Academic Open Internet Journal ISSN 1311-4360

www.acadjournal.com

Volume 21, 2007

Signs of information perfection in distributing critical temperatures in the nature

David Kisets

The National Physical Laboratory of Israel, Danziger “A” bldg., Hebrew University

Givat-Ram, Jerusalem, 91904, Israel.

davein@netvision.net.il

Abstract - This paper presents results of exploring, based on mathematical analysis, to what extent allocations of critical temperatures of the Elements and blood as examples of steady quantitative manifestations in the nature conform to such fundamental concepts of mathematical relation as harmony, balance and information optimality. The conformity is considered as a sign of perfection, and the concepts are interpreted by corresponding constants on the numerical ratio scale. The recently formulated Law of Information Constants has also been used in the exploration as an integrated estimation criterion. It is demonstrated that regarding temperature scale the sought conformity exists for parameters of statistical distribution of boiling and melting temperatures of the Elements, and for critical blood temperatures. The existence of specific temperature point on the blood temperature scale is discovered, which meets the requirements both the information optimality and harmony. The author believes that this paper marks the beginning of studying the informational perfection by means of the joint usage of the above-mentioned concepts in various steady quantitative manifestations and regularities.

Keywords: informational optimality – mathematical harmony and balance – melting and

                     boiling temperatures – blood temperature – relations of critical temperatures.

“God, however, has chosen the most perfect, that is to say, the one which is at the same time the simplest in hypotheses and the richest in phenomena, as might be a geometric line whose construction would be easy but whose properties and effects would be very remarkable and of wide reach.”      (Leibniz , 1686)

Introduction

Hypothetically, quantitative manifestations and regularities in the nature are tending to be steady and informatively perfect if aiming to match the conceptions of informational optimality, harmony and balance. The optimality is considered here in terms of information theory [1], using the method of eliminating the information redundancy based on the recently discovered principle of information cyclicity [2]. The mathematical harmony and balance represent here golden section and equality respectively.

    The display of a kind of perfection is known, for instance, in mathematically harmonious relations occurring in various fields of mankind activity, mainly in art, architecture and music [3]. Recently the proof of existence of so-called information optimality (informatively optimal relation) - another characteristic of the perfection has also been revealed when discussing the information nature of optimal classifications [4], e.g. in the regularity in appearing the significant historical events discovered by Meece [5] and in the classification of words-use frequency on the basis of Cobuild Wordlist [6] in linguistics, as well as in classifying interpolation errors in metrology [7].

    The idea of exploring possible combination of optimal and harmonious relations in the nature reasonably arises (particularly with the discovery of direct realization of harmonious relation in genetic code [8, 9]). However, on a broad scale there are no signs of enough attention to this topic, and therefore one of purposes of this paper is to initiate a professional discussion of the subject on the very significant example of critical temperatures in the nature.

    If taking perfection by the aim and result of creation or by the vector of evolution in the nature, relations of critical temperatures or their distributions, as being physical indicators of certain constancy and completeness, are quite suitable for the consideration concerned. Together with absolute temperature (-273.16°C), by critical temperatures are named here: melting and boiling temperatures of the Elements in the mineral kingdom, and normal blood temperatures in the animal kingdom. In both cases the critical temperatures are distributed within relatively narrow ranges: approximately from -253°C to 5660°C for melting and boiling temperatures of 93 Elements, the temperatures of which so far are known to the full [10, 11], and from 0°C to 100°C in the animal kingdom where the normal blood temperature of warm-blooded creatures (being the most promoted ones) approximately ranges from 36.7°C to 44.4°C [12]. The narrowness of ranges of critical temperatures is analogous to the result of decrease of thermodynamic entropy.

    The conceptions of informational optimality, harmony and balance have been diversely applied for the two temperature systems under consideration, i.e. the statistical approach has been used for melting and boiling temperatures, and non-statistical - for blood temperature. Before considering the temperatures, the brief discussion of specific constants underlying the used conceptions is given in the next section.

