is the root-mean-square momentum of particle. The result (5) can be expressed in terms of either frequency or wavelength
m u 2 p 2 1/ 2 2 1/ 2 h
(10)
m u 2 p 2 1/ 2 2 1/ 2 k
(11)
when the definition of stochastic Planck and Boltzmann factors are introduced as h p 2 1/ 2 (12)
k p 2 1/ 2
(13)
EED
At the important scale of EKD vacuum, corresponding to photon gas one obtains the universal constants of Planck and Boltzmann that in view of (10)-(11) become [28]
h h k m k c k2 1/ 2 6.626 1034 J-s
(14)
k k k m k c k2 1/ 2 1.3811023 J/K
(15)
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(21)
that is related to De Pretto number 8338 [32].
Because at thermodynamic equilibrium the mean velocity of each particle (oscillator) vanishes = 0, the energy of the particle can be expressed as
m u 2 p 2 1/ 2 2 1/ 2
(20)
ECD
( e , e , L e ) (105 , 10 3 , 101 ) m 7
5
3
( c , c , L c ) (10 , 10 , 10 ) m 9
7
5
EMD ( m , m , L m ) (10 , 10 , 10 ) m
(22a) (22b) (22c)
If one applies the same (atom, element, system) = ( , , L ) relative sizes in (22) to the entire spatial scale of Fig.1, then the resulting cascades or hierarchy
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by the invariant Maxwell-Boltzmann distribution function
of overlapping statistical fields will appear as schematically shown in Fig.2.
10 10 10
10 10 10
19 17 15
-15 -17 -19
10 10 10 10 10 10
10 10 10 10 10 10
23 21 19 17 15 13
-13 -15 -17
10 10 10
25
10 10 10
15
23 21
13 11
10 10 10
-11
10 10 10
-21
-13 -15
10 10 10
10 10 10
10 10 10
27 25 23
10 10 10
29 27
10 10 10
31 29 27
10 10 10
33 31
10 10 10
dN u
COSMIC SCALE
35 33 31
25
11
-9 -11 -13
10 10 10
10 10 10
11 9 7
-7 -9 -11
10 10 10
10 10 10
9 7 5
-5 -7 -9
10 10 10
10 10 10
7 5 3
-3 -5 -7
10 10 10 10 10 10
5 3 1 -1 -3 -5
10 10 10 10
3 1 -1
10
-23
-23 -25
0
uc
-23 -25 -27
10 10 10
-29
10 10 10
0.4
-31
10 10 10
-31 -33
10 10 10
0.2
-31
-35
PLANCK SCALE
0.0
uai
MOLECULAR TRANSITION
ATOMIC TRANSITION
0.1
200
vc = ue
uaj
umj
400
600
800
1000
1200
va = um
vm = uc
1600 1800
2000
2200
2400
vs = ua
VELOCITY m/s
the velocity potential lead to the invariant Bernoulli equation [32] (ρ ) t
( ρ )
2
p cons tan t 0
2
(25)
Comparison of (25) with the Hamilton-Jacobi equation of classical mechanics [2]
Recently it was shown [32] that the energy spectrum of stationary isotropic turbulence is governed by the invariant Planck energy distribution law [42]
S t
(S)
2
2m
U0
(26)
resulted in the introduction of the invariant action and quantum mechanic wave function as [32, 45]
3
(23) d h / kT e 1 At large wave numbers, the universal Planck law (23) gives the Kolmogorov 5/ 3 law [32]. The energy of each oscillator (eddy) is = h . Preliminary examinations of the three-dimensional energy spectrum E(k, t) for isotropic turbulence appear to support such a correspondence [43, 44]. Under the condition of thermodynamic equilibrium, the speed of particles will be governed
S ( x, t) ρ
ISSN: 1790-2769
umi
A central question is what causes the stability of the “particle”. The question of particle stability is addressed by examination of the invariant Schrödinger equation [32]. The conservation equations (5) and (7) for an incompressible and irrotational v flow with
6 Quantum Mechanical Foundation of Turbulence
u 3
EMD
Fig.3 Maxwell-Boltzmann speed distribution viewed as stationary spectra of cluster sizes for EED, ECD, and EMD scales at 300 K.
scale of planetary dynamics (astrophysics) 1017 and the latter from the scale of cosmology or galacticdynamics 1035 m.
