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Philosophers

Mortimer Adler
Rogers Albritton
Alexander of Aphrodisias
Samuel Alexander
William Alston
Anaximander
G.E.M.Anscombe
Anselm
Louise Antony
Thomas Aquinas
Aristotle
David Armstrong
Harald Atmanspacher
Robert Audi
Augustine
J.L.Austin
A.J.Ayer
Alexander Bain
Mark Balaguer
Jeffrey Barrett
William Barrett
William Belsham
Henri Bergson
George Berkeley
Isaiah Berlin
Richard J. Bernstein
Bernard Berofsky
Robert Bishop
Max Black
Susanne Bobzien
Emil du Bois-Reymond
Hilary Bok
Laurence BonJour
George Boole
Émile Boutroux
F.H.Bradley
C.D.Broad
Michael Burke
Lawrence Cahoone
C.A.Campbell
Joseph Keim Campbell
Rudolf Carnap
Carneades
Nancy Cartwright
Gregg Caruso
Ernst Cassirer
David Chalmers
Roderick Chisholm
Chrysippus
Cicero
Randolph Clarke
Samuel Clarke
Anthony Collins
Antonella Corradini
Diodorus Cronus
Jonathan Dancy
Donald Davidson
Mario De Caro
Democritus
Daniel Dennett
Jacques Derrida
René Descartes
Richard Double
Fred Dretske
John Dupré
John Earman
Laura Waddell Ekstrom
Epictetus
Epicurus
Herbert Feigl
Arthur Fine
John Martin Fischer
Frederic Fitch
Owen Flanagan
Luciano Floridi
Philippa Foot
Alfred Fouilleé
Harry Frankfurt
Richard L. Franklin
Bas van Fraassen
Michael Frede
Gottlob Frege
Peter Geach
Edmund Gettier
Carl Ginet
Alvin Goldman
Gorgias
Nicholas St. John Green
H.Paul Grice
Ian Hacking
Ishtiyaque Haji
Stuart Hampshire
W.F.R.Hardie
Sam Harris
William Hasker
R.M.Hare
Georg W.F. Hegel
Martin Heidegger
Heraclitus
R.E.Hobart
Thomas Hobbes
David Hodgson
Shadsworth Hodgson
Baron d'Holbach
Ted Honderich
Pamela Huby
David Hume
Ferenc Huoranszki
Frank Jackson
William James
Lord Kames
Robert Kane
Immanuel Kant
Tomis Kapitan
Walter Kaufmann
Jaegwon Kim
William King
Hilary Kornblith
Christine Korsgaard
Saul Kripke
Thomas Kuhn
Andrea Lavazza
Christoph Lehner
Keith Lehrer
Gottfried Leibniz
Jules Lequyer
Leucippus
Michael Levin
Joseph Levine
George Henry Lewes
C.I.Lewis
David Lewis
Peter Lipton
C. Lloyd Morgan
John Locke
Michael Lockwood
E. Jonathan Lowe
John R. Lucas
Lucretius
Alasdair MacIntyre
Ruth Barcan Marcus
James Martineau
Storrs McCall
Hugh McCann
Colin McGinn
Michael McKenna
Brian McLaughlin
John McTaggart
Paul E. Meehl
Uwe Meixner
Alfred Mele
Trenton Merricks
John Stuart Mill
Dickinson Miller
G.E.Moore
Thomas Nagel
Otto Neurath
Friedrich Nietzsche
John Norton
P.H.Nowell-Smith
Robert Nozick
William of Ockham
Timothy O'Connor
Parmenides
David F. Pears
Charles Sanders Peirce
Derk Pereboom
Steven Pinker
Plato
Karl Popper
Porphyry
Huw Price
H.A.Prichard
Protagoras
Hilary Putnam
Willard van Orman Quine
Frank Ramsey
Ayn Rand
Michael Rea
Thomas Reid
Charles Renouvier
Nicholas Rescher
C.W.Rietdijk
Richard Rorty
Josiah Royce
Bertrand Russell
Paul Russell
Gilbert Ryle
Jean-Paul Sartre
Kenneth Sayre
T.M.Scanlon
Moritz Schlick
Arthur Schopenhauer
John Searle
Wilfrid Sellars
Alan Sidelle
Ted Sider
Henry Sidgwick
Walter Sinnott-Armstrong
J.J.C.Smart
Saul Smilansky
Michael Smith
Baruch Spinoza
L. Susan Stebbing
Isabelle Stengers
George F. Stout
Galen Strawson
Peter Strawson
Eleonore Stump
Francisco Suárez
Richard Taylor
Kevin Timpe
Mark Twain
Peter Unger
Peter van Inwagen
Manuel Vargas
John Venn
Kadri Vihvelin
Voltaire
G.H. von Wright
David Foster Wallace
R. Jay Wallace
W.G.Ward
Ted Warfield
Roy Weatherford
C.F. von Weizsäcker
William Whewell
Alfred North Whitehead
David Widerker
David Wiggins
Bernard Williams
Timothy Williamson
Ludwig Wittgenstein
Susan Wolf