Constants of perfection - information constants

It is convenient to consider the constants expressing informational optimality, mathematical harmony and balance, in terms of dimensional ratios [13]. In so doing, a dimension S divides into two (A and B) or proportional number of parts matching these constants as follows:

       1)  the informational optimality when the ratio of parts equals 1/2π, where π = 3.1416.

 Importantly, here the optimality is of different purposes: it may be the optimality of uncertainty-tolerance ratio: ρ_oa = B/S = 1/2π (optimum accuracy coefficient) or classification optimality: ρ_oc = B/A = 1/2π (optimum classification coefficient). In both cases ρ_o = 1/2π » 0.159 indicates on a redundancy absence or on a minimum possible information sufficiency (1 - ρ_o). The deriving of optimum accuracy and classification coefficients is briefly dealt with in the Appendix 1;

       2) the harmony (harmonious relation): f_o = B/A = A/S = 0.618 – the fundamental mathematical constant known as the measure of harmony (named also golden section or golden mean or golden proportion or divine proportion);

3) the balance (equality):  λ = A/S = B/S = 0.5  or λ_o = A/B = 1.

    The coefficients ρ_o,  f_o, and λ  (or λ_o) together with the base of decimal numeration z_o =10, proven (Appendix 2) as the informatively optimal numeration system [14], represent so-called information constants, and the approximate equation ρ_o* f_o* z_o* λ_o = ρ_o* f_o* z_o = 1 has been called the Law of Information Constants [15]. In the considering aspect it might be also called the Law of Informational Perfection.

    Being optimum, the accuracy coefficient ρ_oa possesses the universality for determining the informatively permissible deviation from any of above constants. The deviation equal to ± 0.5 ρ_oa multiplied by the respective constant can reasonably be attributed to any of the constants; this limits possibilities of its practical use within the optimal range. Thus, the coefficients with their permissible (about 8%) deviations are determined as follows:

ρ_o (1 ± 0.5 ρ_oa) = 0.159 ± 0.013,                                                                           (1)                                                                   

f_o (1 ± 0.5 ρ_oa) = 0.618 ± 0.049,                                                                            (2)                                                                

λ (1 ± 0.5 ρ_oa) = 0.5 ± 0.04  or  λ_o (1 ± 0.5 ρ_oa) = 1 ± 0.08                                    (3)

    Henceforth these values are considered as tolerated ones in terms of the criterion of acceptability in determining the conformity of critical temperatures to meet the requirements of informational optimality, harmony and balance.

    Clearly, there are two possibilities in analyzing a system for the informational perfection: 1) with determining whether each of information constants is within permissible tolerance, and 2) with their product using the Law of Information Constants.

Relations of critical temperatures of the Elements

First, before analyzing the relations of critical temperatures of the Elements with the dimensional approach, the system of best (in terms of informational perfection) parameters of normalized statistical distribution serving as the analytical model is suggested below.

Analytical model

In the field of statistics it is easy to demonstrate the existence of approximate (2.9% estimation error) harmonious relation: 2σ(x)/µ(x) = f_o between double standard deviation (standard interval or standard tolerance) 2σ(x) and expectation µ(x) in the distribution of normalized (sum =1) quantities (x) when it is characterized by σ(x) = ρ_o and µ(x) = λ. By the way, the ratio 2σ(x)/µ(x) being double coefficient of variability may be called the form index (γ) of a distribution.

    Such optimum, harmonious and balanced system is being characterized by the triple perfection quality and in this sense represents informatively unique statistical distribution. On this basis a distribution can be analyzed whether it possesses such triple quality or not.                                                                      

    Thus, resting on expressions (1), (2) and (3), by analogy if the mean µ_e and the experimental standard deviation σ_e of some real normalized distribution lie within the ranges: 0.46 £ µ_e £ 0.54 and 0.146 £ σ_e £ 0.172 respectively, then likely the harmonious relation between 2σ_e and µ_e, i.e. their averaged ratio γ_e = 2σ_e /µ_e within the range 0.569 £ γ_e £ 0.667 can be expected.

