vol III Development:
Chapter 3: Cybernetics
page 2: Entropy
Thermodynamics
After confused beginnings, physicists realised that there are two important quantitites related to heat and motion, energy and entropy. Energy is conserved in closed systems. Entropy is conserved in reversible processes. These two quantities serve as fundamental constraints on the behaviour of our world. Conservation of energy - Wikipedia, Reversible process (thermodynamics) - Wikipedia
Thermodynamic entropy
The industrial revolution really took off when people learned to use heat engines to turn the energy released by fire into mechanical motion. Engineers who design heat engines are interested in the efficiency of their machines, that is proportion of the energy in the fuel that is converted into mechanical energy. Industrial Revolution - Wikipedia
In his book Reflections on the Motive Power of Fire Sadi Carnot devised an idealised cycle (the Carnot Cycle) which showed that the theoretical maximum efficiency of a heat engine depended only on temperatures. If T1 is the temperature of the heat entering the heat engine and T2 the temperature of the heat leaving the engine, the theoretical maximum efficiency of the 'Carnot cycle' is (T1 - T2)/T1. Such a machine can only work at 100% efficiency if its heat output temperature T2 is absolute zero! This fact has been the bane of designers of heat engines ever since. Carnot cycle - Wikipedia, Carnot
The Carnot cycle is bound by the two conservation laws. First, the conservation of energy requires that the sum of the heat energy and the mechanical energy leaving the engine a must be equal to the total heat energy entering the engine. Second, because the Carnot cycle is a reversible process, it conserves entropy. Since the entropy of mechanical energy is zero all the entropy that enters the engine from a hot input must leave with the cold output. Since the Carnot cycle is reversible, it may also act as a refrigerator. An input of mechanical energy and low temperature heat is output as high temperature heat..
Global heat engines
Long before mechanical heat engines were invented, the temperature difference between the heat of the sun and the coldness of space drove the winds and the ocean currents that control the climate on Earth.
Statistical mechanics
Thermodynamics deals with macroscopic phenomena that we can see and feel. We now know that these phenomena derive from the motions of invisible particles. Although the ancient Greek Democritus believed that matter was made of atoms, evidence for atoms only began to emerge at the beginning of the nineteenth century. Toward the end of that century Ludwig Boltzmann laid the foundations statistical mechanics to explain the macroscopic properties of matter in terms of atomic particles colliding with one another according to the laws of Newtonian dynamics. Sylvia Berryman: Democrtus, Atomic theory - Wikipedia, Ludwig Boltzmann - Wikipedia, Newtons Laws of Motion - Wikipedia, Cercignani: Ludwig Boltzmann: The Man Who Trusted Atoms
Boltzmann discovered that the thermodynamic entropy of a gas was a function of the number of states of motion available to the gas. He expressed this in the equation:
S = k log W.
where S represents entropy, k is a constant of proportionality called Boltzmann's constant and W is a count of the number of states of motion available to the gas, which is in effect the number of different ways the energy in the gas can be distributed among its particles. Boltzmann constant - Wikipedia
Statistical mechanics, like thermodynamics, connects entropy to energy. In thermodynamics the connection is through temperature, so the formula for entropy reads:
S = Q/T
where T is temperature and Q energy. What this is telling us is that the entropy flow associated with a flow of energy Q varies inversely as the temperature. High temperature corresponds to low entropy.
On the other hand, statistical mechanics relates entropy to energy by counting microscopic states. It is, in effect, a numbers game. The transfinite numbers, ℵ0, ℵ1, ℵ2, .
. . also give us a scale of entropy, and because they are scales of ordering,
they show us how entropy is generated. Boltzmann's discovery was based around
finding a way to count states that matched the way states actually multiply.
