# Essay on Kolmogorov Law of Minus 5 over 3 Viewed with Golden Ratio

Authors: Ming Li; Wei Zhao
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

The golden ratio is an astonishing number in high-energy physics, neutrino physics, and cosmology. The Kolmogorov −5/3 law plays a role in describing energy transfer of random data or random functions. The contributions of this essay are in twofold. One is to express the Kolmogorov −5/3 law by using the golden ratio. The other is to represent the fractal dimension of random data following the Kolmogorov −5/3 law with the golden ratio. It is our hope that this essay may be helpful to provide a new outlook of the Kolmogorov −5/3 law from the point of view of the golden ratio.


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## Body

## 1. Instruction

Letφ be the golden ratio. It equals to (1+5)/2. Its inverse 1/φ=(5-1)/2 is called the golden mean. Both are irrational numbers. Approximately, they are
(1)φ≈1.618,1φ≈0.618.The numberφ has wide applications to various fields, ranging from physics (Wurm and Martini [1], King [2], Ding et al. [3], Feruglio and Paris [4]), to cosmology (Livio [5], Boeyens [6]).In addition to the golden ratio, fractal is a mathematical model attracting interests of scientists and physicists in the field of high-energy physics (Ghosh et al. [7]). While studying energy transfer, the Kolmogorov -5/3 law introduced by Kolmogorov [8] plays a role in the field (Qian [9], Brun et al. [10, Chapter 7], Hillebrandt and Kupka [11, Chapter 4], Gomes-Fernandes et al. [12, page 309], Lumley and Yaglom [13], Warhaft [14], Geipel et al. [15]). Motivated by those, this essay aims at exhibiting the Kolmogorov -5/3 law and the self-similarity, which is an important fractal property (Mandelbrot [16, 17], Cattani et al. [18, 19]), from the point of the golden ratio. It is our expectation that this essay may be helpful to describe the nature of random data that follows the Kolmogorov -5/3 law.The rest of paper is organized as follows. The preliminaries are briefed in Section2. The results that the Kolmogorov -5/3 law and the self-similarity are explained from a view of the golden ratio are given in Section 3, which is followed by conclusions.


## 2. Preliminaries

### 2.1. Golden Ratio

Consider a straight line in Figure1. Arrange three points A, B, and C, there such that the ratio below
(2)ACCB=ABACFigure 1
Illustration of a straight line for the golden ratio.equals toφ. Then, we say that the straight line is cut in extreme and mean ratio ([5], Ackermann [20], Kaygn et al. [21]).There are various ways to synthesize the numberφ. According to the definition of extreme and mean ratio, one has
(3)x=x+1x.Multiplyingx on the both sides of the above yields
(4)x2-x-1=0.Solving (4) produces
(5)x1=1+52,x2=1-52.Conventionally,x1 is denoted by φ, and x2 is denoted by -1/φ.


### 2.2. Fractal Dimension

Let the autocorrelation function (ACF) of a random functionx(t) be rxx(τ), where rxx(τ)=E[x(t)x(t+τ)]. Then, rxx(τ) for τ→0 represents the small-scaling phenomenon of x(t). Following Davies and Hall [22], if rxx(τ) is sufficiently smooth on (0, ∞) and if
(6)rxx(0)-rxx(τ)~c1|τ|αfor|τ|⟶0,
where c1 is a constant and 0<α≤2 is the fractal index of x(t), the fractal dimension, which is denoted by D, of x(t) is in the form
(7)D=2-α2.Note that fractal dimension1≤D<2 is a measure to characterize the local self-similarity or local roughness of x(t) (Mandelbrot [17], Gneiting and Schlather [23]).Denote the power spectrum density (PSD) function ofx(t) by Sxx(ω). Denote by F the operator of Fourier transform. Then,
(8)Sxx(ω)=F[rxx(t)]=∫-∞∞rxx(t)e-jωtdt,j=-1.
In the domain of generalized functions, we have (Kanwal [24], Gelfand and Vilenkin [25], Li and Lim [26])
(9)F(|τ|α)=-2sin(πα2)Γ(1+α)|ω|-(α+1),
where Γ is the Gamma function. Therefore, we have an analogy of (6) in the frequency domain in the form
(10)F(|τ|α)~c2|ω|-(α+1)forω⟶∞,
where c2 is a constant (Chan et al. [27]).