    Now it is possible to proceed to critical temperatures of the Elements as such.

Statistical system of critical temperatures of the Elements

The system of melting (t_m) and boiling (t_b) temperatures of the Elements (Fig.1) together with the absolute temperature (t₁) serves as an example of such statistical regularities in the nature where (and it will be proven below), beyond question, the triple perfection quality exists.

Fig. 1: Distribution of melting (squares) and boiling (triangles) temperatures of

              chemical elements in their alphabetic order (from Actinium to Zirconium)

    The ratios α_i = (t_b – t_m)/( (t_b – t₁) and β_i = (t_m – t₁)/( (t_b – t₁) were calculated as relative dimensional arguments in determining normalized experimental statistical parameters. The critical temperatures of initial group of 93 Elements have been treated to find out and exclude informatively insignificant ones. Possessing quantitative symmetry, a mean (µ_e) according to classification approach predetermines informatively significant components of a system in the limits from µ_e* ρ_oc to [ µ_e + µ_e (1 - ρ_oc)]. Therefore, the following conditions are true for excluding non-informative Elements:

µ_e(α)/2π ³ α_i ³ µ_e(α)*(2 – 1/2π),                                                                      (4) 

µ_e(β)/2π ³ β_i ³ µ_e(β)*(2 – 1/2π)                                                                       (5)

    The calculations performed using routine statistical expressions for determining the means and experimental standard deviations have led to conclusion that, excepting Helium, inert gases (Group Zero of Periodic Table of the Elements), as well as Gallium (Group Thirteen) and Astatine (Group Seventeen) do not meet these requirements and, thus, they ought to be excluded from the consideration. Formulas and results of calculation for the rest 86 Elements are as follows:

µ_e (α) = (1/86) = 0.486,                                                                             (6)

µ_e (β) = (1/86) = 0.514 ;                                                                            (7)

σ_e (α) = {(1/85) - µ_e (a)]²}^1/2 = 0.159,                                                       (8) 

σ_e (β) = {(1/85) - µ_e (b)]²}^1/2 = 0.159;                                                      (9) 

γ_e (α) = 2σ_e (α)/µ_e ( α) = 0.654,                                                                           (10)

γ_e (β) = 2σ_e (β) /µ_e ( β) = 0.619.                                                                          (11)

    The closeness of normalized statistical parameters of the system of melting and boiling points of the Elements to information constants (within established tolerances) is shown in Fig. 2. In ascertaining the fact of informational perfection, among others one can note the remarkable result concerning informational optimality.

   Fig. 2: The positioning of normalized statistical parameters of the system of melting

               and boiling temperatures of the Elements within permissible ranges for

               information constants.

    The applying of the Law of Information Constants when using accepted here statistical designations yields the following results:

σ_e (α)* γ_e (α)*10*[2 µ_e (α)] = 1.011,                                                                    (12)

σ_e (β)* γ_e (β)*10*[2 µ_e (β)] = 1.012                                                                     (13)

    Thus, by means of integrated measure of informational perfection the systematic perfection of critical temperatures may also be considered proven (the deviation from the law is about 1%).

Relations of critical temperatures of blood

The temperature scale of any substance that is vital for the normal functioning of some biological or technical structure represents the system that includes specific critical temperatures needed for its existence, the temperatures of its physical states, and, if any, temperature range(s) in which the vitality has to be provided. Blood temperature represents an example of such a system. Temperature scales of blood, water and mercury are presented on Fig. 3.

Fig.3: Temperature scales of blood, water and mercury

    Temperature scale of physically stable blood approximately stretches from the absolute zero of temperature scale (t₁ = -273.16°C) up to the temperature of boiling water (t₆ = 100°C). The blood temperature of cold-blooded ranges approximately from the triple point of water (t₃ = 0°C) upward in accordance with the temperature of the surrounding medium. The normal blood temperature of warm-blooded, approximately ranging from t₄ = 36.7°C to t₅ = 44.4°C, contains specific temperature t_o, which hypothetically matches the informational optimality and mathematical harmony for the scale of blood.

