Quantumlike Chaos in Prime Number Distribution and in Turbulent Fluid Flows
| Scientific Paper | |
|---|---|
| Title | Quantumlike Chaos in Prime Number Distribution and in Turbulent Fluid Flows |
| Read in full | Link to paper |
| Author(s) | A Mary Selvam |
| Keywords | quantum-like chaos in prime numbers, fractal structure of primes, quantification of prime number distribution, prime numbers and fluid flows |
| Published | 2001 |
| Journal | Apeiron |
| Volume | 8 |
| Number | 3 |
| No. of pages | 36 |
| Pages | 29-64 |
Read the full paper here
Abstract
Recent studies by mathematicians and physicists have identified a close association between the distribution of prime numbers and quantum mechanical laws governing the subatomic dynamics of quantum systems such as the electron or the photon. It is now recognised that Cantorian fractal space-time fluctuations characterise dynamical systems of all space-time scales ranging from the microscopic subatomic dynamics to macro-scale turbulent fluid flows such as atmospheric flows. The spacing intervals of adjacent prime numbers also exhibit fractal (irregular) fluctuations generic to dynamical systems in nature. The apparently irregular (chaotic) fractal fluctuations of dynamical systems, however, exhibit self-similar geometrical pattern and are associated with inverse power-law form for the power spectrum. Self-similar fluctuations imply long-range space-time correlations identified as self-organized criticality. A cell dynamical system model for atmospheric flows developed by the author gives the following important results: (a) Self-organized criticality is a signature of quantum-like chaos (b) The observed self-organized criticality is quantified in terms of the universal inverse power-law form of the statistical normal distribution (c) The spectrum of fractal fluctuations is a broadband continuum with embedded dominant eddies. The cell dynamical system model is a general systems theory applicable to all dynamical systems (real world and computed) and the model concepts are applied to derive the following results for the observed association between prime number distribution and quantum-like chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different sets of adjacent prime number spacing intervals show that the power spectra follow the model predicted universal inverse power-law form of the statistical normal distribution. The prime number distribution therefore exhibits self-organized criticality, which is a signature of quantum-like chaos. (iii) The continuum real number field contains unique structures, namely, prime numbers, which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows.
Overview
Published in Apeiron in July 2001 by A. M. Selvam, then retired from the Indian Institute of Tropical Meteorology in Pune, this paper takes a model built for atmospheric turbulence and applies it, without modification, to the distribution of the prime numbers. Selvam's "cell dynamical system" model was developed to explain the fractal fluctuations of atmospheric flows; its central claim is that such fluctuations are not merely chaotic but quantum-like, because the model makes eddy amplitudes additive while their squares represent probability densities — the same structural relation found in the wave mechanics of the electron or photon. She holds the model to be "a general systems theory applicable to all dynamical systems in nature", independent of the chemical or physical details of any particular system.
The departure from the mainstream account lies in the direction of explanation. Where the established quantum-chaos literature relates prime distribution to the spectra of quantum systems through the Riemann zeta function and random matrix theory, Selvam proposes instead that the continuum real number field is itself a dynamical substratum from which discrete structures — the primes — emerge in exactly the way that dominant eddies emerge from a turbulent continuum. Prime numbers are, on this reading, "analogous to the dominant eddies in the eddy continuum in turbulent fluid flows". The paper's concrete claims are that the model's f distribution reproduces the observed prime number density, that periodogram analysis of prime spacing intervals in 3–1000 yields dominant periodicities matching the model's Fibonacci-based prediction, and that the variance and phase spectra track one another — an appearance of Berry's phase in a purely arithmetic object.
The argument
The cell dynamical system model
The model rests on Townsend's picture of a large eddy as an envelope enclosing smaller turbulent eddies. If W and w* are the r.m.s. circulation speeds of large and turbulent eddies of radii R and r, Selvam takes
- W2 = (2/π)(r/R)w*2.
