Mathematical irrational numbers not so physically irrational
aa r X i v : . [ phy s i c s . d a t a - a n ] J a n Mathematical irrational numbers not so physically irrational
Y. J. Zhao
Advanced Materials Laboratory, Fudan University, Shanghai 200433, China
Y. H. Gao
School of Machinery and Electricity, Henan Instituteof Science and Technology, Xinxiang 453003, China
J. P. Huang ∗ Department of Physics and Surface Physics Laboratory (National key laboratory),Fudan University, Shanghai 200433, China (Dated: October 23, 2018)
Abstract
We investigate the topological structure of the decimal expansions of the three famous naturally occurringirrational numbers, π , e , and golden ratio, by explicitly calculating the diversity of the pair distributions ofthe ten digits ranging from 0 to 9. And we find that there is a universal two-phase behavior, which collapsesinto a single curve with a power law phenomenon. We further reveal that the two-phase behavior is closelyrelated to general aspects of phase transitions in physical systems. It is then numerically shown that suchcharacteristics originate from an intrinsic property of genuine random distribution of the digits in decimalexpansions. Thus, mathematical irrational numbers are not so physically irrational as long as they havesuch an intrinsic property. Keywords: two-phase behavior; irrational numbers; phase transition
PACS numbers: 05.70.Fh; 89.70.Cf; 64.90.+b; 64.60.A- ∗ Corresponding author. Tel: +86 21 55665227. Fax: +86 21 55665239. E-mail address: [email protected] π , e , and golden ratio. The π is the ratio of a circle’s circumference to itsdiameter in Euclidean geometry. The e is the unique irrational number such that the value of thederivative of f ( x ) = e x at the point x = 0 is exactly 1. Two quantities are seen to be in thegolden ratio if the ratio between the sum of the two quantities and the larger one is the same as theratio between the larger one and the smaller. The golden ratio is expressed as (1 + √ / . Thethree naturally occurring irrational numbers have abundant of uses in physics, mathematics, andengineering. However, our question is if there is common interesting physical senses behind thethree naturally occurring irrational numbers. Our answer is true, as to be addressed in this work.Pair distributions are widely used, both experimentally and theoretically, to treat physical prob-lems, as a major descriptor for various microstructures [1, 2, 3, 4, 5], in which a pair distributionfunction describes the density of inter-atomic or inter-particulate distances. We shall investigatepair distributions of the ten digits (namely, 0, 1, · · · , 9) in the decimal expansions of the threeirrational numbers, with a focus on their diversity, so that one could get insight into the connectionbetween statistical physics and topological structures of numbers.The information entropy of a set of pair distributions provides a measure of the diversity ofthe set. The greater the diversity of the pair distributions in a set, the greater the entropy, whilea set with regular patterns has a small value for the entropy. In general, information entropyhas the following characters [6]: (i) Changing the value of one of the probabilities by a very smallamount should only change the entropy by a small amount (continuity); (ii) The result should keepunchanged if the probabilities are re-ordered (symmetry); (iii) If all the probabilities are equallylikely, information entropy should be maximal (maximum); (iv) Adding or removing an event withprobability zero does not contribute to the entropy; and (v) The amount of entropy should be thesame independently of how the process is regarded as being divided into parts (additivity). Here(v) characterizes the information entropy of a system with sub-systems, and it demands that theinformation entropy of a system can be calculated from the information entropy of its sub-systems2f we know how the sub-systems interact with each other.For the first digits in the decimal expansions of the three irrational numbers, π , e and goldenratio, we calculate their information entropy H ( N ) of the pair distributions according to H ( N ) = − X i =0 X j ( p i,j log p i,j ) , with p i,j = N i,j / P j N i,j , where N i,j denotes the number of pairs with step size j (which is thedistance between every pair of two i ′ s) for the digit i ranging from 0 to 9.Figure 1 displays the phase diagram for the existence of two distinct phases for the three nat-urally occurring irrational numbers, π , e , and golden ratio, by plotting order parameter H (infor-mation entropy of the pair distributions of the ten digits, ranging from 0 to 9) as a function of thenumber N of the first digits in the decimal expansions. The inset of Fig. 1 shows a log-log plot ofinformation entropy H versus N for the same three irrational numbers, and a power-law relationappears, as indicated by a straight line. Such a single-curve collapse for the three irrational num-bers suggests that they are governed by the same rule due to the same degree of disorder, whichis, on the other hand, implied by Fig. 2. In fact, the two-phase behavior shown for the ten digitsholds for each digit as well. In this work, we focus on the two-phase behavior for the ten digits asa whole.We analyze the diversity of the pair distributions