His method of counting microstates in a gas of N identical particles is
represented by the equation Boltzmann's computation of the number of complexions W
W = N! / Πi Ni!
where Ni particles are in each microstate i. This approach is analogous to the use of permutation to generate the transfinite numbers. Boltzmann's entropy formula - Wikipedia
Entropy as Boltzmann understood it is one of the most fundamental quantities that we can measure. Anything countable, like the natural numbers, gives us a scale of entropy. It has no dimension, like mass, length or time. It is simply a number. The observation known as the second law of thermodynamics, that 'entropy never decreases' gives direction to the growth of entropy, which has been called 'the arrow of time'. In our macroscopic world time moves in the direction of increasing entropy, that is increasing number of states. If energy is conserved, the amount of energy per state must decrease, which means that the temperature of the universe must decrease. This is known as the heat death of the universe. Entropy
(arrow of time) - Wikipedia
Why are there so many particles in the Universe?
The current hot big bang hypothesis postulates an apparently impossible beginning for the Universe. The initial singularity, is a structureless state of zero size endowed with infinite energy and temperature. This singularity is predicted by the general theory of relativity. It is believed to have expanded very rapidly, like a time reversed back hole, differentiating into space-time and the vast number and variety of fundamental particles that we currently observe. Hawking & Ellis: The Large Scale Structure of Space-Time
These particles subsequently coalesced to into the various larger nuclei, atoms and molecules that we find in the current Universe. The primordial gases, hydrogen and helium, broke up into local clouds which coalesced through gravitational collapse into galaxies and smaller clouds of gas that continued to collapse and and became hot enough to initiate the thermonuclear reactions that fuelled the first generation of stars. These stars created some of the heavier elements, became supernovas and dispersed themselves through space to become second and third generation stars. Some of the matter in the initial clouds collapsed into the supermassive black holes to be found at the centre of most galaxies. Much of the material in the Universe exists now as dark energy and dark matter which cannot be directly detected except by their gravitational contribution to the peculiar dynamics of galaxies.
The principal evidence for this scenario is the cosmic microwave radiation released soon after the universe was formed and high energy accelerator experiments which reproduce conditions close to the beginning. These experiments have enabled us to catalogue the fundamental particles that constitute the universe. We will return to detailed consideration of the early universe in Physics, chapter 4. Weinberg: Cosmology, Peacock: Cosmological Physics
The principal cybernetic consideration here is that the entropy of the initial singularity was probably very close to zero. Since the information carried by a state is equal to the entropy of the space in which the state exists, it could not have had the variety necessary to define the structure of the Universe. This must therefore have been created by an evolutionary process, which we discuss on page 10 below.
One constraint on the nature of the Universe is the fact that we exist and are able to scientifically investigate the nature of our habitat. This is called the anthropic principle. Some maintain that this shows that the initial conditions of the Universe were precisely tuned to lead to the evolution of intelligent life. If, as suggested here, the only initial condition of the Universe was that it simply existed, we are led to conclude that existence and intelligence are equivalent, an ancient theological position. Barrow & Tipler: The Anthropic Cosmological Principle, Aquinas, Summa: I, 14, 1: Is there knowledge in God?
(revised 8 January 2019)
|
Copyright:
You may copy this material freely provided only that you quote fairly and provide a link (or reference) to your source.
Further readingBooks
Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)
Barrow, John D., and Frank J. Tipler, The Anthropic Cosmological Principle, Oxford University Press 1996 'This wide-ranging and detailed book explores the many ramifications of the Anthropic Cosmological Principle, covering the whole spectrum of human inquiry from Aristotle to Z bosons. Bringing a unique combination of skills and knowledge to the subject, John D. Barrow and Frank J. Tipler-two of the world's leading cosmologists-cover the definition and nature of life, the search for extraterrestrial intelligence, and the interpretation of the quantum theory in relation to the existence of observers.'
Amazon
back |
Carnot, Sadi, and Translated by R H Thurston; edited and with an introduction by E Mendoza, Reflections on the Motive Power of Fire: and other papers on the second law of thermodynamics by E Clapeyron and R Clausius., Peter Smith Publisher 1977 Reflections: Everyone knows that heat can produce motion. ... in these days when the steam-engine is everywhere so well known. ... To develop this power, to appropriate it to our uses, is the object of heat engines. ... Notwithstanding the work of all kinds done by steam-engines, notwithstanding the satisfactory condition to which they have been brought today, their theory is very little understood, and the attempts to improve them are still directed almost by chance. ... In order to consider in the most general way the principle of the production of motion by heat, it must be considered independently of any mechanism or any particular agent. It is necessary to establish principles applying not only to steam-engines but to all imaginable heat engines, whatever the working substance and whatever the method by which it is operated. ... [Here enters the seed of entropy] The production of motive power is then due in steam-engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body, that is, to its reestablishment of equilibrium - an equilibrium considered as destroyed by any cause whatever, by chemical action such as combustion, or by any other.' pages 3-7.