    A dependence of physiological activity from blood temperature is decreasing in favor of internal thermal auto-regulation. This has resulted in the transition of blood temperature from the relatively wide range typical of cold-blooded creatures to the narrow range of warm-blooded and has been accompanied by the optimization and harmonization regarding critical points of blood temperature scale. For warm-blooded animals the shortening of normal blood temperature within the range (t₆ – t₃) had resulted in achieving the temperature tolerance Dt_w = (t₅ – t₄) (Fig. 3).

Specific point of blood temperature scale

A heat exchange between surrounding medium and blood is equivalent to the change of thermodynamic entropy, which is materially less for the blood temperature auto-regulation. Thus, one can note again that formal analogy between thermodynamic and logic entropy allows to consider a study of temperature scale with the information approach reasonable.

    Informational optimality and mathematical harmony are inherent in some parts of the blood temperature scale. For example, we can consider two parts, the first of which bears upon the physical temperature scale of water and blood and is being hypothetically characterized by the ratio (t₆ – t_o)/(t₆ – t₁) = r_o. The second part bears upon liquid blood and is hypothetically being characterized by the ratio (t₆ – t_o)/(t₆ – t₃) = f_o. The existence of specific temperature t_o satisfying both the optimality and harmony and located within the range of Dt_w (Fig. 3) can serve as the evidence in favor of the hypotheses. The proof of that is easily obtained by using the Law of Information Constants to the above-mentioned parts for the determination of t_o as follows:

t_o  = arg {[(t₆ – t_o)/(t₆ – t₁)]*[ (t₆ – t_o)/(t₆ – t₃)]*10 =1} = 38.9 °C                             (14)

    The temperature t_o is being characterized by the minimum estimation error d_min » 0.0025 (Fig. 4) and thus illustrates high optimization level; in this case the error (d) as function of temperature has been calculated as follows:

δ = {1 - ô[(t₆ – t)/(t₆ – t₁)]*[ (t₆ – t)/(t₆ – t₃)]*10ô} /1                                              (15)

 Fig. 4: Estimation error as function of temperature characterizes the degree of

                    optimality of t_o

It is remarkable in analyzing human temperature variation effects [16] that the relative difference between the critical human body temperature 44°C, which almost certainly characterizes the occurrence of death, and its normal temperature 37°C is very close (about 0.04% estimation error) to ρ_o = 1/2π.  

Discussion

Some features of use of the Law of Information Constants and statistical parameters should be taken in attention when estimating the information perfection of a system. Since the Law definitely possesses an uncertainty principle, its usage for integrated estimating the informational perfection predetermines the situation of satisfactory result for some combinations of information constants when one of them somewhat goes out permissible deviations. In the theoretically worst situation, when all information coefficients are of maximum or minimum permissible levels according to conditions (1), (2) and (3) the estimation error is about 24%. In a real practice, by analogy with information constants, the limitation of maximum estimation error of 8%, i.e. the permissible deviation of 1 ± 0.08 is possibly being realistic. Clearly, using the Law, the obtained results for critical temperatures of the Elements (1% estimation error) and blood (0.25% estimation error) are characterized by much higher accuracy that shows the high level of informational perfection of these systems.

Two peculiarities of statistical estimation

    1. The expectation of harmonious relation 2σ_e /µ_e within permissible range, when arguments σ_e and µ_e are within their permissible ranges, is limited. The limitations for such expectation by the error E = [(γ_e - f_o)/ f_o]*100% = [(2σ_e /µ_e – 0.618)/0.618]*100% within permissible ranges for σ_e and µ_e were calculated. Acceptable results are obtained for those combinations of σ_e and µ_e pairs that provide the error E within the permissible range (±8%). Calculations show such an availability of about 67% of all possible the combinations.