V
//
wc
According to Fig.2, starting from the hydrodynamic scale (103 , 101 , 101 , 103 ) after seven generations of statistical fields one reaches the electro-dynamic scale with the element size 1017 , and exactly after seven more generations one reaches the Planck length scale ( G / c3 )1/ 2 10 35 m , where G is the gravitational constant. Similarly, exactly seven generations of statistical fields separate the hydrodynamic scale (103 ,101 , 10 1 , 10 3 ) from the
8h
ua
ucj
-33
Hierarchy of statistical fields with ( , , L ) from cosmic to Planck scales [41].
(24)
du
-29
Fig.2
dN
um ECD
CLUTSER TRANSITION
-27 -29
2
m u / 2kT
uci
-25 -27
2kT
EED
0.6
Namj N ac 10 10 10
)3 / 2 u 2 e
-3
-19 -21
m
that was derived directly from the invariant Planck distribution function [32]. By (24), one arrives at a hierarchy of embedded Maxwell-Boltzmann distribution functions for …EED, ECD, and EMD scales as shown in Fig.3.
13
9
4 (
N
29
,
( x, t) S ( x, t)
(27)
The wave function (27) was recently applied to derive [32] the invariant time-dependent Schrödinger equation [46] i o
t
2 2m
2 U 0
(28)
It is now clear that the potential energy [32]
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p ρ V2 n U U
(see Fig.2) the hydrodynamic field represents the physical space [51] or the Casimir vacuum [53] with its fluctuations. Also, at cosmic scales 1035 with = g (Figs.1, 2), the wave function g will correspond to the wave function of the universe [54, 55]. The wave-particle duality of galaxies has been established by their observed quantized red shifts [56]. Also, the dual and seemingly incompatible objective versus subjective natures of , emphasized by de Broglie [1-3], can now be resolved. This is because the objective part of (27) is associated with the density and accounts for the particle localization while the subjective part of is associated with the complex velocity potential that accounts for the observed action-at-a-distance as well as renormalization and thereby the success of Born’s [57] probabilistic interpretation of .
(29)
in (28) acts as Poincaré stress [47-49] and is responsible for the stability of “particles” or de Broglie wave packets. The invariant Planck energy distribution law (23) and the invariant Schrödinger equation (28) fully establish the quantum mechanical foundation of turbulence. The formal derivation [32] of the Schrödinger equation (28) from the Bernoulli equation (25) provides for a new paradigm of the physical foundation of quantum mechanics. Soon after the introduction of his equation, Schrödinger himself [50] tried to identify some type of ensemble average interpretation of the wave function. Clearly, according to (27) the statistical ensemble nature of naturally arises from the velocity potential . To reveal the universal (Fig.1) physical foundation of quantum mechanics one starts by noting that the statistically stationary size of “clusters” is governed by the MaxwellBoltzmann distribution function (24) [32]. Next, one views the transfer of a cluster from a small rapidly oscillating eddy (j) to a large slowly oscillating eddy (i) as transition from the high energy level (j) to the low energy level (i), see Fig.3, as schematically shown in Fig.4,
7 Invariant Definitions of System, Element, and Atomic Lengths and Times In view of the model described in Fig.1, the spatial coordinate for the statistical field at scale will be defined as [29]
x ln N A
Eddy-j
Therefore, the spatial distance of each statistical field is measured on the basis of the number of “atoms” of that particular statistical field N A . With definition (31) the counting of numbers must begin with the number zero naturally since it corresponds to one atom. The characteristic lengths of the (system, element, atom) = (L , , ) at scale are defined as
ej
cluster c ji
molecule
mji
Eddy-i ei
Fig.4 Transition of cluster c ij from eddy-j to eddy-i leading to emission of molecule m ij .
