Quantum-like Chaos in Prime Number Distribution and in Turbulent Fluid Flows

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.

Keywords:  quantum-like chaos in prime numbers, fractal structure of primes, quantification of prime number distribution, prime numbers and fluid flows

Numerical solutions obtained using Equation (2), which is basically a numerical integration procedure, involve iterative computations with feedback and amplification of round-off error of real number finite precision arithmetic. The Equation (2) also represents the relationship between continuum number field and embedded discrete (finite) number fields. Numerical solutions for non-linear dynamical systems represented by Equation (2) are sensitively dependent on initial conditions and give apparently chaotic solutions, identified as deterministic chaos. Deterministic chaos therefore characterises the evolution of discrete (finite) structures from the underlying continuum number field.
    Historically, sensitive dependence on initial conditions of non-linear dynamical systems was identified nearly a century ago by Poincare (Poincare, 1892) in his study of three-body problem, namely the Sun, Earth and the Moon. Non-linear dynamics remained a neglected area of research till the advent of electronic computers in the late 1950s. Lorenz, in 1963 showed that numerical solutions of a simple model of atmospheric flows exhibited sensitive dependence on initial conditions implying loss of predictability of the future state of the system. The traditional non-linear dynamical system defined by Equation (2) is commonly used in all branches of science and other areas of human interest. Non-linear dynamics and chaos soon (by 1980s) became a multidisciplinary field of intensive research (Gleick, 1987). Sensitive dependence on initial conditions implies long-range space-time correlations. The observed irregular fluctuations of real world dynamical systems also exhibit such non-local connections manifested as fractal or self-similar geometry to the space-time evolution. The universal symmetry of self-similarity ubiquitous to dynamical systems in nature is now identified as self-organized criticality (Bak, Tang and Wiesenfeld, 1988). A symmetry of some figure or pattern is a transformation that leaves the figure invariant, in the sense that, taken as a whole it looks the same after the transformation as it did before, although individual points of the figure may be moved by the transformation (Devlin, 1997). Self-similar structures have internal geometrical structure, which resemble the whole. The space-time organization of a hierarchy of self-similar space-time structures is common to real world as well as the numerical models (Equation 2) used for simulation. A substratum of continuum fluctuations self-organizes to generate the observed unique hierarchical structures both in real world and the continuum number field used as the tool for simulation. A cell dynamical system model developed by the author [Mary Selvam, 1990; Selvam and Suvarna Fadnavis, 1998; 1999a;b] for turbulent fluid flows shows that self-similar (fractal) space-time fluctuations exhibited by real world and numerical models of dynamical systems are signatures of quantum-like mechanics. The model concepts are independent of the exact details, such as, the chemical, physical, physiological, etc., properties of the dynamical systems and therefore provide a general systems theory (Peacocke, 1989; Klir, 1993; Jean, 1994) applicable for all dynamical systems in nature. The model concepts are applicable to the emergence of unique prime number spectrum from the underlying substratum of continuum real number field.
    Recent studies indicate a close association between number theory in mathematics, in particular, the distribution of prime numbers and the chaotic orbits of excited quantum systems such as the hydrogen atom [Keating, 1990; Cipra, 1996; Klarreich, 2000]. Mathematical studies also indicate that Cantorian fractal space-time characterises quantum systems [Ord, 1983; Nottale, 1989; El Naschie, 1993]. The fractal fluctuations exhibited by prime number distribution and microscopic quantum systems belong to the newly identified science of non-linear dynamics and chaos. Quantification of the apparently irregular (chaotic) fractal fluctuations will help compute (predict) the space-time evolution of the fluctuations. The cell dynamical system model concepts described below (Section 2) provide a theory for unique quantification of the observed fractal fluctuations in terms of the universal inverse power-law form of the statistical normal distribution.

