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

A. M. Selvam

(Retired) Indian Institute of Tropical Meteorology, Pune 411 008, India

email: [email protected]

web sites: http://amselvam.webs.com/index.html

http://members.tripod.com/~amselvam


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.

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

1. Introduction

The continuum real number field (infinite number of decimals between any two integers) represented as Cartesian co-ordinates [Mathews, 1961; Stewart and Tall, 1990; Devlin, 1997; Stewart, 1998] is the basic computational tool in the simulation and prediction of the continuum dynamics of real world dynamical systems such as fluid flows, stock market price fluctuations, heart beat patterns, etc. Till the late 1970s, mathematical models were based on Newtonian continuum dynamics with implicit assumption of linearity in the rate of change with respect to (w. r. t) time or space of the dynamical variable under consideration. The traditional mathematical model equations were of the form
(1)
Constant value was assumed for the rate of change (dX/dt)n of the variable Xn at computational step n and infinitesimally small time or space intervals dt. Equation (1) will be linear and can be solved analytically provided the rate of change (dX/dt)n is constant. However, dynamical systems in nature exhibit irregular (fractal) fluctuations on all space and time scales and therefore the assumption of constant rate of change fails and Equation (1) does not have analytical solution. Numerical solutions are then obtained for discrete (finite) space-time intervals such that the continuum dynamics of Equation (1) is now computed as discrete dynamics given by
(2)


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.

2. Cell Dynamical System Model Concepts

The model concepts are based on Townsend’s [Townsend, 1956] visualization of large eddies as envelopes enclosing turbulent eddy (small-scale) fluctuations (Figure 1).



Figure1: Visualisation of the formain of large eddy (ABCD) as envelope enclosing smaller scale eddies. By analogy, the continuum number field domain(Cartesian co-ordinates ) may also be obtained from successive integration of enclosed finite number field domains.


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

(3)


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.

2.1 Cell Dynamical System Model Predictions

A summary of the important theoretical results derived from Equation (3) [Mary Selvam, 1990; Selvam and Suvarna Fadnavis, 1998; 1999a; b], which are applicable to the present study, is given in the following. (a) The fractal structure of the continuum flow pattern is resolved into an overall logarithmic spiral trajectory RoR1R2R3R4R5 with the quasiperiodic Penrose tiling pattern for the internal structure and is equivalent to a hierarchy of vortices (Figure 2).



 
 

Figure 2 : The quasiperiodic Penrose tiling pattern with five-fold symmetry traced by the small eddy circulations internal to dominant large eddy circulation in turbulent fluid flows.


    The successively larger eddy radii (ORo,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 (RoR1R2R3R4R5) will reveal a continuum of periodicities with progressive increase dq in phase angle q  (theta) as shown in Figure 3.


Figure 2: The equiangular logarithmic spiral given by (R/r) = eaq   where aandq are each equal to 1/z  for each length step growth. The eddy length scale ratio z is equal to R/r  . The crossing angle a is equal to the small increment dq   in the phase angle q . Traditional power spectrum analysis will resolve such a spiral flow trajectory as a continuum of eddies with progressive increase dq   in phase angle q .


(c)The broadband power spectrum will have embedded dominant wavebands (RoOR1, R1OR2, R2OR3, R3OR4, R4OR5, etc.) the bandwidth increasing with period length (Figure 2). The peak periods En in the dominant wavebands is be given by the relation

En = Ts (2+t) tn

(4)


where t is the golden mean equal to (1+Ö 5)/2  [approximately equal to 1.618 ] and Ts , 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 En 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

(5)


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

1/k ~ 2.62

(6)


    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 Wn+1 and Wn 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

statistical normalized standard deviation t = 0,1,2,3, etc.

(7)


    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

t = (log L / log T50 ) – 1

(8)


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 Ts is equal to the unit number class interval (spacing interval between adjacent primes) and T50 is obtained as

T50 = (2+t)t0  ~ 3.6 spacing interval between two adjacent primes

(9)


    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).