Scientists

David Albert
Michael Arbib
Walter Baade
Bernard Baars
Jeffrey Bada
Leslie Ballentine
Gregory Bateson
John S. Bell
Mara Beller
Charles Bennett
Ludwig von Bertalanffy
Susan Blackmore
Margaret Boden
David Bohm
Niels Bohr
Ludwig Boltzmann
Emile Borel
Max Born
Satyendra Nath Bose
Walther Bothe
Jean Bricmont
Hans Briegel
Leon Brillouin
Stephen Brush
Henry Thomas Buckle
S. H. Burbury
Melvin Calvin
Donald Campbell
Sadi Carnot
Anthony Cashmore
Eric Chaisson
Gregory Chaitin
Jean-Pierre Changeux
Rudolf Clausius
Arthur Holly Compton
John Conway
Jerry Coyne
John Cramer
Francis Crick
E. P. Culverwell
Antonio Damasio
Olivier Darrigol
Charles Darwin
Richard Dawkins
Terrence Deacon
Lüder Deecke
Richard Dedekind
Louis de Broglie
Stanislas Dehaene
Max Delbrück
Abraham de Moivre
Paul Dirac
Hans Driesch
John Eccles
Arthur Stanley Eddington
Gerald Edelman
Paul Ehrenfest
Manfred Eigen
Albert Einstein
George F. R. Ellis
Hugh Everett, III
Franz Exner
Richard Feynman
R. A. Fisher
David Foster
Joseph Fourier
Philipp Frank
Steven Frautschi
Edward Fredkin
Lila Gatlin
Michael Gazzaniga
Nicholas Georgescu-Roegen
GianCarlo Ghirardi
J. Willard Gibbs
Nicolas Gisin
Paul Glimcher
Thomas Gold
A. O. Gomes
Brian Goodwin
Joshua Greene
Dirk ter Haar
Jacques Hadamard
Mark Hadley
Patrick Haggard
J. B. S. Haldane
Stuart Hameroff
Augustin Hamon
Sam Harris
Ralph Hartley
Hyman Hartman
John-Dylan Haynes
Donald Hebb
Martin Heisenberg
Werner Heisenberg
John Herschel
Basil Hiley
Art Hobson
Jesper Hoffmeyer
Don Howard
William Stanley Jevons
Roman Jakobson
E. T. Jaynes
Pascual Jordan
Ruth E. Kastner
Stuart Kauffman
Martin J. Klein
William R. Klemm
Christof Koch
Simon Kochen
Hans Kornhuber
Stephen Kosslyn
Daniel Koshland
Ladislav Kovàč
Leopold Kronecker
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
David Layzer
Joseph LeDoux
Gilbert Lewis
Benjamin Libet
David Lindley
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
Owen Maroney
Humberto Maturana
James Clerk Maxwell
Ernst Mayr
John McCarthy
Warren McCulloch
N. David Mermin
George Miller
Stanley Miller
Ulrich Mohrhoff
Jacques Monod
Emmy Noether
Alexander Oparin
Abraham Pais
Howard Pattee
Wolfgang Pauli
Massimo Pauri
Roger Penrose
Steven Pinker
Colin Pittendrigh
Max Planck
Susan Pockett
Henri Poincaré
Daniel Pollen
Ilya Prigogine
Hans Primas
Henry Quastler
Adolphe Quételet
Lord Rayleigh
Jürgen Renn
Juan Roederer
Jerome Rothstein
David Ruelle
Tilman Sauer
Jürgen Schmidhuber
Erwin Schrödinger
Aaron Schurger
Sebastian Seung
Thomas Sebeok
Claude Shannon
David Shiang
Abner Shimony
Herbert Simon
Dean Keith Simonton
B. F. Skinner
Lee Smolin
Ray Solomonoff
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
Max Tegmark
Libb Thims
William Thomson (Kelvin)
Giulio Tononi
Peter Tse
Francisco Varela
Vlatko Vedral
Mikhail Volkenstein
Heinz von Foerster
Richard von Mises
John von Neumann
Jakob von Uexküll
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
Herman Weyl
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
Stephen Wolfram
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek
Konrad Zuse
Fritz Zwicky

