 ≈ 9.5 unit spacing intervals
Data and method
Selvam used zeta zeros from Odlyzko's tables at AT&T: the first 100 000 zeros, and blocks of 10 000 zeros beginning at the 1012th, 1021st and 1022nd zero (the latter two accurate only to about 10−6). Five data groups (zeros1a, zeros1b, zeros3, zeros4, zeros5) were formed, with individual series ranging from 50 to 10 000 values, drawn from both the start and the middle of each file.
The analysis uses Jenkinson's (1977) quasi-continuous periodogram, constructed over 10 000 geometrically spaced periodicities Lm = 2exp(0.001m), estimating Amcos(2πνmS − φm). Cumulative percentage contribution to total variance was accumulated from the high-frequency end, T50 located, and spectra plotted as cumulative variance against t alongside the statistical normal distribution. Phase spectra were plotted as cumulative percentage of total phase rotation.
Results
Selvam reports that the spacing intervals plotted directly (a sample of 100 zeros from the 80 000th) show the irregular zig-zag characteristic of fractal fluctuations. For the power spectra:
- Almost all variance spectra follow the statistical normal distribution, with chi-square goodness of fit significant at the 5 % level or better.
- The observed T50 values cluster very close to the predicted 3.6 unit spacing intervals, across all data sets and all four zero ranges.
- The phase spectrum is close to normal but the fit is not statistically significant in a majority of cases.
- Berry's phase — the correspondence between variance and phase spectra — is statistically significant for individual dominant wavebands, particularly at longer periodicities.
- Short dominant periodicities up to about 5 spacing units occur most frequently and are most often statistically significant, consistent with T50 ≈ 3.6.
Why the zeros lie on the critical line
The paper closes with a physical reading of ζ(s) = 1 + 1/2s + 1/3s + …. Selvam proposes that the individual fractions 1/2, 1/3, 1/4, … represent the length-scale ratios r/R of enclosed primary eddy to large eddy, i.e. the probabilities of occurrence of the primary perturbation at successive growth stages; and since r/R also represents the variance or eddy energy, the fractions raised to the complex power s = x + iy are fractional probabilities at the phase angle given by the Argand-diagram coordinates. The zeta function therefore "represents the energy spectrum of quantum systems at any location (x, y)", and the zeros on the critical line are eddy energy minima; a 90° rotation would give the maxima. As for why x = 1/2: an eddy circulation is bidirectional and bimodal, formation and dissipation, and "since manifestation of energy in phenomenological form occurs only in one-half cycle, the corresponding energy levels occur at x = 1/2".
Assessment
The strongest thing in this paper is a genuine, quantitative, falsifiable prediction that appears to be borne out. T50 ≈ 3.6 was derived from the model before the data were examined — it is (2 + τ)τ0 with τ the golden mean — and it was then tested across roughly sixty independent data series drawn from four widely separated regions of the zeta zero sequence, spanning heights from the first zeros up to the 1022nd. That the same value recurs is not nothing. The methodology is also unusually transparent for a paper of this kind: data sources are given as URLs with their stated accuracies, every series is identified by starting index and length, the results table (Figure 6) reports failures as well as successes with an explicit S/N marking, and Selvam states plainly which of her tests do not reach significance — the phase spectra fail the normality test in a majority of cases, and she says so rather than burying it. The recovery of k = 1/τ2 = 0.382 against a measured von Kármán constant of 0.38 is a real point of contact with an independent measurement, even if the agreement is to two figures only.
The difficulties are correspondingly serious. The most consequential is that the paper does not compare its result to the established prediction for the same data. Zeta zero spacings are the best-studied case of GUE (Gaussian Unitary Ensemble) statistics in all of mathematics: Odlyzko's own computations on precisely these tables showed agreement with the random-matrix pair correlation to several decimal places, and it is that agreement, not any general "fractal" character, which underwrites the quantum-chaos interpretation Selvam invokes in her introduction. She cites Berry and Cipra for the connection but never sets her spectrum against the GUE prediction, never computes the nearest-neighbour spacing distribution, and never asks whether a GUE spectrum would itself produce T50 ≈ 3.6 — which, if it did, would make her result a rederivation rather than a discovery. Without that comparison the paper cannot distinguish its model from the accepted one.
The second difficulty concerns what the analysis can detect. Zeta zero spacings are, after unfolding, a stationary point process with a mean near unity, and the periodogram of such a series has broadband structure by construction. The paper's test is that cumulative variance against t follows a normal distribution — but t is defined from T50, which is itself read off the same cumulative curve, so the horizontal axis is fitted to the data before the fit is assessed. A chi-square goodness-of-fit statistic computed on a cumulative curve is not valid in the usual way, since successive points are strongly dependent; the effective degrees of freedom are far fewer than the number of plotted points, and reported significance at the 5 % level cannot be taken at face value. The paper does not present a null distribution — no surrogate data, no shuffled spacings, no comparison against white noise or a Poisson process — so there is no way to know how often T50 ≈ 3.6 would arise from a series with no structure at all.
Third, the golden-mean waveband prediction is weak as a test. The predicted peaks 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1, 64.9 are compared against wavebands binned as 2–3, 3–4, 4–6, 6–12, 12–20, 20–30, 30–50, 50–80 — bins wide enough that almost any geometric progression with ratio near 1.6 would fall inside them, and the bins were evidently chosen to contain the predictions. That the shortest periods dominate is expected for any broadband spectrum and is not evidence for the specific values. Similarly, "Berry's phase" here means only that phase and variance spectra track each other; the geometric phase of Berry (1984) is a holonomy acquired under adiabatic cyclic transport of a quantum state, and the two have no established relation beyond the shared word.
The most serious problem is the final section. The reading of the terms of ζ(s) as eddy length-scale ratios is offered without derivation: the series is a sum over all integers with no dynamical content, its terms are not probabilities (they do not sum to one, and for Re s ≤ 1 the series does not converge at all), and the analytic continuation that actually defines ζ(s) in the critical strip — where every non-trivial zero lies — is not the series Selvam interprets. Her explanation of why x = 1/2 ("energy manifests in only one-half cycle") is a verbal analogy, not an argument; it makes no contact with the functional equation ζ(s) = 2sπs−1sin(πs/2)Γ(1−s)ζ(1−s), which is the actual reason the critical line sits at Re s = 1/2, being the axis of symmetry s ↔ 1 − s. Presenting a heuristic where a symmetry argument exists and is elementary weakens the whole section, and a reader could reasonably take it as an attempted physical intuition for the Riemann hypothesis, which it is not.
Finally, the paper's largest claim — that variance representing probability density is the Born rule and therefore that macroscopic turbulence is "quantum-like" — rests on the Central Limit Theorem, which delivers real Gaussian statistics. Quantum probabilities come from squaring a complex amplitude, and it is precisely the interference between complex phases, with no classical counterpart, that distinguishes quantum from classical statistics. The paper's eddies superpose with real amplitudes and cannot produce destructive interference, so the analogy stops exactly where quantum mechanics begins. The observed violation of Bell inequalities places this beyond analogy: no model whose probabilities arise from a classical variance can reproduce those correlations.
Taken as what it demonstrably is — an empirical report that zeta zero spacings, analysed by a particular periodogram method, show a T50 near 3.6 across many independent samples, together with an unusual scale-free framework for interpreting it — the paper is careful, honest about its negative results, and worth the attention of anyone interested in cross-scale statistical universality. Taken as a physical theory of the zeta function or of quantum mechanics, it asserts far more than it establishes.