2.2 Applications of model concepts to prime number distribution

The incorporation of Fibonacci mathematical series, representative of ramified bifurcations, indicates ordered growth of fractal patterns (Stewart, 1992). The fractal patterns are shown to result from the cumulative integration of enclosed small-scale fluctuations (Selvam and Suvarna Fadnavis, 1998). By analogy it follows that the continuum number field when computed as the integrated mean over successively larger discrete domains, also follows the quasiperiodic Penrose tiling pattern.
It is shown in the following that the steady state emergence of progressively larger fractal structures incorporate unique primary perturbation domains of progressively increasing total number equal to z/lnz where z, the length step growth stage is equal to the length scale ratio of large eddy to turbulent eddy.
In number theory, prime numbers are unique numbers and the prime number theorem (PNT) states that z/lnz gives approximately the number of primes less than or equal to z [Rose, 1995]. Historically, the PNT was postulated just before 1800 by both Legendre (1798) and Gauss (1791 in a personal communication) on numerical evidence and it was finally established by Hadamard and (independently) de la Vallee Poussin in 1896. The PNT states that if p(z) is the number of primes p which satisfy 2<= p <= z then p(z) is approximately equal to z/lnz where ln represents the natural logarithm (Rose, 1995; Allenby and Redfern, 1989).
The cell dynamical system model for turbulent fluid flows predicts, as explained in the following, that the function z/lnz represents the normalized cumulative variance spectrum of the eddies and this spectrum follows statistical normal distribution.
    The important result of the study is that the prime number spectrum is the same as the eddy energy spectrum for quantum-like chaos in atmospheric flows and the spectra follow the universal inverse power-law form of the statistical normal distribution.
    The cell dynamical model concepts and its application to the evolution of prime number spectrum is explained in the following.
    Large eddies are envelopes of enclosed turbulent eddy circulations, 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 given as (Equation 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

(10)
The steady state emergence of fractal structure A is

(11)


The outward and upward growing large eddy carries only a fraction f of the primary perturbation equal to

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

from equation (5)
from equation (10)
from equation (3)
Therefore

(12)


    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 .

Therefore the ratio P equal to A/f gives the number of units of the unique domain of surface air at level z.

(13)


    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.

3. Data and Analysis

    The actual prime number tables (the first 1000 primes) were obtained from the web site: http://www.utm.edu/research/primes. The first 1000 prime numbers were used for the study. The prime numbers were also computed using the Prime Number Theorem proposed in 1799 by Gauss, namely the total number of primes p(z) equal to or less than the number z is approximately equal to z/ln z . The computed prime number density distribution is equal to 1/ln z .
The computed f distribution (Equation 12), the actual prime number density distribution and the computed prime number density distributions are shown in Figure 4.



Figure 4: The cumulative prime number(actual) density and the corresponding f distribution have a maximum approximately equal to 0.6 for the number z equal to 2p which represents one complete eddy cycle . The eddy length scale ratio z represents the phase for the eddy continuum dynamics in turbulent fluid flows. A complete dominant eddy cycle(z = 2p) is a self-sustaining soliton-like structure.



 

    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.

3.1 The Frequency Distributions of Prime numbers, f Distribution and the Statistical Normal Distribution

The values of actual prime number distribution, the corresponding values computed using the relation z/lnz (Prime Number Theorem) which give the number of primes less than or equal to z and the f distribution follow statistical normal distribution (Selvam and Suvarna Fadnavis, 2001) as described in the following. The frequency distributions were computed in terms of the normalised standard deviation as explained in the following for prime number (calculated) distribution. The number of primes p less than z are calculated for a range of n values from x1 = z1 to xn = zn . The cumulative percentage number of primes pc is calculated as equal to (pm / pn)*100 where m = 1,2, ...n for each class interval X = (xm+xm+1 ) / 2. The number of primes pt = pm+1 - pm in each class interval X is also calculated. The normalized standard deviate t is then equal to (Xbar - X)/s  where Xbar is the mean of the prime number distribution. The corresponding standard deviation of the X versus pt distribution is then calculated as equal to s .
    The prime number (actual and computed) frequency distribution and also the corresponding f distribution for values of z from 3 to 1000 at unit intervals are shown in Figure 5. The statistical normal distribution is also plotted in the Figure 5. It is seen that the prime number (actual and computed) distributions and the corresponding f distribution closely follow statistical normal distribution.



Figure 5: Prime number(actual and computed) distribution and corresponding f distribution  follow closely the statistical normal distribution.