Presentations

Biosemiotics
Free Will
Mental Causation
James Symposium
 
Gilbert N. Lewis

The American chemist Gilbert Lewis discovered the covalent bond in 1916 and introduced a novel diagram to explain the bonding, with double dots for the electrons arranged in pairs. He was the first to purify heavy water (deuterium dioxide) and should have shared the Nobel Prize awarded to his student, Harold Urey. Another student, Glenn Seaborg, also won a Nobel Prize and Novel Prize winner Linus Pauling became famous developing Lewis' theory of the covalent bond.

In late 1926, Lewis wrote an article on Albert Einstein's light quanta, at a time when the "founders" of quantum mechanics, Max Born, Werner Heisenberg, and Pascual Jordan, were not yet convinced that light quanta were real and involved in "quantum jumps.". Lewis renamed light quanta "photons" by analogy with electrons.

Lewis published a letter in Nature called "The Conservation of Photons" (which, unfortunately are not, like electrons, conserved).

WHATEVER view is held regarding the nature of light, it must now be admitted that the process whereby an atom loses radiant energy, and another near or distant atom receives the same energy, is characterised by a remarkable abruptness and singleness. We are reminded of the process in which a molecule loses or gains a whole atom or a whole electron but never a fraction of one or the other. When the genius of Planck brought him to the first formulation of the quantum theory, a new kind of atomicity was suggested, and thus Einstein was led to the idea of light quanta which has proved so fertile.
Lewis is, of course, wrong about the "definite path."
Indeed, we now have ample evidence that radiant energy (at least in the case of high frequencies) may be regarded as travelling in discrete units, each of which passes over a definite path in accordance with mechanical laws.

Had there not seemed to be insuperable objections, one might have been tempted to adopt the hypothesis that we are dealing here with a new type of atom, an identifiable entity, uncreatable and indestructible, which acts as the carrier of radiant energy and, after absorption, persists as an essential constituent of the absorbing atom until it is later sent out again bearing a new amount of energy. If I now advance this hypothesis of a new kind of atom, I do not claim that it can yet be proved, but only that a consideration of the several objections that might be adduced shows that there is not one of them that can not be overcome.

It would seem inappropriate to speak of one of these hypothetical entities as a particle of light, a corpuscle of light, a light quantum, or a light quant, if we are to assume that it spends only a minute fraction of its existence as a carrier of radiant energy, while the rest of the time it remains as an important structural element within the atom. It would also cause confusion to call it merely a quantum, for later it will be necessary to distinguish between the number of these entities present in an atom and the so-called quantum number. I therefore take the liberty of proposing for this hypothetical new atom, which is not light but plays an essential part in every process of radiation, the name photon.