L ln N AS
Such transitions (Fig.4) will be accompanied with emission of a “sub-particle” or “molecule” to carry away the excess energy [51]
ji j i h( j i )
ln N AE
0
(30)
(32)
,
N A 1
where (N AS , N AE ) respectively refer to the number
in harmony with Bohr’s theory of atomic spectra [52]. Therefore, the reason for the quantum nature of “molecular” energy spectra in equilibrium isotropic turbulent fields is that transitions must occur between eddies whose energy levels must satisfy the criterion of stationarity imposed by the Maxwell-Boltzmann distribution function [51]. The hydrodynamic model of the Schrödinger equation and the stochastic models of electrodynamics [1-17] are in harmony with the quantum mechanical foundation of turbulence (23) and (28). At the Planck scale ( G / c3 )1/ 2 10 35 m
ISSN: 1790-2769
(31)
of atoms in the system and the element. In view of the definitions in (31)-(32), the following schematic representation of the hierarchy of length scales may be obtained
... L1 ____________ 1 _____ 01
L ______ ... (33)
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In the following it will be shown that N ES N AE1
The size of the element and atom of adjacent statistical fields within the cascade (Fig.1) will be related by
N AE N AS1
(34)
could be identified as the fractal dimension at scale . By (31) and (37), the system length at scale becomes
N A N AE1
(35)
N L ln N AS ln(N AE ) ES
N ES ln(N AE )
(38)
As a result of (32) and (34)-(35), system, element, and zero lengths of the adjacent statistical fields are related to each other as
By definitions in (32) the result (38) can also be expressed as
L1
L N ES
0 1
The relationship between the element and the system lengths of adjacent scales in (32) suggests that
(36)
ln N AE L1 N ES1 1
in accordance with (33). A modified definition of dimension is next identified on the basis of analogy with the classical definition of dimension in Euclidean space. We consider the Cartesian coordinate in Euclidean space (x, y, z) (N x , N y , N z ) and note that with
N ES1 ln N AE 1 ln N AE 1 ln N AS1
N AS1 N AE1
Nz
Ny
N ES1
(41)
will be defined as the unit or “measure” such that by (31) and (32) one can introduce the “dimensionless” or “measureless” coordinate
y Nx
x
x
Fig.5 Discrete Cartesian coordinates in threedimensional Euclidean space.
x
ln N A
(42)
ln N AE
that by (37) gives
, the total number of atoms in the volume is V N xyz N x N y N z . Hence, a three-
x
dimensional Cartesian coordinates means that there are three degrees of freedom that can accommodate atoms at equal intervals under a constant measure in a Euclidian geometry. Following this same model, since each element at scale contains N AE1 atoms that constitute N ES independent
N E ln N AE ln N AE
N E
(43)
Therefore, the natural numbers 1, 2, 3, … of a Euclidean space at scale refer to integral numbers of this constant “measure” defined in (32). Hence, without specification of a “measure” the natural numbers will have no meaning in mathematics in the same spirit that numbers have no meaning without specification of their “dimension” in physical sciences. The conventional operations of addition, subtraction, division and multiplication will apply to such measureless numbers. However, such a Euclidian space with a constant measure will transform into a generalized Riemannian space if the
elements of the system at lower scale each of which will in turn decompactify into N AE atoms (Fig.2), the total number of atoms of the system will be given by
ISSN: 1790-2769
(40)
in accordance with (37). In view of the physical nature of the statistical fields shown in Fig.1, it is reasonable to expect that the mathematics required for their description should involve dimensional entities like other branches of physical science. Therefore the element length
z
N ES
N ES1
such that
discrete spectrum shown in Fig.5
N AS N AE
(39)
(37)
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measure becomes a variable. The distribution
atomic "time" ( , , t ) for the statistical field at
of particles amongst various energy levels of such statistical fields and their connection to quantum mechanics was recently described in terms of a scale invariant Planck distribution function [32]. With the “measureless” coordinate (42) the cascade (33) will become normalized [29]
scale [29]
...