The relationship between root mean square (r.m.s.) circulation speeds W and w_* respectively of large and turbulent eddies of respective radii R and r is then given as

The dynamical evolution of space-time fractal structures is quantified in terms of ordered energy flow between fluctuations of all scales in Equation (3), because the square of the eddy circulation speed represents the eddy energy (kinetic). A hierarchical continuum of eddies is generated by the integration of successively larger enclosed turbulent eddy circulations and therefore the eddy energy (kinetic) spectrum follows statistical normal distribution according to the Central Limit Theorem [Ruhla, 1992; see Section 2.1(e) below]. Therefore, square of the eddy amplitude or the variance represents the probability. Such a result that the additive amplitudes of eddies, when squared, represent the probability densities is observed for the subatomic dynamics of quantum systems such as the electron or photon (Maddox 1988). Townsend’s visualisation of large eddy structure as quantified in Equation (3) leads to the most important result that the self-similar fractal fluctuations of atmospheric flows are manifestations of quantum-like chaos.

    The successively larger eddy radii (OR_o,OR₁, etc.) and the corresponding circulation speeds (W₁, W₂ etc.) follow the Fibonacci mathematical series. A brief summary of details of Penrose tiling pattern relevant to the present study is given in the following.
    Historically, the British mathematician Roger Penrose discovered in 1974 the quasiperiodic Penrose tiling pattern, purely as a mathematical concept. The fundamental investigation of tilings, which fill space completely, is analogous to investigating the manner in which matter splits up into atoms and natural numbers split up into product of primes. The distinction between periodic and aperiodic tilings is somewhat analogous to the distinction between rational and irrational real numbers, where the latter have decimal expansions that continue forever, without settling into repeating blocks [Devlin, 1997]. Even earlier Kepler saw a fundamental mathematical connection between symmetric patterns and 'space filling geometric figures' such as his own discovery, the rhombic dodecahedron, a figure having 12 identical faces [Devlin, 1997]. The quasiperiodic Penrose tiling pattern has five-fold symmetry of the dodecahedron. Recent studies [Seife, 1998] show that in a strong magnetic field, electrons swirl around magnetic field lines, creating a vortex. Under right conditions, a vortex can couple to an electron, acting as a single unit. Vortex geometrical structure is ubiquitous in macro-scale as well as microscopic subatomic dynamical fluctuation patterns.(b) Conventional continuous periodogram power spectral analyses of such spiral trajectories in Figure 2 (R_oR₁R₂R₃R₄R₅) will reveal a continuum of periodicities with progressive increase dq in phase angle q  (theta) as shown in Figure 3.

(c)The broadband power spectrum will have embedded dominant wavebands (R_oOR₁, R₁OR₂, R₂OR₃, R₃OR₄, R₄OR₅, etc.) the bandwidth increasing with period length (Figure 2). The peak periods E_n in the dominant wavebands is be given by the relation

where t is the golden mean equal to (1+Ö 5)/2  [approximately equal to 1.618 ] and T_s , the primary perturbation time period, for example, is the annual cycle (summer to winter) of solar heating in a study of atmospheric interannual variability. The peak periods E_n are superimposed on a continuum background. For example, the most striking feature in climate variability on all time scales is the presence of sharp peaks superimposed on a continuous background [Ghil, 1994].
(d)The ratio r/R also represents the increment dq in phase angle q (Equation 3 and Figure 3) and therefore the phase angle q represents the variance [Mary Selvam, 1990]. Hence, when the logarithmic spiral is resolved as an eddy continuum in conventional spectral analysis, the increment in wavelength is concomitant with increase in phase. The angular turning, in turn, is directly proportional to the variance (Equation 3). Such a result that increments in wavelength and phase angle are related is observed in quantum systems and has been named 'Berry's phase' [Berry, 1988]. The relationship of angular turning of the spiral to intensity of fluctuations is seen in the tight coiling of the hurricane spiral cloud systems.
(e)The overall logarithmic spiral flow structure is given by the relation

where the constant k is the steady state fractional volume dilution of large eddy by inherent turbulent eddy fluctuations. The constant k is equal to 1/t ² (~0.382) and is identified as the universal constant for deterministic chaos in fluid flows [Mary Selvam, 1990]. Since k is less than half, the mixing with environmental air does not erase the signature of the dominant large eddy, but helps to retain its identity as a stable self-sustaining soliton-like structure. The mixing of environmental air assists in the upward and outward growth of the large eddy. The steady state emergence of fractal structures is therefore equal to

