# Empirical Evidence for a Law of Information Growth

## Abstract

**:**

## Introduction

_{i}the entropy is given by

_{i}ln p

_{i}erg/degree,

_{i},

_{i}log

_{2}p

_{i}bits,

## System Growth and Evolution

_{o}x(1 - x/X)

_{o}is the initial growth rate. The time from initiation to termination of this growth can be estimated from the slope of the line through the sigmoid inflection point, extrapolated from initiation to saturation values of x. From the appendix it is ∆t = 4/r

_{o}. The saturation at X is generally due to some natural limit of the process. For example, in the case that x represents the population number of a newly introduced species it may be food limits in the area of growth or the attraction of predators as x increases. In biological growth X may be the size ascribed to the limits of physical stress. In technology it can be taken to be an economic limit or fulfillment of the capabilities of a new approach.

_{k+1}/∆t

_{k}∝ R

_{k}

^{-k}, where R

_{k}= X

_{k}/X

_{k-1}.

_{k}> 1 so that the series of escalations converges. If the R change slowly with cycle number then, from the appendix, the total time from the end of the sequence of escalations, back to the k-th transition t

_{k}, is

_{k(past)}= log (Const.) - k logR

_{k(past)}versus k, this is a straight line: the “Evolutionary Trajectory” of the phenomena being studied.

## Natural Evolution

^{10}BP, where we use the paleontologists notation “BP,” for years Before the Present.

^{9}BP. As with almost all the events we consider, there is a range of argued values for its age, due to differing criteria and sampling uncertainties. However, as we will be using the logarithm of that age this variance is greatly reduced on the scale we will employ. We recognize, of course, that the formation and reproduction of the first replicating molecule, and then cell, were major informational storage and transmission developments.

^{8}BP.

## Post-Organic Evolution

- TABLE I: PHYLOGENY OF MANKIND
- Event number 3
- Descent category: CLASS - Mammal
- Date (BP): 1.5-2 (10)
^{8}

- Event number 4
- Descent category: SUPERFAMILY- Hominoid
- Date (BP): 2.5-4.0 (10)
^{7}

- Event number 5
- Descent category: FAMILY - Hominid
- Date (BP): 4 -10 (10)
^{6}

- Event number 6
- Descent category: GENUS - Homo
- Date (BP): 1.5-2 (10)
^{6}

- Event number 7
- Descent category: SPECIES - H. sapiens
- Date (BP): 2-5 (10)
^{5}

- Event number 8
- Descent category: SUBSPECIES - H. s. sapiens
- Date (BP): 7-10 (10)
^{4}

- Event number 9: Civilization
- Date (BP): 1- 2 (10)
^{4}

- Event number 10: Writing
- Date (BP): 3 - 5 (10)
^{3}

- Event number 11: Printing
- Date (BP)* 5.45 (10)
^{2}

- Event number 12:
- Digital Communication & Computing

## Information

The relation between entropy and information means that this is true of the latter as well as the former. Despite this the extent of The Evolutionary Trajectory suggests that there is a fundamental behavior that is being observed. . We have been led to it by the logistic extension of equation (6) but once presented it can be regarded as a purely an empirical result, pertaining to evolution on earth. The questions of whether a theoretical foundation can be found or whether the relation applies elsewhere, and at what rate, are, at this point, conjecture.“Processes that generate order are in no sense driven by the growth of entropy.”[10]

## Discussion

## APPENDIX

#### Derivation of The Evolutionary Trajectory

^{2}x/dt

^{2}= 0 gives, for the maximum values,

**P**denotes the product of terms. Note that there are no terms with j>k since, as indicated in the text, these appear later, i.e., only when their escalations occur. In the product of factors with j<k, all terms have x > X

_{j}or, more likely, x » X

_{j}so the unity can be dropped Equation (A4) then becomes

_{max}= [k/(k+1)] X

_{k}

_{k+1}/∆t

_{k}, the transition times ratio of two successive escalations. This results in a product of two terms:

_{k+1}/∆t

_{k}= {#} {(R

_{k}/R

_{k+1}) [(R

_{k+1}-1)/(R

_{k}- 1)] R

^{-k}

_{k+1}}

^{k+2}k

^{k}/(k+1)

^{2(k+1)}

_{k}= X

_{k}/X

_{k-1}.

^{-k}. We therefore obtain a particularly simple dependence of elapsed times on cycle number, k. Defining r = R

^{-1}< 1, and using

**D**in place of delta,we have

**D**t

_{k+}

_{1}/

**D**t

_{k}≈ r

^{k}≈ (X

_{k-1}/X

_{k})

^{k}

_{k}is

**S**denotes summation and C = (

**D**t

_{1})/(1-r) is not a function of k. Since r

_{k}is less than unity, equation (A9) indicates that the series of escalation times converges. In fact, from (A10) we see that when k becomes large the limit is t

_{∞}= C, giving

_{∞}- t

_{k}= C r

^{k}

_{∞}so that t

_{∞}- t

_{k}= t

_{past}(k). Then

_{past}(k) = log C - k logR

_{past}and cycle number.

_{o}, in equation (A5), since equation (A2) indicates that the transition time is inversely proportional to this term.

_{∞}since, for two transitions N and k

_{past}= t

_{N}- t

_{k}= C r

^{N}(1 + r

^{k - N})

_{past}.