Since the square of circulation speed is eddy kinetic energy, this makes the energy spectrum a cumulative integration of enclosed fluctuations, and by the Central Limit Theorem the resulting spectrum should follow the statistical normal distribution. Because additive amplitudes yield squared probabilities, Selvam identifies this as the signature of quantum-like behaviour.
Four consequences are drawn. First, the flow trajectory is an overall logarithmic spiral with a quasiperiodic Penrose tiling internal structure, the successive radii and speeds following the Fibonacci series. Second, dominant peak periods obey En = Ts(2 + τ)τn, with τ = (1 + √5)/2 the golden mean and Ts the primary perturbation period. Third, the ratio r/R equals the phase-angle increment dθ, so wavelength and phase increase together — the relation Selvam identifies with Berry's phase. Fourth, W/w* = (1/k) log z with z = R/r, where k ≈ 1/τ2 ≈ 0.382 is the fractional volume dilution of the large eddy; Selvam notes this coincides with the observed value 0.38 of the Von Karman constant in the atmospheric boundary layer.
Carrying the model into the number field
The bridge to arithmetic is an interpretive move rather than a derivation. The speeds W and w* are reread as "units of computations of respective yardstick lengths R and r"; integrating w* units of a finite yardstick r produces a larger computational domain WR, larger than w*r because of the uncertainty in measuring with a finite yardstick. The continuum number field of the Cartesian plane is thus obtained by successive integration of enclosed finite number-field domains, and the same Fibonacci/Penrose structure is asserted to hold for "the means of ensembles of successively larger number field domains".
The key quantity is the fraction f of the primary perturbation carried outward by the growing large eddy, f = (W/w*)(r/R). Chaining the model relations gives
- f = √(2/πz) ln z.
Selvam notes that in atmospheric flows this same f describes the observed vertical profiles of cloud liquid water content and of aerosol concentration — it is the signature at height z of air originating at the surface. By analogy it is taken to be the cumulative prime number density at z. The steady-state fractal emergence A = WR/w*r then gives
- P = A/f = z/ln z,
which is precisely the Prime Number Theorem's estimate for π(z). Selvam presents this as the model deriving the PNT: z/ln z "represents the normalized cumulative variance spectrum of the eddies", so the prime number spectrum and the eddy energy spectrum are the same object.
Data and results
The first 1000 primes were taken from the University of Tennessee at Martin prime pages. Four findings are reported.
Density. The computed f distribution, the actual prime density and 1/ln z are plotted together; their shapes are close, and all three peak at about 0.6 at z = 2π. Selvam reads z = 2π as one complete eddy circulation — "a stable self-sustaining soliton-like structure".
Normality. Plotted as cumulative percentage frequency against normalised standard deviate t, the actual prime distribution, the z/ln z distribution and the f distribution all track the statistical normal distribution closely over z = 3 to 1000.
Spectra. Using Jenkinson's quasi-continuous periodogram over 10,000 geometrically spaced periodicities, the frequency of primes in unit class intervals from 3 to 1000 was analysed. The goodness of fit between the variance spectrum and the normal distribution is significant at ≤ 5%; the phase spectrum is close but not significantly so, though variance and phase do agree significantly within individual dominant wavebands (illustrating Berry's phase). Of 110 dominant wavebands, 14 are significant at ≤ 5%. The period at which cumulative variance reaches 50% is T50 = 3.242 spacing intervals, against the model prediction T50 = (2 + τ)τ0 ≈ 3.6. Model-predicted dominant periods of 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1 and 64.9 spacing intervals (n = −1 to 6) are said to agree closely with those found. The significant period 2 is identified with the twin primes.
Spiral. Since the zth prime is approximately z ln z, and the model's spiral gives dθ = 1/z, the zth prime is assigned an angular phase difference of 1/z radians from the (z − 1)th. Plotting the first 20 and first 100 primes at those angles produces spiral arms that Selvam likens to phyllotaxis — the arrangement of leaves on a stem, florets in composite flowers, scales on a pine cone.