    Logarithmic wind profile relationship such as Equation 5 is a long-established (observational) feature of atmospheric flows in the boundary layer, the constant k, called the Von Karman ’s constant has the value equal to 0.38 as determined from observations [Wallace and Hobbs, 1977]. In Equation 5, W represents the standard deviation of eddy fluctuations, since W is computed as the instantaneous r.m.s. (root mean square) eddy perturbation amplitude with reference to the earlier step of eddy growth. For two successive stages of eddy growth starting from primary perturbation w_* the ratio of the standard deviations W_n+1 and W_n is given from Equation 5 as (n+1)/n. Denoting by s the standard deviation of eddy fluctuations at the reference level (n=1) the standard deviations of eddy fluctuations for successive stages of eddy growth are given as integer multiple of s, i.e., s, 2s, 3s, etc. and correspond respectively to

    The conventional power spectrum plotted as the variance versus the frequency in log-log scale will now represent the eddy probability density on logarithmic scale versus the standard deviation of the eddy fluctuations on linear scale since the logarithm of the eddy wavelength represents the standard deviation, i.e. the r.m.s. value of eddy fluctuations (Equation 5). The r.m.s. value of eddy fluctuations can be represented in terms of statistical normal distribution as follows. A normalized standard deviation t = 0 corresponds to cumulative percentage probability density equal to 50 for the mean value of the distribution. Since the logarithm of the wavelength represents the r.m.s. value of eddy fluctuation the normalized standard deviation t is defined for the eddy energy as

where L is the period in units of time or space scale used in the analyses and T₅₀ is the period up to which the cumulative percentage contribution to total variance is equal to 50 and t = 0 . The variable logT₅₀ also represents the mean value for the r.m.s. eddy fluctuations and is consistent with the concept of the mean level represented by r.m.s. eddy fluctuations. Spectra of time series of any dynamical system, for example, meteorological parameters when plotted as cumulative percentage contribution to total variance versus t should follow the model predicted universal spectrum. The literature shows many examples of spectra of pressure, wind and temperature whose shapes display a remarkable degree of universality [Canavero and Einaudi, 1987]. The theoretical basis for formulation of the universal spectrum is based on the Central Limit Theorem in Statistics, namely, if an overall random variable is the sum of very many elementary random variables, each having its own arbitrary distribution law, but all of them being small, then the distribution of the overall random variable is Gaussian [Ruhla, 1992]. Therefore, when the spectra of space-time fluctuations of dynamical systems are plotted in the above fashion, they tend to closely (not exactly) follow cumulative normal distribution.
The period T₅₀ up to which the cumulative percentage contribution to total variance is equal to 50 is computed from model concepts as follows. The power spectrum, when plotted as normalized standard deviation t versus cumulative percentage contribution to total variance represents the statistical normal distribution (Equation 8), i.e., the variance represents the probability density. The normalized standard deviation value 0 corresponds to cumulative percentage probability density P equal to 50 from statistical normal distribution characteristics. Since t represents the eddy growth step n (Equation 7), the dominant period T₅₀ up to which the cumulative percentage contribution to total variance is equal to 50 is obtained from Equation 4 for value of n equal to 0 . In the present study of periodicities in prime number spacing intervals, the primary perturbation time period T_s is equal to the unit number class interval (spacing interval between adjacent primes) and T₅₀ is obtained as

    Prime numbers with spacing intervals up to 3.6 or approximately 4 contribute up to 50% to the total variance. This model prediction is in agreement with computed value of T₅₀ (Section 3.3).