Let us postulate for the photon the following properties: (1) In any isolated system the total number of photons is constant. (2) All radiant energy is carried by photons, the only difference between the radiation from the wireless station and from an X-ray tube being that the former emits a vastly greater number of photons, each carrying a very much smaller amount of energy. (3) All photons are intrinsically identical. As the molecules of hydrogen differ from one another in direction and energy of translation, and in direction and amount of rotation, so two photons, as seen by a single observer, differ in direction of motion, in energy, and in polarisation. If we were moving with rapid acceleration toward a wireless station, its photons would appear to possess increasing amounts of energy, and would pass over the whole spectral scale through the visible and into the ultraviolet. At a certain instant, for example, they would be indistinguishable from the photons emitted by excited sodium atoms. (4) The energy of an isolated photon, divided by the Planck constant, gives the frequency of photons which is therefore by definition strictly monochromatic; although two photons coming even from similar atoms would never have precisely the same frequency. (5) All photons are alike in one property which has the dimensions of action or of angular momentum, and is invariant to a relativity transformation. (6) The condition that the frequency of a photon emitted by a certain system be equal to some physical frequency existing within that system, is not in general fulfilled, but comes nearer to fulfilment the lower the frequency is.

The serious objections to the idea of the conservation of photons are met in a consideration of the thermodynamics of radiation and of the laws of spectroscopy. According to the classical thermodynamics of radiation, the energy of hohlraum at a given temperature is determined solely by the volume. If we define the number of photons in a small spectral interval by the amount of energy in that interval divided by hv, then, by Wien's displacement law, the number of photons remains constant in any reversible adiabatic process.

Another error: the expanding walls increase the wavelengths and reduce the energy in each photon.
Also in the irreversible adiabatic process of free expansion from a given volume to a larger volume (both with perfectly reflecting walls) the number of photons remains constant, for neither the energies nor the frequencies are changed. If the original radiation, corresponding to a definite temperature, freely expands, let us say, to sixteen times the first volume, then, according to the thermodynamics of Wien and Planck, it may be brought to a new temperature equilibrium by introducing an infinitesimal black body. Calculating from their equations, we find that in this process the number of photons is doubled. If this is so, there obviously can be no conservation law for photons. However, if we analyse carefully the thermodynamics of radiation, we find that Wien and Planck have tacitly employed a postulate which is supported by no experimental facts; namely, if an infinitesimal black body is introduced into a hohlraum, the radiation will come to a certain temperature, and then no further change will ensue when a large black body of the same temperature is introduced.

Dispensing with this postulate, and adding a new variable, the number of photons, the variables which have previously been deemed sufficient to define that state of a system, we obtain a greatly enlarged science of thermodynamics. In this new thermodynamics, which included as true and stable equilibria such states of equilibrium as those to which Einstein has applied the terms "aussergewohnlich" and "improprement dit" (Ann. Phys.,38, 881, 1912; Jour. de Phys.,3, 277, 1913) the familiar laws of radiation and of physical and chemical equilibrium become special cases, true only for an unlimited supply of photons. Even so fundamental a process as the flow of heat must involve two factors, the amount of energy and the number of photons transferred. A fuller account of this new thermodynamics will shortly be published.

Turning to spectroscopy, we find that the principle of the conservation of photons is in obvious conflict with existing notions of the radiation process. We must assume that in an elementary process of radiation one, and only one, photon is lost by the emitting atom. Suppose that an atom which is in the 4-2 state drops to the 3-3, then to the 2-2, then to the 1-1. It thus loses three photons, but the same atom dropping directly from the 4-2 state to the 1-1 loses only one photon. If, therefore, we are to admit the conservation of photons, we must say that the atom does not pass from precisely the same initial to the same final state by the two paths, but rather that either the 4-2 or the 1-1 states must be multiple. Even if the inner quantum number is given, as well as the total and the azimuthal quantum numbers, the atomic states must still be regarded as not completely specified. Indeed, numerous examples have been found (see the review by Ruark and Chenault, Phil. Mag.,50, 937, 1925) of superfine structure which is not yet accounted for.