3.2 Spectra of prime number distribution

In the quantum-like chaos in atmospheric flows the function z/lnz represents the variance spectrum of the fractal structures as shown below.
    The length scale ratio z equal to R/r represents the relative variance (Equation 3). The relative upward mass flux of primary perturbation equal to W/w* is proportional to lnz (Equation 5). Therefore z/lnz represents the cumulative variance normalized to upward flow of primary perturbation. The cumulative variance or energy spectrum of the eddies is therefore represented by z/lnz distribution.
By concept (Equation 3) large eddies are but the integrated mean of inherent turbulent eddies and therefore the eddy energy spectrum follows statistical normal distribution according to the Central Limit Theorem (Section 2.1(e) above). The prime number spectrum, which is equivalent to the variance (energy) spectrum of eddies follows statistical normal distribution as seen in Figure 5. Earlier studies using various meteorological data sets have shown that atmospheric eddy energy spectrum follow statistical normal distribution [Selvam and Suvarna Fadnavis, 1998].

3.3 Power Spectral Analysis: Analyses Techniques, Data and Results

The broadband power spectrum of space-time fluctuations of dynamical systems can be computed accurately by an elementary, but very powerful method of analysis developed by Jenkinson (1977) which provides a quasi-continuous form of the classical periodogram allowing systematic allocation of the total variance and degrees of freedom of the data series to logarithmically spaced elements of the frequency range (0.5, 0). The periodogram is constructed for a fixed set of 10000(m) periodicities Lm which increase geometrically as Lm=2 exp(Cm) where C=.001 and m=0, 1, 2,....m . The data series Yt for the N data points was used. The periodogram estimates the set of Amcos(2pnmS-fm) where Am, nm and fm denote respectively the amplitude, frequency and phase angle for the mth periodicity and S is the time or space interval. In the present study the frequency of occurrence of primes in unit number class intervals ranging from 3 to 1000 was used. The cumulative percentage contribution to totavariance was computed starting from the high frequency side of the spectrum. The period T50 at which 50% contribution to total variance occurs is taken as reference and the normalized standard deviation tm values are computed as (Equation 8).

tm = (log Lm / log T50)-1

    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).


Figure 6: The variance and phase spectra along with statistical normal distribution



Figure 7a:  Illustration of Berry 's phase in quantum-like chaos in prime number distribution. The phase and variance spectra are the same for prime number spacing intervals up to 10.



Figure 7b:  Illustration of Berry 's phase in quantum-like chaos in prime number distribution. The phase and variance spectra are the same for prime number spacing intervals from 10 to 50.


    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.



Figure 8: Continuous periodogram analysis results : Dominant (normalised variance greater than 1) statistically significant wavebands.


    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.

3.4 Spiral Pattern of Prime Number Distribution in the x-y Plane

The zth prime is approximately equal to zln z [Allenby and Redfern, 1989]. In the following it is shown that the prime numbers are arranged in a spiral pattern in the x-y plane. The eqiangular logarithmic spiral shown at Figure 3 is given by the relation

R/r = eaq

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

ln z = aq = (1/z)(1/z)

zln z = 1/z = r/R = dq

The zth 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


Figure 10


4. Conclusions

In mathematics, Cantorian fractal space-time fluctuations is now associated with reference to quantum systems [Ord, 1983; Nottale, 1989; El Naschie, 1993; 1998]. 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 [Cipra, 1996; Berry, 1992; Cipra http://www.maths.ex.ac.uk/~mwatkins/zeta/cipra.htm]. The spacing intervals of adjacent prime numbers exhibit fractal fluctuations generic to diverse dynamical systems in nature. The irregular (chaotic) fractal fluctuations however, exhibit self-similar geometry manifested as inverse power-law form for power spectra. Self-similar fluctuations imply long-range correlations or non-local connections identified as self-organized criticality. A cell dynamical system model for atmospheric flows developed by the author shows that self-organized criticality is a signature of quantum-like chaos. The cell dynamical system model is a general systems theory applicable to all dynamical systems in nature. The model concepts show that quantum-like chaos in dynamical systems incorporates prime number distribution functions in the quantification of self-organized criticality. The model also provides unique quantification for the observed self-organized criticality in terms of the universal statistical normal distribution.
    The important result of the present study is that power spectra of different data sets of spacing intervals of adjacent prime numbers follow the model predicted universal inverse power-law form of the statistical normal distribution, a signature of self-organized criticality. The prime number distribution therefore exhibits quantum-like chaos.

Acknowledgements

The author is grateful to Dr. A. S. R. Murty for his keen interest and encouragement during the course of the study.