    In number field domain, the above equation can be visualized as follows. The r.m.s. circulation speeds W and w_* are equivalent to units of computations of respective yardstick lengths R and r. Spatial integration of w_* units of a finite yardstick length r, i.e., a computational domain w_*r, results in a larger computational domain WR [Mary Selvam, 1993]. The computed domain WR is larger than the primary domain w_*r  because of uncertainty in the length measurement using a finite yardstick length r, which should be infinitesimally small in an ideal measurement. The continuum number field domain (Cartesian co-ordinates) may therefore be obtained from successive integration of enclosed finite number field domains (Mary Selvam, 1993) as shown in Figure 1.
Cartesian co-ordinates represent the complex number field. Historically, Gauss (1799) clearly regarded a complex number as a pair of real numbers. The idea was originally stated in a little known work of a Danish surveyor Wessel (1797) and later by Gauss. In 1806, the French mathematician Argand described a complex number x+iy as a point in the plane and this description was given the name 'Argand Diagram' [Stewart and Tall, 1990].
    The above visualization (Figure 1) will help apply concepts developed for continuum atmospheric flow dynamics to evolution of unique structures such as the distribution of prime numbers in real number field continuum, as explained in the following.
    Fractal structures emerge in atmospheric flows because of mixing of environmental air into the large eddy volume by inherent turbulent eddy fluctuations. The steady state emergence of fractal structures A is equal to [Selvam and Suvarna Fadnavis, 1999a; b]

    The spatial integration of enclosed turbulent eddy circulations as given in Equation (3) represents an overall logarithmic spiral flow trajectory with the quasiperiodic Penrose tiling pattern (Figure 2) for the internal structure [Selvam and Suvarna Fadnavis, 1999a; b] and is equivalent to a hierarchy of vortices (Section 2 above). The incorporation of Fibonacci mathematical series, representative of ramified bifurcations indicates ordered growth of fractal patterns and signifies non-local connections characteristic of quantum-like chaos. By analogy, the means of ensembles of successively larger number field domains follow a logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern (Figure 2) for the internal structure.

where z is equal to the eddy length scale ratio R/r and k is equal to the steady state fractional volume dilution of large eddy by turbulent eddy fluctuations and is given as

because the fractional outward mass flux of primary perturbation equal to W/w_* occurs in the fractional turbulent eddy cross-section r/R.

    In atmospheric flows a fraction equal to f of surface air is transported upward to level z and represents the upward transport of moisture, which condenses as liquid water content in clouds, and also aerosols of surface origin. The observed vertical profile of liquid water content inside clouds is found to follow the f distribution [Mary Selvam and Ramachandra Murty, 1985; Mary Selvam, 1990]. The vertical profile of aerosol concentration in the atmosphere also follows the f distribution [Sikka et al., 1988]. The fraction f carries the unique signature of surface air (primary perturbation) at the level z.
    The f distribution represents, at level z, the signature of unique primary perturbation originating from the underlying substratum. The f distribution therefore corresponds to the cumulative prime number density distribution corresponding to number z .

    In number theory, the Prime Number Theorem states that z/ln z where ln is the natural logarithm, represents approximately the number of primes less than or equal to z. Prime numbers are unique numbers, i.e., which cannot be factorized [Stewart, 1996]. Therefore P represents the cumulative unique domain lengths of the primary perturbation carried up to the level z. In the next Section (3.0) the following model predictions (Section 2.0) are verified. (a) The f distribution represents the actual and computed prime number density distribution. (b) The power spectra (variance and phase) of prime number distribution follow the universal and unique inverse power-law form of the statistical normal distribution. Inverse power-law form for power spectra signify self-similarity or long-range correlations inherent to the eddy continuum. (c) The broadband eddy continuum exhibits dominant periodicities in close agreement with model predicted periodicities (Equation 4). (d) The variance and phase spectra follow each other closely, particularly for the dominant eddies, thereby exhibiting 'Berry's phase' characterising quantum systems.

    The shape of the actual prime number density distribution is close to and resembles f distribution. Further, the maximum value (approximately equal to 0.6) for these two distributions occurs for z value equal to 2p. The eddy length scale ratio z represents the phase (Section 2) and therefore the maximum values for f and also (by analogy), for the prime number distributions occur for one complete cycle of eddy circulation. Such a closed self-sustaining circulation is similar to a soliton, a stable self-sustaining eddy structure.