I had hoped to be able to derive certain familiar selection principles from the conservation of photons. Here I have not as yet succeeded, and can only state that if we assume the existence of a number of atomic states with nearly the same energy but with different numbers of photons, the new theory is not in conflict with the results of spectroscopy.

The rule that one, and only one, photon is lost in each elementary radiation process, is far more rigorous than any existing selection principle, and forbids the majority of processes which are not supposed to occur. To account for the apparent existence of these processes, it is necessary to assume that atoms are frequently changing their photon number by the exchange of photons of very small energy, corresponding to thermal radiation in the extreme infra-red. The new theory therefore predicts that many atomic processes will be inhibited at very low temperatures, and for this there seems to be some experimental evidence. But the existence of numerous extraneous factors obscures the issue. In order to simplify matters, a molecular stream might be passed through the centre of a tube cooled to a very low temperature, so as to reduce to a minimum the amount of thermal radiation. The theory would predict that in such circumstances certain processes within the stream, such as fluorescence or the emission of light from activated atoms, would be profoundly changed. Experiments in this direction are now in progress.

Symmetry of Time
A few years later, Lewis was awarded the Gold Medal of the Society of Arts and Sciences. He gave an address in 1930 on "The Symmetry of Time in Physics." He distinguished our common idea of unidirectional time (psychological and the result of consciousness and memory) from the symmetrical time of Newtonian mechanics.

He said that throughout the sciences of physics and chemistry, symmetrical time everywhere suffices. This is of course not correct, because unidirectional time emerges in thermodynamics and statistical mechanics. It is even more important i kinetic theory, if we examine the interactions of particles.

Lewis hoped that the four-dimensional theories of spacetime of Hermann Minkowski and Albert Einstein could restore symmetric time, so we could not distinguish cause from effect.

This of course is the idea of a deterministic universe, in which information is conserved, that all times are visible in the eyes of a Laplacian super-intelligence. He found the second law of thermodynamics to be "in direct defiance to the law of symmetry of time."

Lewis hypothesized that to a believer in symmetric time he called "Dr.X"it would be...

a great satisfaction to read in a paper of Willard Gibbs that “the impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability”; and later to follow the development of this thesis by Boltzmann until near the end of the famous lectures on “Gastheorie” he found Boltzmann saying, “Hence, for the universe, both directions of time are indistinguishable, as in space there is no up or down.”

Boltzmann’s qualifications of this statement seemed unnecessary to Dr. X, who now definitely included thermodynamics among those branches of physics which require symmetrical time only. In his note book we read, “The statistical interpretation of thermodynamics offered by Gibbs and Boltzmann affords for the first time an understanding of entropy. The process irreversible in time does not exist. This corollary of the law of symmetry in time itself leads to further important consequences. Thence we may prove to those who are still skeptical the atomic structure of matter, as follows: if we imagine two continuous media to diffuse into one another, such a diffusion would in principle be a phenomenon which by no physical means could be reversed, but if two streams composed of discrete particles should diffuse, then, although it might be a matter of great difficulty to recapture the particles and restore each to its own kind, yet in principle the process is reversible and indeed, according to Boltzmann, the separation will occur spontaneously if the system be left to itself for a sufficiently long period.”

Dr. X adds a remark of much subtlety. “While we recognize the particulate nature of matter, we allow each particle to have a position and a velocity chosen from a whole continuum of possible values. Thus while we claim that an isolated system repeatedly returns nearly to its initial condition, we can not say that it returns exactly to that condition. If we start with a number of molecules all moving in precisely the same direction, we can not claim that after some disturbance they ever again move quite parallel to one another. This implies a sort of irreversibility, and while I am not sure that it is a contradiction to symmetrical time, I confess that I should be better satisfied if we could claim the exact recurrence of an initial state.” It is a pity that Dr. X did not live to see the universal acceptance of quantum theory, which assigns to an isolated system not an infinite continuum of states, but a finite number of discrete states. Thus every particular state exactly recurs within finite time. This modern picture is far simpler than that of Boltzmann, especially as we are going to see that each particular state occurs as often as every other. Hence molecular statistics furnishes quite elementary problems in the theory of probability, like the tossing of coins or the shuffling of cards.