## 2.1. Golden Ratio

Consider a straight line in Figure1. Arrange three points A, B, and C, there such that the ratio below
(2)ACCB=ABACFigure 1
Illustration of a straight line for the golden ratio.equals toφ. Then, we say that the straight line is cut in extreme and mean ratio ([5], Ackermann [20], Kaygn et al. [21]).There are various ways to synthesize the numberφ. According to the definition of extreme and mean ratio, one has
(3)x=x+1x.Multiplyingx on the both sides of the above yields
(4)x2-x-1=0.Solving (4) produces
(5)x1=1+52,x2=1-52.Conventionally,x1 is denoted by φ, and x2 is denoted by -1/φ.


## 2.2. Fractal Dimension

Let the autocorrelation function (ACF) of a random functionx(t) be rxx(τ), where rxx(τ)=E[x(t)x(t+τ)]. Then, rxx(τ) for τ→0 represents the small-scaling phenomenon of x(t). Following Davies and Hall [22], if rxx(τ) is sufficiently smooth on (0, ∞) and if
(6)rxx(0)-rxx(τ)~c1|τ|αfor|τ|⟶0,
where c1 is a constant and 0<α≤2 is the fractal index of x(t), the fractal dimension, which is denoted by D, of x(t) is in the form
(7)D=2-α2.Note that fractal dimension1≤D<2 is a measure to characterize the local self-similarity or local roughness of x(t) (Mandelbrot [17], Gneiting and Schlather [23]).Denote the power spectrum density (PSD) function ofx(t) by Sxx(ω). Denote by F the operator of Fourier transform. Then,
(8)Sxx(ω)=F[rxx(t)]=∫-∞∞rxx(t)e-jωtdt,j=-1.
In the domain of generalized functions, we have (Kanwal [24], Gelfand and Vilenkin [25], Li and Lim [26])
(9)F(|τ|α)=-2sin(πα2)Γ(1+α)|ω|-(α+1),
where Γ is the Gamma function. Therefore, we have an analogy of (6) in the frequency domain in the form
(10)F(|τ|α)~c2|ω|-(α+1)forω⟶∞,
where c2 is a constant (Chan et al. [27]).


## 3. Results

The Kolmogorov-5/3 law implies that the PSD of a random function has the asymptotic behavior expressed by
(11)Sxx(ω)~c|ω|-5/3forω⟶∞,wherec is a constant (Monin and Yaglom [28, 29]). The above well characterizes the local irregularity of turbulence function x(t) (Tropea et al. [30]).In order to connect the Kolmogorov-5/3 law with the golden ratio, we write
(12)53+e=φ.
From the previous discussions, we have
(13)e=φ-53=1+52-53=5-12-23=1φ-23.
Thus, the Kolmogorov -5/3 law expressed by (11) may be rewritten by
(14)Sxx(ω)~c|ω|-(φ-e)forω⟶∞.From (10), we see that the fractal index may be expressed by using the golden ratio as
(15)α=φ-e-1.
Thus, we attain the fractal dimension from the point of view of the golden ratio in the form
(16)D=2-φ-e-12.
From the previous discussions, we have the following theorems.Theorem 1.
Letx(t) be the random function that obeys the Kolmogorov -5/3 law expressed by (11). Then, its PSD may be expressed using the golden ratio in (14).Theorem 2.
Letx(t) be the random function that obeys the Kolmogorov -5/3 law expressed by (11). Then, its fractal dimension may be represented by using the golden ratio as that in (16). The concrete value of D in (16) is 5/3.Finally, we note that the significance of the present results lies in expressing the Kolmogorov-5/3 law, which plays a role in energy transfer, by using the golden ratio, which attracts interests of researchers and scientists in high-energy physics, neutrino physics, cosmology, and many others.


## 4. Conclusions

We have explained our results in expressing the Kolmogorov-5/3 law with the golden ratio. In addition, we have expressed the fractal dimension of random data obeying the Kolmogorov -5/3 law based on the golden ratio.



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*Source: 680678-2013-09-25.xml*