of the digits ranging from 0 to 9, which isquantitatively expressed by their information entropy H [6], and discover not only a single-curvecollapse but also the surprising existence of a sudden change region (Region B), as shown in Fig. 1.For N within Region A, the value of H is roughly zero; we interpret this as a uniformity phasein which the diversity of the pair distributions of the digits does not predominate. For Region C, H increases significantly; we interpret this as an out-of-uniformity phase in which the diversitydominates and power-law phenomenon appears.While phase transitions have been demonstrated to exist in various mathematical systems [7],our findings for the irrational-number problem are identical to what is known to occur in all phase-transition phenomena in physical systems. The distinguishing characteristic of a phase transitionis an abrupt sudden change in one or more physical properties at a critical threshold κ c of somecontrol parameter κ . The change in behavior at κ c can be quantified by an order parameter φ ( κ ) .For the irrational-number problem, we find that the order parameter φ ( κ ) is given by the infor-3ation entropy H ( N ) . Region B bridges Regions A and C, and denotes a sudden change region,indicating a sudden increase of the diversity of the pair distributions. Next, we interpret these twoirrational-number phases according to Regions A and C.In Region A, the entropy is roughly zero; we interpret this to be the irrational-number unifor-mity phase, because the diversity of the pair distributions of the ten digits does not predominate. Inthe uniformity phase, there is almost no diversity, and information entropies fluctuate around their uniformity values, suggesting that most of the pair distributions donot have diversity, but behaveas uniformity (or none).In Region C, the entropy is large and increases step by step. We interpret this to be the out-of-uniformity phase, because increasing entropy suggests increasing diversity. So, in the out-of-uniformity phase, the prevalent uniformity has changed, and the diversity is now being driven tonew degree, which is consistent with the fact that more pair distributions of the ten digits cometo appear with longer step sizes as N increases. A power law is any polynomial relationship thatexhibits the property of scale invariance, and has been explicitly used to characterize a staggeringnumber of natural patterns [8, 9, 10]. The observation of a power-law relation in the data for π , e , and golden ratio, points to a specific kind of mechanisms that underly the irrational-numberphenomenon in question, and indicates a deep connection between the irrational numbers. Such aconnection originates from the same degree of disorder, as shown in Fig. 2. Figure 2 presents theoccurrence percentage of the distance between two consecutive digits in the decimal expansionsof the three irrational numbers ( π , e , and golden ratio) and a set of genuine random numbers.(Here “genuine random numbers” mean those random numbers or quasi-random numbers thathave a very high degree of randomness. Throughout this work, the phrase “random numbers”is simply used to indicate “genuine random numbers”. Similarly, in this work “genuine randomdistribution” is used to represent the distribution with a very high degree of randomness.) Theserandom numbers were generated by the commercial software Mathematica. The symbol lines forthe three irrational numbers are almost overlapped, which are further nearly overlapped with thatof the set of random numbers. This shows that the three irrational numbers possess the same degreeof disorder as a group of random numbers. In other words, the above-mentioned characteristicsrelated to the topological structure of the decimal expansions of π , e , and golden ratio actuallyoriginate from an intrinsic property of genuine random distribution of the digits in the decimal4xpansions.Our results suggest that there is a link between a mathematical system with many digits (theirrational number) and the ubiquitous phenomenon of phase transitions that occur in physicalsystems with many units.We hope that our work will stimulate further studies of number physics. Here we have revealeda universal two-phase behavior for the three famous naturally occurring irrational numbers, π , e ,and golden ratio. We should claim that the unique results obtained herein for the three irrationalnumbers (as shown in Fig. 1) also hold for many other irrational numbers like √ and √ . It isalso worth mentioning that the results obtained from Fig. 1 (e.g., the single-curve collapse) donot work for some other irrational numbers, e.g., √ . . So far, there is no efficient method tosort such irrational numbers clearly, except for investigating their decimal expansions one by one.Nevertheless, we could safely conclude that our findings are a universal behavior for numerousirrational numbers as long as they have an intrinsic property of genuine random distribution of thedigits in decimal expansions, and raise the possibility that the topological structure of irrationalnumbers are related to general aspects of phase transitions in physical systems. In this sense, suchmathematical irrational numbers seem to be not so physically irrational .5
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