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

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.

1. Instruction

Let 𝜑 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 1𝜑≈1.618,𝜑≈0.618. (1)

The number 𝜑 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 Section 2. 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 Figure 1. Arrange three points A, B, and C, there such that the ratio below AC=CBABAC (2)

Figure 1 

Illustration of a straight line for the golden ratio.

equals to 𝜑. 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 number 𝜑. According to the definition of extreme and mean ratio, one has 𝑥=𝑥+1𝑥. (3)

Multiplying 𝑥 on the both sides of the above yields 𝑥²−𝑥−1=0. (4)

Solving (4) produces 𝑥₁=1+√52,𝑥₂=1−√52. (5)

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 function 𝑥(𝑡) be 𝑟_𝑥𝑥(𝜏), where 𝑟_𝑥𝑥(𝜏)=𝐸[𝑥(𝑡)𝑥(𝑡+𝜏)]. Then, 𝑟_𝑥𝑥(𝜏) for 𝜏→0 represents the small-scaling phenomenon of 𝑥(𝑡). Following Davies and Hall [22], if 𝑟_𝑥𝑥(𝜏) is sufficiently smooth on (0, ∞) and if 𝑟_𝑥𝑥(0)−𝑟_𝑥𝑥(𝜏)∼𝑐₁|𝜏|^𝛼for|𝜏|⟶0, (6) 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 𝛼𝐷=2−2. (7)

Note that fractal dimension 1≤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 of 𝑥(𝑡) by 𝑆_𝑥𝑥(𝜔). Denote by F the operator of Fourier transform. Then, 𝑆_𝑥𝑥(𝜔)=F𝑟_𝑥𝑥(𝑡)=^∞_−∞𝑟_𝑥𝑥(𝑡)𝑒^{−𝑗𝜔𝑡}𝑑𝑡,𝑗=√−1. (8) In the domain of generalized functions, we have (Kanwal [24], Gelfand and Vilenkin [25], Li and Lim [26]) F|𝜏|^𝛼=−2sin𝜋𝛼2Γ(1+𝛼)|𝜔|^−(𝛼+1), (9) where Γ is the Gamma function. Therefore, we have an analogy of (6) in the frequency domain in the form F|𝜏|^𝛼∼𝑐₂|𝜔|^−(𝛼+1)for𝜔⟶∞, (10) 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 𝑆_𝑥𝑥(𝜔)∼𝑐|𝜔|^−5/3for𝜔⟶∞, (11)

where 𝑐 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 53+𝑒=𝜑. (12) From the previous discussions, we have 5𝑒=𝜑−3=1+√52−53=√5−12−23=1𝜑−23. (13) Thus, the Kolmogorov −5/3 law expressed by (11) may be rewritten by 𝑆_𝑥𝑥(𝜔)∼𝑐|𝜔|^{−(𝜑−𝑒)}for𝜔⟶∞. (14)

From (10), we see that the fractal index may be expressed by using the golden ratio as 𝛼=𝜑−𝑒−1. (15) Thus, we attain the fractal dimension from the point of view of the golden ratio in the form 𝐷=2−𝜑−𝑒−12. (16) From the previous discussions, we have the following theorems.

Theorem 1. Let 𝑥(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. Let 𝑥(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.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under the Project Grant nos. 61272402, 61070214, and 60873264, and by the 973 plan under the Project Grant no. 2011CB302800.