    2. The appropriate level of confidence that the expanded uncertainty in determining µ_e is within the range of optimality is to be ensured. In case the postulate concerning the normal distribution of n normalized components of a system fits the reality, the following conditions are true:

k σ_e / £ 0.5 µ_e ρ_o,  or  k σ_e / £ 0.08 µ_e;                                                    (16)

n ³ 625 (k σ_e)² ,                                                                                                 (17)

where σ_e / = σ_e (µ_e) is the standard uncertainty of the mean; k = the coverage factor for certain level of confidence. For example, when σ_e = 1/2π and widely used 95% level of confidence (k = 2), n equals 63.

    For critical temperatures of the Elements the distributions of arguments α_i and β_i in determining experimental statistical parameters have been analyzed by c²-criterion according to the standard procedure [17] that has resulted in assuming their approximate conformity to expected normal distribution. Thus, according to formulas (16) and (17) the estimation of coverage factor and confidence level (C) for the found σ_e = 0.159 and n = 86 has led to k = 2.33 and C = 98%. All this illustrates satisfactory results in terms of both the informational and statistical approach.

Perfect interrelation of temperature scales of water, blood and mercury

There has been an assumption that the temperature range of liquid blood and of water (t₆ – t₃) » 100°C supposedly belongs to some temperature hierarchy, i.e. may represent the information redundancy regarding the greater range (100°C)/r_o = 628.32°C. It proves to be the case and to be considered as an additional result of the work. The extended in such a way scale (from –273.16°C to 356°C) with 0.13% estimation error matches the temperature scale of mercury (Fig. 3).

    It is noteworthy (and could be perceived as the astonishing juggling in the nature) that the temperature scale of the liquid part of blood and water is encompassed by the part of the temperature scale of only one chemical element in its liquid state, i.e. mercury - the unique medium traditionally used in thermometry. Together temperature scales of water, blood and mercury are looked as if they form the optimized and harmonized system with interconnected sub-ranges that meet the requirements of optimality (through r_o) and harmony (through f_o ); and the system is also balanced (through l) around t_o (Table 1).

Table 1. Informational conception of blood and mercury temperature scales

______________________________________________________________________

Information coefficients                  Ratio of  ranges                                Deviation

Symbol             Constant             Expression             Result                       (1 – R/C)*100

_______________(C)_________________________ (R)_________________________

f_o1                  0.618             (t₇ – t₂)/(t₇ – t₁)            0.626                             -1.3 %

  f_o2                  0.618             (t₆ – t₁)/(t₇ – t₁)            0.594                             +3.9 %

  f_o3                0.618             (t₆ – t₄)/(t₆ – t₃)            0.635                              -2.7 %

---------------------------------------------------------------------------------------------------------------------

r_o1                  1/2p               (t₆ – t₃)/(t₇ – t₁)            0.159                               0 %

---------------------------------------------------------------------------------------------------------------------

                             0.5                (t₇ – t_o)/(t₇ – t₁)            0.505                               -1 %

                             0.5                (t_o – t₁)/(t₇ – t₁)            0.497                              +0.6 %                                                             

Symbols:

f_o1   =  the harmony of mercury temperature scale

f_o2   =  the harmonious relation of blood/water and mercury temperature scale

f_o3   =  the harmonious relation of the temperature scales of warm-blooded and cold-

                  blooded

r_o1  =  the optimality of liquid blood temperature regarding the temperature scale of

                  mercury

l =  the balance of mercury temperature scale in regard to the temperature of

          warm-blooded.

    Table 1 illustrates quite good calculated results in determining information coefficients of the examined system of critical temperatures in relation to their permissible deviations, defined by expressions (1), (2) and (3).

Conclusions

1.  Over viewing obtained results, one can assert that on the example of critical temperatures of the Elements and of blood the fundamental concepts of informational optimality, harmony and balance have very concrete embodiment both in mineral kingdom and animal kingdom.

2.    Of course, the hypothesis that quantitative manifestations and regularities in the nature as a whole are tending to be steady and informatively perfect could not be substantiated by studying the critical temperatures only. However, in the author’s opinion, the proposed concepts taken both singly and jointly (as the Law of Information Constants), might serve as powerful means for studying, understanding, classifying and modeling various quantitative manifestations and regularities in the nature and, very likely, in various areas of human activity. This, for instance, can likely be useful for the improvement of technical, technological, medical, social and other systems.  