Amazon
back |
Cercignani, Carlo, Ludwig Boltzmann: The Man Who Trusted Atoms, Oxford University Press, USA 2006 'Cercignani provides a stimulating biography of a great scientist. Boltzmann's greatness is difficult to state, but the fact that the author is still actively engaged in research into some of the finer, as yet unresolved issues provoked by Boltzmann's work is a measure of just how far ahead of his time Boltzmann was. It is also tragic to read of Boltzmann's persecution by his contemporaries, the energeticists, who regarded atoms as a convenient hypothesis, but not as having a definite existence. Boltzmann felt that atoms were real and this motivated much of his research. How Boltzmann would have laughed if he could have seen present-day scanning tunnelling microscopy images, which resolve the atomic structure at surfaces! If only all scientists would learn from Boltzmann's life story that it is bad for science to persecute someone whose views you do not share but cannot disprove. One surprising fact I learned from this book was how research into thermodynamics and statistical mechanics led to the beginnings of quantum theory (such as Planck's distribution law, and Einstein's theory of specific heat). Lecture notes by Boltzmann also seem to have influenced Einstein's construction of special relativity. Cercignani's familiarity with Boltzmann's work at the research level will probably set this above other biographies of Boltzmann for a very long time to come.' Dr David J Bottomley
Amazon
back |
Hallett, Michael, Cantorian Set Theory and Limitation of Size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson.
Amazon
back |
Hawking, Steven W, and G F R Ellis, The Large Scale Structure of Space-Time, Cambridge UP 1975 Preface: Einstein's General Theory of Relativity . . . leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.'
Amazon
back |
Huang, Kerson, Statistical Mechanics, John Wiley 1987 'Preface: ... The purpose of this book is to teach statistical mechanics as an integral part of theoretical phyiscs, a discipline that aims to describe all natural phenomena on the basis of a single unifying theory. This theory, at present, is quantum mechanics. ... Before the subject of statistical mechanics proper is presented, a brief but self contained discussion of thermodynamics and the classical kinetic theory of gases is given. The order of this devlopment is imperative, from a pedagogical point of view, for two reasons. First, thermodynamics has successfully described a large part of macroscopic experience, which is the concern of statistical mechanics. It has done so not on the basis of molecular dynamics but on the basis of a few simple and intuitive postulates stated in everyday terms. If we first falimiarize ourselves with thermodynamics, the task of statistical mechanics reduces to the explanation of thermodynamics. Second, the classical kinetic theory of gases is the only known special case in which thermodynics can be derived nearly from first principles, ie, molecular dynamics. A study of this special case will help us to understand why statstical mecahnics sorks.'
Amazon
back |
Khinchin, Aleksandr Yakovlevich, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.'