Thus, one arrives at a hierarchy of time elements for the statistical fields shown in Fig.1 as
L _________________1 ______ 0
= L /w =
(48a)
= /v = t
(48b)
t = l /u =
(48c)
. . . e > c > m > p > s > . . .
L1 ___ 11 01
(49)
(44)
Clearly, the most fundamental and universal physical time is the time associated with the tachyon fluctuations [51, 59] of Casimir vacuum [53] at the Planck scale t in ETD field shown in Fig.1.
It is important to note that in (44), the interval (11 , 01 ) is considered to be unobservable at the
8 Distributions of Spacings between Zeros of Riemann Zeta Function
...
scale and is only revealed after re-scaling as a result of decompactification of 0 . Also, rather
The hydrodynamic model of Schrödinger equation and the stochastic models of electrodynamics [1-17] also provide a new paradigm for the physical foundation of quantum mechanics in general and the nature of the particle energy spectra in particular. According to the new paradigm, the energy spectra given by de Broglie- Schrödinger wave mechanics or Heisenberg [60] matrix mechanics will correspond to different size clusters, de Broglie wave packets, as shown in Fig.3 with the associated transitions shown in Fig.4. In this section some of the implications of the new paradigm mentioned above to the important problem of Riemann Hypothesis that lies at the boundary between mathematics and physics will be examined. In particular, the recent work of Montgomery [61] and Odlyzko [62] connecting the normalized spacings between the zeros of Riemann zeta function to the normalized spacings between the egienvalues of Gaussian Unitary Ensemble GUE resulting in what is known as Montgomery-Odlyzko law [63, 64] will be addressed. The pair correlation of Montgomery [61] was subsequently recognized by Dyson to correspond to that between the energy levels of heavy elements [63-64] and thus to the pair correlations between eigenvalues of Hermitian matrices [65]. Hence, a connection was established between quantum mechanics on the one hand and quantum chaos [66] on the other hand. However, the exact nature of the connections between these seemingly diverse fields of quantum mechanics, random matrices, and the distribution of zeros of Riemann zeta function is yet to be understood. In a recent investigation on the implications of the model in Fig.1 to nonstandard analysis [67] a
than the factor of 10 between different decimal generations in the classical arithmetic, the interval (1 , 0 ) opens to an infinite interval (L1 , 01 ) at the first lower scale of -1 as shown in (33) and (44). It is interesting to examine the connection of the result (43) with the classical definition of fractal dimension [58]
D
ln N(r) ln(r)
(45)
where N(r) is the total number of unit shapes and r is the size of the coarse–graining. From (37)-(43), one can introduce a scale-invariant definition of fractal dimension as [67]
D N AE N ES1
ln N AS1 ln N AE1
(46)
that by (45) leads to the invariant definition of coarse-graining
r
1 N AE1
(47)
Since each element could contain very large number of “atoms”, the fractal dimensions of typical statistical field shown in Fig.1 could be exceedingly large 107 . The above definitions for the length (L , , ) and velocity (w , v , u ) result in the following definitions of the system, element, and
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It is important to emphasize again the mathematical model being described namely that every point, “atom”, of the statistical field of EED shown in Fig.3 will decompactify into a completely new statistical field composed of a spectrum of molecular clusters associated with an internal EMD field. In a recent study [51], it was argued that the Maxwell-Boltzmann distribution function could be viewed as the distribution of the size of clusters of particles associated with different speeds. Therefore, one could view the hierarchy of cluster sizes as a hierarchy of energy levels. The principle of detailed balance insures the stochastically stationary states of all such cluster sizes, de Broglie wave packets, even though all possible molecular transitions between various levels (Fig.4) take place continuously. According to (25)-(29) Schrödinger equation has a hydrodynamic origin [4-11, 32] leading to stationary states governed by the steady form of (28). Therefore, energy levels of atomic clusters, such as the heavy element uranium, that are stabilized by the Poincaré stress (29) are given by the eigenvalues of the steady form of (28) that by Hiesenberg’s matrix mechanics [60] will be associated with the eigenvalues of Hermitian matrices [65]. Thus, the model suggests that quantum mechanical systems composed of hierarchy of embedded statistical fields under equilibrium condition have particles with Gaussian velocity distribution, Maxwell-Boltzmann speed distribution, and Planck energy distribution [32]. Now, if one associates “atoms” with “numbers” and cluster sizes with distances, then the distribution of the spacings between various size clusters will be expected to be governed by the Maxwell-Boltzmann distribution function. The comparison between normalized MaxwellBoltzmann distribution and the normalized spacings between zeros of Riemann zeta function n starting from n = 1012 from calculations of Odlyzko [62] are shown in Fig.7.
