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 Belsham Henri Bergson George Berkeley Isaiah Berlin Richard J. Bernstein Bernard Berofsky Robert Bishop Max Black Susanne Bobzien Emil du BoisReymond Hilary Bok Laurence BonJour George Boole Émile Boutroux F.H.Bradley C.D.Broad Michael Burke C.A.Campbell Joseph Keim Campbell Rudolf Carnap Carneades 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 John Martin Fischer Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. Franklin 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 William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan Jaegwon Kim William King Hilary Kornblith Christine Korsgaard Saul Kripke Andrea Lavazza Keith Lehrer Gottfried Leibniz Jules Lequyer Leucippus Michael Levin 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.NowellSmith 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 JeanPaul Sartre Kenneth Sayre T.M.Scanlon Moritz Schlick Arthur Schopenhauer John Searle Wilfrid Sellars Alan Sidelle Ted Sider Henry Sidgwick Walter SinnottArmstrong 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 William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Scientists Michael Arbib Walter Baade Bernard Baars Gregory Bateson John S. Bell Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Donald Campbell Anthony Cashmore Eric Chaisson JeanPierre Changeux Arthur Holly Compton John Conway John Cramer E. P. Culverwell Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Max Delbrück Abraham de Moivre Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Albert Einstein Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher Joseph Fourier Philipp Frank Lila Gatlin Michael Gazzaniga GianCarlo Ghirardi J. Willard Gibbs Nicolas Gisin Paul Glimcher Thomas Gold A.O.Gomes Brian Goodwin Joshua Greene Jacques Hadamard Patrick Haggard Stuart Hameroff Augustin Hamon Sam Harris Hyman Hartman JohnDylan Haynes Donald Hebb Martin Heisenberg Werner Heisenberg John Herschel Art Hobson Jesper Hoffmeyer E. T. Jaynes William Stanley Jevons Roman Jakobson Pascual Jordan Ruth E. Kastner Stuart Kauffman Martin J. Klein Simon Kochen Hans Kornhuber Stephen Kosslyn Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé PierreSimon Laplace David Layzer Benjamin Libet Seth Lloyd Hendrik Lorentz Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau James Clerk Maxwell Ernst Mayr John McCarthy Ulrich Mohrhoff Jacques Monod Emmy Noether 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 Adolphe Quételet Juan Roederer Jerome Rothstein David Ruelle Erwin Schrödinger Aaron Schurger Claude Shannon David Shiang Herbert Simon Dean Keith Simonton B. F. Skinner Roger Sperry John Stachel Henry Stapp Tom Stonier Antoine Suarez Leo Szilard Max Tegmark William Thomson (Kelvin) Giulio Tononi Peter Tse Vlatko Vedral Heinz von Foerster John von Neumann John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss John Wheeler Wilhelm Wien Norbert Wiener Eugene Wigner E. O. Wilson H. Dieter Zeh Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Presentations Biosemiotics Free Will Mental Causation James Symposium 
The Information Interpretation of Quantum Mechanics
There are presently several "interpretations" of quantum mechanics. Many, perhaps most, are attempts to eliminate the element of chance or indeterminism that is involved in the socalled collapse of the wave function.
The Information Interpretation is simply "standard quantum physics" plus information being recorded irreversibly. Unlike the Copenhagen Interpretation, we offer several visualizations of what is going on in quantum reality,
The Information Interpretation is based on three simple premises:
When you hear or read that electrons are both waves and particles, think "eitheror" 
first a wave of possibilities, then an actual particle.
There is only one world.
Information physics claims there is only one world, the quantum world, and the "quantum to classical transition" occurs for any large macroscopic object that contains a large number of atoms. For large enough systems, independent quantum events are "averaged over." The uncertainty in position and momentum of the object becomes less than the observational uncertainty It is a quantum world. Ontologically it is indeterministic. Epistemically, common sense and experience incline us to see it as deterministic Δv Δx ≥ h / m goes to zero as h / m goes to zero. The classical laws of motion, with their implicit determinism and strict causality, emerge when objects are large enough so that microscopic events can be ignored, but this determinism is fundamentally statistical and causes are only probabilistic, however near to certainty. Information philosophy interprets the wave function ψ as a "possibilities" function. With this simple change in terminology, the mysterious process of a wave function "collapsing" becomes more understandable. The wave function ψ evolves to explore all the possibilities (with mathematically calculable probabilities). When a single actuality is realized, the probability for all the nonactualized possibilities goes to zero ("collapses") instantaneously. But they could never reconcile the macroscopic irreversibility needed for the second law Information physics is standard quantum physics. It accepts the Schrödinger equation of motion, the principle of superposition, the axiom of measurement (now including the actual information "bits" measured), and  most important  the projection postulate of standard quantum mechanics (the "collapse" that so many interpretations deny). The "conscious observer" of the Copenhagen Interpretation is not required for a projection, for the wavefunction to "collapse", for one of the possibilities to become an actuality. What the collapse does require is an interaction between systems that creates information that is irreversible and observable, though not necessarily observed. Among the founders of quantum mechanics, almost everyone agreed that irreversibility is a key requirement for a measurement. Irreversibility introduces thermodynamics into a proper formulation of quantum mechanics, and this is a key element of our information interpretation. But this requirement was never reconciled with classical statistical mechanics, which says that collisions between material particles are reversible. Even quantum statistical mechanics claims collisions are reversible because the Schrödinger equation is time reversible. Note that Maxwell's equations of electromagnetic radiation are also time reversible. We have shown that it is the interaction of light and matter, both on their own time reversible, that is the origin of irreversibility.