In the main, however, the problems of thermodynamics to-day are not far different from those discussed by Boltzmann and Dr. X. In the note book of the latter we read, “The earth is constantly receiving energy from the sun, and in consequence water is continuously flowing over Niagara Falls, but these descriptive statements can not be called laws of physics. When we turn to the processes studied in the laboratory we find that when a hot and cold body are brought together, it is almost certain that the twro temperatures will become equalized until no discernible difference remains. If we mix two mutually soluble liquids, we may expect the concentration to become nearly uniform. I have learned that it is possible to perform an operation upon the brains of mice so that they respond to no external stimuli, but can still run aimlessly about. If a large number of these mice are placed in one end of a box, that end is now heavier than the other; but this distinction rapidly disappears as the mice, in their random movements, cover with greater uniformity the bottom of the box, so that we may no longer discern any tendency of the box in one direction or the other. I claim that in all these cases there is no phenomenon irreversible in time, and indeed nothing more formidable occurs than in the proverbial case of a needle dropped into a haystack.”

Before analyzing further these problems, we may consider a very interesting discussion of one-way time by Professor Eddington, in “The Nature of the Physical World.” He arrives at a compromise, first by stating that one-way time does not occur in physics outside of thermodynamics, and then by reducing the principle of the increase of entropy from a “primary” to a “secondary” law, which does not prevent him, however, from deducing therefrom a “running down of the universe.” To this compromise I can not agree.* The first statement will be answered by the cases which will be discussed in the following sections, and the second can not be regarded as consistent with the new conception of thermodynamics.

We must be cautious about extending to the whole cosmos the rules which we have obtained from limited experiments in our small laboratories. The chance of obtaining valid results from such an extrapolation is very small, but it can be made in a purely formal way. If the universe is finite, as is now frequently supposed, then the formal application of our existing ideas of thermodynamics and statistics leads directly to the following statement: The precise present state of the universe has occurred in the past and will recur in the future, and in. each case within finite time. Whether the universe actually is running down is, of course, another matter. All we can say is that such an assumption obtains no support *from thermodynamics. Let us, however, turn from the behavior of the universe, about which we know almost nothing, to the phenomena of the laboratory, about which we know a little more. Even in this limited domain it is going to be difficult enough to persuade ourselves that such a phenomenon as an explosion is wholly compatible with the thesis of symmetrical time. If a statement runs counter to a fixed habit of thought which has become nearly instinctive, it may be accepted by many, but believed by few. The use of one-way time

Turning now to the irreversible thermodynamic process, we shall choose an illustration which is not quite so complicated as an explosion, but involves all essentials. A chemist has spent days in preparing a flask of nearly pure alcohol. This he places in a water bath, and then by accident the flask overturns and the alcohol diffuses through the water. His disappointment is in no way allayed by the fact that none of his material is really lost, nor by the belief that ultimately the molecules of alcohol will accidentally come together to give once more a nearly pure sample. That the chemist would be obliged to wait an unconscionable time for this chance restoration must be given no weight. If it occurred every ten minutes, the principle would be the same. It would still be necessary for him to devise rapid analytical methods to ascertain just when the fortunate event occurred. There is no question but that the accident has involved an element of loss which typifies the irreversible process (which is also spoken of as a process of dissipation, or degradation), but we shall see that this loss in no way implies a dissymmetry of time, nor indeed that it has any temporal implications whatever. Without losing any of the characteristics of the reversible process, we may next examine one of the simplest of systems. Suppose that we have a cylinder closed at each end, and with a middle wall provided with a shutter. In this cylinder are one molecule each of three different gases, A, B and C, and the cylinder is in a thermostat at temperature T. In dealing with the individual molecules we are perhaps arrogating to ourselves the privileges of Maxwell's demon; but in recent years, if I may say so without offense, physicists have become demons.