Assessment
What is genuinely attractive here is the ambition and the concreteness of the model's predictions. Selvam does not merely assert an analogy: she writes down a specific functional form, f = √(2/πz) ln z, a specific set of predicted periodicities from the golden mean, and a specific value T50 ≈ 3.6, and then tests each against actual prime data. The agreement of T50 = 3.242 with the predicted 3.6, and the appearance of period 2 (twin primes) as a dominant significant waveband, are the kind of falsifiable statements that distinguish a model from a metaphor. The general-systems framing — that self-similar fluctuations, inverse power-law spectra and long-range correlation are one phenomenon appearing at every scale — is a serious position with a real literature behind it, and the observation that the dilution constant k = 1/τ2 ≈ 0.382 matches the empirically measured Von Karman constant 0.38 is a striking numerical coincidence within the atmospheric domain where the model was built.
The difficulties are substantial and begin with the central derivation. The chain that produces P = z/ln z is not a derivation of the Prime Number Theorem from fluid dynamics; it is an algebraic rearrangement of definitions in which ln z has already been inserted by hand via the logarithmic velocity profile W/w* = (1/k) log z. Any model containing a logarithmic profile and a ratio will produce z/ln z; the arithmetic content of the PNT — which comes from the non-vanishing of the zeta function on the line Re(s) = 1 — plays no part. The identification of f with "the cumulative prime number density" is made by analogy ("by analogy it follows that"), with nothing forcing it. Likewise the reinterpretation of circulation speeds as "units of computations of yardstick lengths" is an assertion; no correspondence is established between anything physical in the flow and anything arithmetic in the integers.
The statistical case is weaker than presented. A sample of 1000 primes, analysed with 10,000 geometrically spaced trial periods, will inevitably throw up many apparent peaks; reporting 110 dominant wavebands of which 14 are significant "at ≤ 5%" is close to what one would expect by chance from that many tests, and no correction for multiple comparisons is made. The model predicts a geometric series of periods with ratio τ ≈ 1.618, so the predicted periods are dense enough on a log scale that a found peak will usually be near one. The claimed agreement between predicted 3.6 and found 3.242 is a 10% discrepancy with no error bar. Most tellingly, the paper never shows that the results distinguish the model from any alternative: no null hypothesis is constructed from shuffled or surrogate data, and no comparison is made with what the same periodogram would return for a random sequence of comparable density.
Two further steps are asserted rather than shown. The Penrose tiling and Fibonacci structure are imported wholesale into the number field on the strength of the remark that tiling space is "analogous to investigating the manner in which matter splits up into atoms and natural numbers split up into product of primes" — a suggestive sentence, not an argument. And the spiral figures are a construction rather than a discovery: by defining the zth prime's angular position as the accumulated sum of 1/z, any increasing sequence whatever will trace a spiral in the plane, so the resulting figure carries no information specific to the primes. Selvam does cite prior reports of spiral patterns in prime arrangements (Schroeder), but those arise from a different construction.
Finally, the paper does not engage the established mathematics it is closest to. The connection between prime distribution and quantum spectra is a developed subject — the Montgomery–Odlyzko law, in which the pair-correlation of zeta zeros matches the eigenvalue statistics of the Gaussian Unitary Ensemble, is the sharpest quantitative result in this area and is verified to enormous precision. Selvam cites popular accounts (Cipra, Klarreich, Berry) but neither derives that pair-correlation from the cell dynamical model nor explains why her spectrum should coexist with it. Similarly, the claim that "the prime number distribution therefore exhibits self-organized criticality" would be far more convincing if tested against the known fluctuation statistics of π(z) − li(z), whose behaviour is constrained by the zeta zeros. On its own terms the paper is internally consistent and its author is clear about which steps are analogical; taken as a claim about number theory rather than as a heuristic, it does not yet meet the standard of the field it addresses.