Appendix 1: Determining of optimum information coefficients (information cyclicity)

The determination of optimum information coefficients, i.e. accuracy coefficient and classification coefficient, is carried out applying the conception of weights, which in the normalized form conditionally are treated as formal analogs of probabilities.

Optimum accuracy coefficient

A dimension S may be divided onto n uniform constituents (S/n) with weights K_j = 2 (n + 1 – j)/n (n + 1), representing a linear diagram (DK = K_j+1 – K_j = const); and the number j of informative constituents is determined using improved entropy function [2] as follows:

j = exp (-) = exp {-   (18)

    When n ® ¥ the determination of informative parameters with this formula contains the fundamental estimation error of about 2% due to the inequality of weights in the redundant part (from j to n) of the linear diagram of weights. This part consists of m successively reducing subsystems of the complete groups of weights, and thus possesses own summary redundancy. This local redundancy causes both the optimization insufficiency and uncertainty in the main informative part (j=1¸j) of the system, and the redundant number of the subsystem parameters is equal to the product (n - j). The same number of weightiest subsystems parameters may be considered as the limit of possible addition to j, which is the interval of uncertainty in determining the informative parameters. The optimum number of parameters (j_o) is within the limits from j to [j + (n - j)], and with estimation error of about 1% the optimum accuracy coefficient (ρ_oa) may be determined as follows:

ρ_oa = 1 - j_o/n = 1 – [j /n + 0.5(1 - j/n)* ] = 1/2p         (19)

Optimum classification coefficient

Being converted into normalized form, non-dividable parts of dimension S represent the weights K₁ = A/S and K₂ = B/S that, in turn, represent relative contributions to any quality simulated by S.

   The optimum classification coefficient is evaluated for the necessary and sufficient number j_o = exp(-K₁ ln K₁ – K₂ ln K₂) = 1.5 of components that is true for the most uncertain (50% confidence) situation about allowing or ignoring the lesser of two components [18].  Accordingly, the classification coefficient with high estimation accuracy is determined as follows:

r_oc = arg {exp [- () ln () – () ln ()] = 1.5} =

= arg {exp [- () ln () – () ln ()] = 1.5} = 1/2p              (20)

Appendix 2: Information optimality of decimal numeration system

In general, any integer z, which is greater than one, can be used as the base of a numeration system, and the system will employ z different digits. The quality of any quantitative information and, thus, the informational perfection as a whole depends also on the numeration system used. This happened to be true owing to the mathematical theorem known as Benford's Law [19] and the principle of information cyclicity. According to the Benford’s Law, in any numeration system the probability P(d) of any number d from 1 to (z -1) is calculated as follows:

     P(d) = log_z (1 + 1/d)                                                                                      (21)

    The probabilities P(d) form a complete group of independent events, i.e. their sum = 1, and a logarithmic sequence has obvious classification character. If, for instance, z = 10, then d = 1 ÷ 9 repeats for the subgroups: 10 ÷ 90, 100 ÷ 900, etc. owing to the first digits of each subgroup. The different probabilities of the numbers on the one hand, and the infinite diversity of potential numeration systems on the other hand suggest an existence of optimum numeration system that can be determined by examining the logarithmic sequences. The minimum and maximum Benford's probability, i.e. P_min = P[d = (z - 1)] and P_max = P(d = 1) form the specific system of two components, which ratio indicates to the degree of optimality of a numeration system. The optimum ratio (P_min /P_max)_o represents optimum classification coefficient determined as follows:

    (P_min /P_max)_o = P [d = (z - 1)] / P (d = 1) = 1/2p                                                (22)

    In accordance with the above-established criterion, the index of optimum numeration system z_o (that is the logarithm base) is determined as the rounding off number as follows:

     z_o = arg min |log_z [1 + 1/.(z - 1)]/ log_z (1 + 1) - (1/2p )| = 10                           (23)

Therefore, the optimality of decimal system may be considered as proven.

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