    The cumulative percentage contribution to total variance, the cumulative percentage normalized phase (normalized with respect to the total phase rotation) and the corresponding t values were computed. The power spectra were plotted as cumulative percentage contribution to total variance versus the normalized standard deviation t as given above. The period L is in units of number class interval which is equal to one in the present study. Periodicities up to T₅₀ contribute up to 50% of total variance. The phase spectra were plotted as cumulative (%) normalized (normalized to total rotation) phase .The variance and phase spectra along with statistical normal distribution is shown in Figure 6. The 'goodness of fit' between the variance spectrum and statistical normal distribution is significant at <= 5% level. The phase spectrum is close to the statistical normal distribution, but the 'goodness of fit' is not statistically significant. However, the 'goodness of fit' between variance and phase spectra are statistically significant (chi-square test) for individual dominant wavebands (Figures 7a and 7b).

    The cumulative percentage contribution to total variance and the cumulative (%) normalized phase (normalized w. r. t. the total rotation) for each dominant waveband is computed for significant wavebands and shown in Figures 7a and 7b to illustrate Berry's phase, namely the progressive increase in phase with increase in period and also the close association between phase and variance (see Section 2).
The statistically significant (less than or equal to 5% level) wavebands are shown in Figure 8.

    Table 1 gives the list of a total of 110 dominant (normalised variance greater than 1) wavebands obtained from the continuous periodogram analyses for the data set (prime numbers in the interval 3 to 1000 at unit class intervals). The symbol * indicates that the dominant waveband is statistically significant at <= 5% level. There are 14 significant dominant wavebands (Figure 8). The dominant peak periodicities are in close agreement with model predicted dominant peak periodicities, e.g. 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1, and 64.9 prime number spacing intervals for values of n ranging from -1 to 6 (Equation 4). The symbol S indicates that the normalised variance and phase spectra follow each other closely (the 'goodness of fit ' being significant at <= 5% ) displaying Berry 's phase in the quantum-like chaos exhibited by prime number distribution. Earlier study by Marek Wolf (May 1996, IFTUWr 908/96 http://rose.ift.uni.wroc.pl/~mwolf) also shows that the number of Twins (spacing interval 2) and primes separated by a gap of length 4 ("cousins") is almost the same and it determines a fractal structure on the set of primes. The conjecture that there should be approximately equal numbers of prime power pairs differing by 2 and by 4, but about twice as many differing by 6 is proved to be true by Gopalkrishna Gadiyar and Padma (1999, http://www.maths.ex.ac.uk/~mwatkins/zeta/padma.pdf). The dominant perodicities shown above at Figure 8 are consistent with these reported results. The period T₅₀ upto which the cumulative percentage contribution to total variance is equal to 50 is found to be equal to 3.242 spacing interval between two adjacent primes. This periodogram estimate of T₅₀ for the prime numbers in the interval 3 to 1000 is in approximate agreement with model predicted value of T₅₀ approximately equal to 3.6 (Equation 9). The dominant significant period 2 corresponds to twin primes. In number theory [Rose, 1995; Beiler, 1966] the twin prime conjecture states that there are many pairs of primes p, q where q = p + 2 . There are infinitely many prime pairs as z tends to infinity.

Since the eddy length scale ratio z is equal to R/r

The z^th prime number has an angular phase difference equal to 1/z radians from the earlier (z-1)^th prime. The spiral arrangement of the first 20 and 100 primes are shown respectively in Figures 9 and 10. Spiral patterns in the arrangement of prime numbers have been reported earlier by mathematicians (Schroeder, 1986; also shown in the website: http://zaphod.uchicago.edu/~bryan/spiral/index.html).

Figure 9

Table 1: The list of a total of 110 dominant (normalised variance greater than 1) wavebands obtained from the continuous periodogram analyses for the data set (prime numbers in the interval 3 to 1000 at unit class intervals)