logarithmic definition of coordinates and the concept of “dimensionless” or “measureless” numbers
x
x
N ES
(50)
was introduced in terms of the “measure”
defined by
L 1
1
01
e
2 x 1
dx 1
1
(51)
2
In view of the relation (50)-(51), the range ( 1 ,1 ) of the outer coordinate x will correspond to the range ( 1 , 1 ) of the inner coordinate x 1 leading to the coordinate hierarchy schematically shown in Fig.6. + 1 = 2 +1
1 +1
0+ 1
= 2
1
0
1
+ 1
1+ 1
- 1
Fig.6 Hierarchy of normalized coordinates for cascades of embedded statistical fields [67].
The choice of the measure (50) is based on the fact that the equilibrium statistical fields shown on the left hand side of Fig.1 are composed of random fields thus following Gauss’s normal distribution. The hierarchy of embedded statistical fields shown in Fig.1 and 2 could be associated with a hierarchy of numbers in mathematics thus suggesting a new paradigm for the concept of continuum since each atom (point) of scale decompactifies into a finite element, i.e. cluster or Hilbert’s condensation [63], of the lower scale . According to Figs.1 and 3, each atom of the EED field will be composed of an ensemble of molecular clusters and as such also represents an equilibrium molecular dynamic EMD system. Therefore, at thermodynamic equilibrium each point of the external field EED must be in thermal equilibrium with the internal field EMD. Thus, one arrives at a hierarchy of embedded equilibrium statistical fields whose particles must satisfy the invariant Maxwell-Boltzmann speed distribution (24) as shown in Fig.3. The energy spectrum of all such equilibrium statistical fields will be governed by the invariant Planck energy distribution law (23) [32].
ISSN: 1790-2769
fv 1.0
0.8
0.6
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
v
Fig.7 Comparison of distributions of the normalized spacings between zeros of zeta function
n 12 n 12 [62] and the normalized Maxwell-Boltzmann distribution (52).
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The Maxwell-Boltzmann distribution (24) 2
f (u ) 4 (
B m 2 kT
)3/ 2 u 2 e
m ( Bu )2 /2kT
are normalized, they will collapse into a single universal normalized Maxwell-Boltzmann distribution function that applies to all equilibrium statistical fields at all scales. One further notes that such a distribution function may be viewed as the distribution of the size of clusters as a function of the speed and hence energy of the particles.
(52)
shown in Fig.7 corresponds to nitrogen gas m 28 1.66 10 27 kg at the temperature of T = 380 K with the factor B = vβ =(2kT / m)1/2 534 for normalization when speed is made dimensionless through division with the most probable speed vβ u β+1 . It is important to note here that according to the definition of Boltzmann constant (15), Kelvin absolute thermodynamic temperature T corresponds to the mean thermal wavelength of the particles