Begarding each molecule, we shall ask only whether it is in the right or the left half of the cylinder. Obviously eight distributions are possible, such as A and B on the left and C on the right; or B on the left and A and C on the right. According to our ordinary assumptions, each of these distributions is equally probable, or, in other words, the system averages to be in each distribution one eighth of the time. Moreover, each of the eight distributions can be easily described and remembered, so that we are not troubled by a large number of nondescript states. Each distribution occurs over and over, but in no particular order, and in these occurrences there is no trace of dissymmetry with respect to time—there is no “running-down” process here. Yet we may have a typical irreversible process. Suppose that the shutter is closed so as to trap a particular distribution, say all three molecules on the left. We become familiar with this one distribution and wish to study it further, but accidentally the shutter is opened, and instead of the one distribution, we have all eight succeeding one another in a random way. This is a complete analogy to the overturn of the flask of alcohol. If we desire once more to obtain and keep the one distribution in which all the molecules are on the left-hand side of the cylinder, we may . exercise our prerogatives as Maxwell demons by closing the shutter from time to time and determining by spectroscopic means or otherwise which distribution is trapped. In about eight trials we shall obtain the desired result. Unless, however, there is in sentient beings the power to defy the second law of thermodynamics, we shall find that this method of obtaining the desired distribution requires at least as much work as the old-fashioned thermodynamical method of forcing the system into the particular distribution without the aid of demoniacal devices. This classical method consists in slowly pushing a piston from the extreme right of the cylinder as far as the middle wall. In this typical reversible process the work required to overcome the pressure of the three molecules is 3 k T In 2 = k T In 8. At the same time the entropy of the gas is diminished by 3 k In 2.

If we wish to obtain any other one of the particular distributions, from the general distribution, the same amount of work is required. Suppose the particular distribution desired is B on the left, A and C on the right. At the extreme left we have a piston permeable only to B, and at the extreme right a piston permeable only to A and C, and these pistons are moved slowly to the middle wall. We thus obtain the given distribution, and the sum of the work done upon the two pistons is 3 k T In 2. In every case, in passing from the general distribution to a particular known distribution, the gas loses entropy in the amount 3 k In 2. All these processes are completely reversible. If we start with any known distribution and let the proper pistons move outward from the center to the ends of the cylinder, we obtain the general distribution, the system does work in the amount 3 k T In 2, and the entropy of the gas increases by 3 k In 2.

The entropy of the general unknown distribution is greater than the entropy of any one known distribution by 3 k In 2. This, therefore, is the increase in entropy in the irreversible process which occurs when, after trapping any one known distribution, we open the shutter. It is evident, however, that the mere trapping of one distribution makes no change in the entropy, for the shutter may be made as frictionless as we please, and the mere act of opening or closing it will not change the entropy of the system. If we start with the shutter open, with all the eight distributions occurring one after another, and then close the shutter, the system is trapped in one distribution, but there is no change of entropy. Whence we have now reached our most important conclusion. The increase in entropy comes when a known distribution goes over into an unknown distribution. The loss, which is characteristic of an irreversible process, is loss of information. In the simplest case, if we have one molecule which must be in one of two flasks, the entropy becomes less by k In 2, if we know which is the flask in which the molecule is trapped.

Gain in entropy always means loss of information, and nothing more. It is a subjective concept, but we can express it in its least subjective form, as follows. If, on a page, we read the description of a physicochemical system, together with certain data which help to specify the system, the entropy of the system is determined by these specifications. If any of the essential data are erased, the entropy becomes greater; if any essential data are added, the entropy becomes less. Nothing further is needed to show that the irreversible process neither implies one-way time, nor has any other temporal implications. Time is not one of the variables of pure thermodynamics.

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