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 spacetime fluctuations characterise dynamical
systems of all spacetime scales ranging from the microscopic subatomic
dynamics to macroscale 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 selfsimilar geometrical pattern and are associated with inverse
powerlaw form for the power spectrum. Selfsimilar fluctuations imply
longrange spacetime correlations identified as selforganized criticality.
A cell dynamical system model for atmospheric flows developed by the author
gives the following important results: (a) Selforganized criticality
is a signature of quantumlike chaos (b) The observed selforganized
criticality is quantified in terms of the universal inverse powerlaw
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 quantumlike 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 powerlaw form of the statistical normal distribution.
The prime number distribution therefore exhibits selforganized criticality,
which is a signature of quantumlike 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: quantumlike 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 roundoff 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 nonlinear 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 nonlinear dynamical systems was identified
nearly a century ago by Poincare (Poincare, 1892) in his study of threebody
problem, namely the Sun, Earth and the Moon. Nonlinear
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 nonlinear dynamical system defined by Equation
(2) is commonly used in all branches of science and other areas of human
interest.
Nonlinear dynamics and chaos soon (by 1980s) became a
multidisciplinary field of intensive research (Gleick, 1987). Sensitive
dependence on initial conditions implies longrange spacetime correlations.
The observed irregular fluctuations of real world dynamical systems also
exhibit such nonlocal connections manifested as fractal or selfsimilar
geometry to the spacetime evolution. The universal symmetry of selfsimilarity
ubiquitous to dynamical systems in nature is now identified as selforganized
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). Selfsimilar structures have internal
geometrical structure, which resemble the whole. The spacetime organization
of a hierarchy of selfsimilar spacetime structures is common to real
world as well as the numerical models (Equation 2) used for simulation.
A substratum of continuum fluctuations selforganizes 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 selfsimilar (fractal) spacetime
fluctuations exhibited by real world and numerical models of dynamical
systems are signatures of quantumlike 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
spacetime 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 nonlinear dynamics and chaos. Quantification of the
apparently irregular (chaotic) fractal fluctuations will help compute
(predict) the spacetime 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 powerlaw form of the statistical normal distribution.
Figure1: Visualisation of the formain
of large eddy (ABCD) as envelope enclosing smaller scale eddies. By analogy,
the continuum number field domain(Cartesian coordinates
) 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
The dynamical evolution of spacetime
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 selfsimilar fractal fluctuations of atmospheric
flows are manifestations of quantumlike chaos.
Figure 2 : The quasiperiodic Penrose tiling pattern with fivefold symmetry traced by the small eddy circulations internal to dominant large eddy circulation in turbulent fluid flows.
The successively larger
eddy radii (OR_{o},OR_{1},
etc.) and the corresponding circulation speeds (W_{1},
W_{2}
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 fivefold 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 macroscale as well as microscopic subatomic
dynamical fluctuation patterns.(b) Conventional continuous periodogram
power spectral analyses of such spiral trajectories in Figure 2 (R_{o}R_{1}R_{2}R_{3}R_{4}R_{5})
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) = e^{aq} 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 (R_{o}OR_{1}, R_{1}OR_{2}, R_{2}OR_{3}, R_{3}OR_{4}, R_{4}OR_{5}, etc.) the bandwidth increasing with period length (Figure 2). The peak periods E_{n} in the dominant wavebands is be given by the relation
E_{n} = T_{s} (2+t) t_{n}
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 selfsustaining solitonlike 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
Logarithmic wind
profile relationship such as Equation 5 is a longestablished (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
statistical normalized standard deviation t = 0,1,2,3, etc.
The conventional
power spectrum plotted as the variance versus the frequency in loglog
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 T_{50} ) – 1
where L is the period
in units of time or space scale used in the analyses and T_{50}
is the period up to which the cumulative percentage contribution to total
variance is equal to 50 and t = 0 . The variable
logT_{50}
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 spacetime
fluctuations of dynamical systems are plotted in the above fashion, they
tend to closely (not exactly) follow cumulative normal distribution.
The period T_{50}
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 T_{50} 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 T_{50} is obtained as
T_{50} = (2+t)t^{0} ~ 3.6 spacing interval between two adjacent primes
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 T_{50} (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 coordinates) may therefore be obtained
from successive integration of enclosed finite number field domains (Mary
Selvam, 1993) as shown in Figure 1.
Cartesian coordinates 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 nonlocal connections characteristic of quantumlike 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
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 crosssection 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 powerlaw form of the statistical
normal distribution. Inverse powerlaw form for power spectra signify selfsimilarity
or longrange 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.
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 selfsustaining solitonlike 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 selfsustaining circulation is similar to a soliton, a stable selfsustaining eddy structure.
Figure 5: Prime number(actual and computed)
distribution and corresponding f distribution follow
closely the statistical normal distribution.
t_{m} = (log L_{m} / log T_{50})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 T_{50} 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 (chisquare 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 quantumlike 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 quantumlike 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 quantumlike 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 T_{50} 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 T_{50} for the prime numbers in the interval 3 to 1000 is in approximate agreement with model predicted value of T_{50} 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.
R/r = e^{aq}
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 z^{th} prime number has an angular phase difference equal to 1/z radians from the earlier (z1)^{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 10
















































































































































































































































































































































* 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 quantumlike chaos exhibited by prime number distribution