_{∞}or a t

_{N}are not known and a reference date is estimated which is too early by the amount τ, then a MacLaurin expansion of (A12), leads to

_{past}(k) = log C - k logR - τ/(C r

^{k})

## References and Notes

- Schneider, E. D.; Kay, J. J. Order from Disorder: The Thermodynamics of Complexity in Biology. In What is Life? The Next Fifty Years; Murphy, M. P., O’Neill, L., Eds.; Cambridge Univ. Press, 1995; pp. 161–173. [Google Scholar]
- Schroedinger, E. What is Life; Cambridge U. Press, 1944. [Google Scholar]
- Brooks, D. R.; Wiley, E. O. Evolution as Entropy; Univ. of Chicago Press, 1986, 1988. [Google Scholar]Corning, P. A.; Kline, S. J. Thermodynamics, information and life revisited, Part I: To be or entropy. Systems Research and Behavioral Science
**1998**, 15 (4), 273–295. [Google Scholar]Part II: Thermoeconomics and Control Information. ibid**1998**, 15 (6), 453–482. - A recent summary of this is:
Machta, J. Entropy, Information, and Computation. Am. J. Phys
**1999**, 67 (12), 1074–1077. [Google Scholar] - De Solla Price, D. J. DLittle Science, Big Science; Columbia U. Press, 1963. [Google Scholar]
- Sepkoski, J. J., Jr. A Kinematic Model of Phanerozoic Taxonomic Diversity. Paleobiology
**1978**, 4 (3), 223–251. [Google Scholar]Pool, R. Is It Chaos, Or Is It Just Noise? Science**1978**, 243, 223–251. [Google Scholar] - MacArthur, R. H; Wilson, E. O. The Theory of Island Biogeography; Princeton U. Press, 1967. [Google Scholar]
- Coren, R. L. The Evolutionary Trajectory: Information in The History and Future of The Earth; Gordon & Breach Publishers, 1998. [Google Scholar]
- Vermeij, G. J. Evolution and Escalation; Princeton U. Press, 1987. [Google Scholar]
- Layzer, D. Growth and Order in the Universe. In Entropy, Information, and Evolution; Weber, B. H., Depew, D. H., Eds.; MIT Press, 1990; pp. 23–29. [Google Scholar]
- Runnegar, B. Evolution in The Earliest Animals. In Major Events in The History of Life; Schopf, J. W., Ed.; Jones and Bartlet Publ, 1992; pp. 65–93. [Google Scholar]
- Dobzhansky, T. On The Evolutionary Uniqueness of Man. In Evolutionary Biology;
**6**; Dobzhansky, T., Hecht, M. K., Sterre, W. C., Eds.; Appleton-Century Crofts, 1972; pp. 415–430. [Google Scholar] - This list consists of the acknowledged major changes. Because there are also minor subdivisions within this phylogeny it might be regarded as a culling of all the available data for a conformable set
- Lieberman, P. J. Primate Vocalization And Human Linguistic Ability. J. Acoustical Soc. Amer.
**1968**, 44 (16), 1157–1164. [Google Scholar]On Human Speech, Syntax, and Language. Human Evolution**1988**, 3, 3–18. - M. Maxwell, M. Human Evolution; Columbia U. Press, 1984. [Google Scholar]
- Gelb, I. J. Study of Writing; U. Chicago Press, 1952. [Google Scholar]
- McMurtrie, D. C. The Book; Oxford U. Press, 1943. [Google Scholar]
- Himmelfarb, G. Revolution in the Library. American Scholar
**1997**, 1–3. [Google Scholar] - Branson, H. R. A Definition of Information From The Thermodynamics of Irreversible Processes. In Information Theory and Biology; Quastler, H., Ed.; U. of Illinois Press, 1953; pp. 25–41. [Google Scholar]
- Kerr, B. A. Periodic Impacts and Extinctions Reported. Science
**1992**, 257, 1277–1279. [Google Scholar] - Woese, C. R. The Emergence of Genetic Organization. In Exobiology; Ponnamerpuma, C., Ed.; (N. Holland Publ., 1972; pp. 301–341. [Google Scholar]
- Jdanko, A. V. A Cybernetic Systems Approach to Universal Evolution. In Proc. 12th International Congress on Cybernetics-, Namur, Belgium; 1994. [Google Scholar]Haefner, K. Evolution and Information Processing Systems; Springer-Verlag, 1992. [Google Scholar]Schopf, J. W. Major Events in The History of Life; Jones & Bartlett Publishers, 1992; pp. 1286–1297. [Google Scholar]Smith, J.M.; Szathmary, E. The Major Transitions in Evolution; Freeman & Co., 1995. [Google Scholar]
- http://pespmc1.vub.ac.be/SELVAR.html.

^{*}These dates are taken with respect to the calendar year 2000.

© 2001 by MDPI. All rights reserved.

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**MDPI and ACS Style**

Coren, R.L.
Empirical Evidence for a Law of Information Growth. *Entropy* **2001**, *3*, 259-272.
https://doi.org/10.3390/e3040259

**AMA Style**

Coren RL.
Empirical Evidence for a Law of Information Growth. *Entropy*. 2001; 3(4):259-272.
https://doi.org/10.3390/e3040259

**Chicago/Turabian Style**

Coren, Richard L.
2001. "Empirical Evidence for a Law of Information Growth" *Entropy* 3, no. 4: 259-272.
https://doi.org/10.3390/e3040259