Amazon
back |
Peacock, John A, Cosmological Physics, Cambridge University Press 1999 Nature Book Review: ' The intermingling of observational detail and fundamental theory has made cosmology an exceptionally rich, exciting and controversial science. Students in the field — whether observers or particle theorists — are expected to be acquainted with matters ranging from the Supernova Ia distance scale, Big Bang nucleosynthesis theory, scale-free quantum fluctuations during inflation, the galaxy two-point correlation function, particle theory candidates for the dark matter, and the star formation history of the Universe. Several general science books, conference proceedings and specialized monographs have addressed these issues. Peacock's Cosmological Physics ambitiously fills the void for introducing students with a strong undergraduate background in physics to the entire world of current physical cosmology. The majestic sweep of his discussion of this vast terrain is awesome, and is bound to capture the imagination of most students.' Ray Carlberg, Nature 399:322
Amazon
back |
Weinberg, Steven, Cosmology, Oxford University Press, USA 2008 Amazon book description: 'This book is unique in the detailed, self-contained, and comprehensive treatment that it gives to the ideas and formulas that are used and tested in modern cosmological research. It divides into two parts, each of which provides enough material for a one-semester graduate course. The first part deals chiefly with the isotropic and homogeneous average universe; the second part concentrates on the departures from the average universe. Throughout the book the author presents detailed analytic calculations of cosmological phenomena, rather than just report results obtained elsewhere by numerical computation. The book is up to date, and gives detailed accounts of topics such as recombination, microwave background polarization, leptogenesis, gravitational lensing, structure formation, and multifield inflation, that are usually treated superficially if at all in treatises on cosmology. Copious references to current research literature are supplied. Appendices include a brief introduction to general relativity, and a detailed derivation of the Boltzmann equation for photons and neutrinos used in calculations of cosmological evolution. Also provided is an assortment of problems.'
Amazon
back |
Links
Absolute infinite - Wikipedia, Absolute infinite - Wikipedia, the free encyclopedia, 'The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God. He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller objec' back |
Aquinas, Summa: I, 14, 1, Is there knowledge in God?, 'I answer that, In God there exists the most perfect knowledge. To prove this, we must note that intelligent beings are distinguished from non-intelligent beings in that the latter possess only their own form; whereas the intelligent being is naturally adapted to have also the form of some other thing; for the idea of the thing known is in the knower. Hence it is manifest that the nature of a non-intelligent being is more contracted and limited; whereas the nature of intelligent beings has a greater amplitude and extension; therefore the Philosopher says (De Anima iii) that "the soul is in a sense all things." Now the contraction of the form comes from the matter. Hence, as we have said above (Question 7, Article 1) forms according as they are the more immaterial, approach more nearly to a kind of infinity. Therefore it is clear that the immateriality of a thing is the reason why it is cognitive; and according to the mode of immateriality is the mode of knowledge. Hence it is said in De Anima ii that plants do not know, because they are wholly material. But sense is cognitive because it can receive images free from matter, and the intellect is still further cognitive, because it is more separated from matter and unmixed, as said in De Anima iii. Since therefore God is in the highest degree of immateriality as stated above (Question 7, Article 1), it follows that He occupies the highest place in knowledge. back |
Atomic theory - Wikipedia, Atomic theory - Wikipedia, the free encyclopedia, 'In chemistry and physics, atomic theory is a scientific theory of the nature of matter, which states that matter is composed of discrete units called atoms. It began as a philosophical concept in ancient Greece and entered the scientific mainstream in the early 19th century when discoveries in the field of chemistry showed that matter did indeed behave as if it were made up of atoms.' back |
Boltzmann constant - Wikipedia, Boltzmann constant - Wikipedia, the free encyclopedia, 'The Boltzmann constant (k or kB) is the physical constant relating energy at the particle level with temperature observed at the bulk level. Values of k:
1.380 6504(24) × 10−23 J K-1
8.617 343(15) × 10−5 eV K−1
1.380 6504(24) × 10−16 erg K−1.' back |
Boltzmann's entropy formula - Wikipedia, Boltzmann's entropy formula - Wikipedia, the free encyclopedia, 'In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of an ideal gas to the quantity W, which is the number of microstates corresponding to a given macrostate:
S = k ln W
where k is the Boltzmann constant, . . . which is equal to 1.38062 x 10−23 J/K. back |
Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then
the measured result will be one of the eigenvalues λ of A, and
the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'. back |
Carl Hoefer, Causal Determinism (Standord Encyclopaedia of Philosophy), 'We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such an intelligence. (Laplace 1820)' back |