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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)

 
No
Periodicities in unit number class intervals
Peak period
Wave band
1
2.000
2.000 to 2.006 *
2
2.010
2.010..to..2.010 *
3
2.014
2.014..to..2.014
4
2.022
2.022..to..2.022
5
2.034
2.034..to..2.034
6
2.077
2.077..to..2.077
7
2.100
2.098..to..2.100
8
2.113
2.113..to..2.113
9
2.136
2.136..to..2.136
10
2.143
2.141..to..2.145
11
2.149
2.149..to..2.149
12
2.164
2.164..to..2.164
13
2.199
2.197..to..2.199
14
2.235
2.235..to..2.235
15
2.266
2.264..to..2.266
16
2.307
2.305..to..2.310
17
2.333
2.331..to..2.335 *
18
2.356
2.356..to..2.356
19
2.364
2.364..to..2.366
20
2.445
2.443..to..2.448
21
2.467
2.467..to..2.470
22
2.500
2.497..to..2.505 *
23
2.615
2.615..to..2.617
24
2.625
2.623..to..2.628
25
2.657
2.654..to..2.657
26
2.711
2.708..to..2.711
27
2.727
2.724..to..2.730
28
2.749
2.746..to..2.752
29
2.801
2.796..to..2.804 *
30
2.890
2.890..to..2.890
31
2.925
2.925..to..2.925
32
2.936
2.933..to..2.936
33
2.969
2.969..to..2.969
34
2.999
2.987..to..3.014 *
35
3.023
3.023..to..3.023
36
3.050
3.047..to..3.050
37
3.143
3.143..to..3.146
38
3.252
3.252..to..3.252
39
3.288
3.288..to..3.288
40
3.297
3.294..to..3.301
41
3.334
3.327..to..3.341 *
42
3.347
3.347..to..3.351
43
3.361
3.361..to..3.364
44
3.456
3.442..to..3.456
45
3.470
3.470..to..3.473
46
3.498
3.491.to..3.505 *S
47
3.537
3.537.to..3.540
48
3.558
3.558..to..3.558
49
3.666
3.663..to..3.670
50
3.688
3.685..to..3.688
51
3.714
3.707..to..3.722
52
3.748
3.744..to..3.755
53
3.782
3.778..to..3.782
54
3.820
3.816..to..3.823
55
3.881
3.881..to..3.881
56
4.125
4.125..to..4.125
57
4.200
4.196..to..4.204
58
4.247
4.242..to..4.247
59
4.285
4.281..to..4.294
60
4.333
4.320..to..4.346 S
61
4.372
4.367..to..4.376
62
4.402
4.394..to..4.407
63
4.568
4.568..to..4.568
64
4.600
4.596..to..4.605
65
4.670
4.656..to..4.679 *
66
4.717
4.717..to..4.722
67
4.750
4.745..to..4.760
68
4.774
4.769..to..4.779
69
4.939
4.934..to..4.944
70
4.964
4.964..to..4.969
71
4.999
4.984..to..5.014 * S
72
5.034
5.034..to..5.034
73
5.084
5.074..to..5.094
74
5.161
5.161..to..5.161
75
5.197
5.192..to..5.197
76
5.250
5.250..to..5.250
77
5.497
5.491..to..5.502
78
5.813
5.802..to..5.819
79
5.913
5.907..to..5.919
80
5.949
5.943..to..5.960
81
6.002
5.972..to..6.026 *S
82
6.051
6.044..to..6.057
83
6.087
6.081..to..6.087
84
6.124
6.124..to..6.130
85
6.279
6.272..to..6.285
86
6.329
6.323..to..6.329
87
6.496
6.489..to..6.502
88
6.995
6.974..to..7.023 *
89
7.346
7.324..to..7.361
90
7.509
7.472..to..7.539 S
91
7.638
7.615..to..7.653
92
8.102
8.086..to..8.119
93
8.399
8.366..to..8.425
94
8.501
8.484..to..8.526
95
9.986
9.926..to..10.066 * S
96
10.540
10.508..to..10.571
97
10.981
10.915..to..11.036
98
12.977
12.912..to..13.029 S
99
13.212
13.212..to..13.226
100
14.001
13.876..to..14.128 * S
101
15.031
14.897..to..15.152 S
102
15.458
15.412..to..15.505
103
17.050
16.948..to..17.153 S
104
22.113
21.871..to..22.335 S
105
28.793
28.450..to..29.025 S
106
29.998
29.581..to..30.452 S
107
31.223
31.192..to..31.254
108
42.020
41.353..to..42.655 S
109
198.571
192.702..to..204.414 S
110
1534.787
965.984..to..3402.097 S

* indicates that the dominant waveband is significant at <= 5% .

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