Carnot cycle - Wikipedia, Carnot cycle - Wikipedia, the free encyclopedia, 'The Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded by Benoit Paul Émile Clapeyron in the 1830s and 40s. It can be shown that it is the most efficient cycle for converting a given amount of thermal energy into work, or conversely, creating a temperature difference (e.g. refrigeration) by doing a given amount of work.' back |
Conservation of energy - Wikipedia, Conservation of energy - Wikipedia, the free encyclopedia, 'In physics, the law of conservation of energy states that the total energy of an isolated system cannot change—it is said to be conserved over time. Energy can be neither created nor destroyed, but can change form, for instance chemical energy can be converted to kinetic energy in the explosion of a stick of dynamite. back |
Counting - Wikipedia, Counting - Wikipedia, the free encyclopedia, 'Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.' back |
Determinism - Wikipedia, Determinism - Wikipedia, the free encyclopedia, 'Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen.' back |
Entropy - Wikipedia, Entropy - Wikipedia, the free encyclopedia, 'Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when converting energy to work. During this work, entropy accumulates in the system, but has to be removed by dissipation in the form of waste heat.' back |
Entropy (arrow of time) - Wikipedia, Entropy (arrow of time) - Wikipedia, the fre encyclopedia, 'Entropy is the only quantity in the physical sciences that "picks" a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system will increase when no extra energy is consumed.' back |
Heat death of the universe - Wikipedia, Heat death of the universe - Wikipedia, the free encyclopedia, 'The heat death is a possible final state of the universe, in which it has "run down" to a state of no thermodynamic free energy to sustain motion or life. In physical terms, it has reached maximum entropy. The hypothesis of a universal heat death stems from the 1850s ideas of William Thomson (Lord Kelvin) who extrapolated the theory of heat views of mechanical energy loss in nature, as embodied in the first two laws of thermodynamics, to universal operation' back |
Industrial Revolution - Wikipedia, Industrial Revolution - Wikipedia, the free encyclopedia, 'The Industrial Revolution was the transition to new manufacturing processes in the period from about 1760 to sometime between 1820 and 1840. This transition included going from hand production methods to machines, new chemical manufacturing and iron production processes, improved efficiency of water power, the increasing use of steam power, and the development of machine tools.' back |
Information theory - Wikipedia, Information theory - Wikipedia, the free encyclopedia, 'Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E. Shannon to find fundamental limits on compressing and reliably storing and communicating data. Since its inception it has broadened to find applications in many other areas, including statistical inference, natural language processing, cryptography generally, networks other than communication networks — as in neurobiology, the evolution and function of molecular codes, model selection in ecology, thermal physics, quantum computing, plagiarism detection and other forms of data analysis.' back |
Ludwig Boltzmann - Wikipedia, Ludwig Boltzmann - Wikipedia, the free encyclopedia, Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist and philosopher whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms (such as mass, charge, and structure) determine the physical properties of matter (such as viscosity, thermal conductivity, and diffusion). back |
Metre - Wikipedia, Metre - Wikipedia, the free encyclopedia, 'The metre (British spelling), or meter (American spelling), (SI unit symbol: m), is the fundamental unit of length (SI dimension symbol: L) in the International System of Units (SI), which is maintained by the BIPM. [The International Burea of Weights and Measures] Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole (at sea level), its definition has been periodically refined to reflect growing knowledge of metrology. Since 1983, it has been defined as "the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second." ' back |
Newtons Laws of Motion - Wikipedia, Newton's Laws of Motion - Wikipedia, the free encyclopedia, 'Newton's laws of motion are three physical laws that together laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to said forces. . . . The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687' back |
Reversible process (thermodynamics) - Wikipedia, Reversible process (thermodynamics) - Wikipedia, the free encyclopedia, 'In thermodynamics, a reversible process -- or reversible cycle if the process is cyclic -- is a process that can be "reversed" by means of infinitesimal changes in some property of the system without entropy production (i.e. dissipation of energy).' back |
Sylvia Berryman, Democritus (Stanford Encyclopedia of Phlosophy), 'Democritus, known in antiquity as the ‘laughing philosopher’ because of his emphasis on the value of ‘cheerfulness,’ was one of the two founders of ancient atomist theory. He elaborated a system originated by his teacher Leucippus into a materialist account of the natural world. The atomists held that there are smallest indivisible bodies from which everything else is composed, and that these move about in an infinite void space.' back |
|