Information is not a conserved quantity like energy and mass, despite the view of many mathematical physicists, who generally accept determinism. The universe began in a state of equilibrium with minimal information, and information is being created every day, despite the second law of thermodynamics
Classical interactions between large macroscopic bodies do not generate new information. Newton's laws of motion imply that the information in any configuration of bodies, motions, and force is enough to know all past and future configurations. Classical mechanics conserves information.
In the absence of interactions, an isolated quantum system evolves according to the unitary Schrödinger equation of motion. Just like classical systems, the deterministic Schrödinger equation conserves information. And just like classical systems, Schrödinger's unitary evolution is time reversible. Unlike classical systems however, when there is an interaction between material quantum systems, the two systems become entangled and there may be a change of state in either or both systems. This change of state may create new information. Or if there is an interaction between light and matter the evolution is no longer unitary, there is an irreversible collapse of the wave function.
If that information is instantly destroyed, as in most interactions, it may never be observed macroscopically. If, on the other hand, the information is stabilized for some length of time, it may be seen by an observer and considered to be a "measurement." But it need not be seen by anyone to become new information in the universe. The universe is its own observer!
For the information (negative entropy) to be stabilized, the second law of thermodynamics requires that an amount of positive entropy greater than the negative entropy must be transferred away from the new information structure. Exactly how the universe allows pockets of negative entropy to form as "information structures" we describe as the "cosmic creation process." This core twostep process has been going on since the origin of the universe. It continues today as we add information to the sum of human knowledge.
Note that despite the Heisenberg principle, quantum mechanical measurements are not always uncertain. When a system is measured (prepared) in an eigenstate, a subsequent measurement (Pauli's measurement of the first kind) will find it in the same state with perfect certainty.
What then are the possibilities for new quantum states? The transformation theory of Dirac and Jordan lets us represent ψ in a set of basis functions for which the combination of quantum systems (one may be a measurement apparatus) has eigenvalues (the axiom of measurement). We represent ψ as in a linear combination (the principle of superposition) of those "possible" eigenfunctions. Quantum mechanics lets us calculate the probabilities of each of those "possibilities."
Interaction with the measurement apparatus (or indeed interaction with any other system) may select out (the projection postulate) one of those possibilities as an actuality. But for this event to be an "observable" (a John Bell "beable"), information must be created and positive entropy must be transferred away from the new information structure, in accordance with our twostage information creation process.
All interpretations of quantum mechanics predict the same experimental results. Where interpretations differ is in the picture (the visualization) they provide of what is "really" going on in the microscopic world  the socalled "quantum reality." The "orthodox" Copenhagen interpretation of Neils Bohr and Werner Heisenberg discourages such attempts to understand the nature of the "quantum world," because they say that all our experience is derived from the "classical world" and should be described in ordinary language. This is why Bohr and Heisenberg insisted on the path and the "cut" between the quantum event and the mind of an observer. The information interpretation encourages visualization. Schrödinger called it Anschaulichkeit. He and Einstein were right that we should be able to picture quantum reality. But that demands that we accept the reality of quantum possibilities and discontinuous random "quantum jumps," something many modern interpretations do not do. (See our visualization of the twoslit experiment, our EPR experiment visualizations, and Dirac's three polarizers to visualize the superposition of states and the projection or "collapse" of a wave function.) Bohr was of course right that classical physics plays an essential role. His Correspondence Principle allowed him to recover some important physical constants by assuming that the discontinuous quantum jumps for low quantum numbers (low "orbits" in his old quantum theory model) converged in the limit of large quantum numbers to the continuous radiation emission and absorption of classical electromagnetic theory. In addition, we know that in macroscopic bodies with enormous numbers of quantum particles, quantum effects are averaged over, so that the uncertainty in position and momentum of a large body still obeys Heisenberg's indeterminacy principle, but the uncertainty is for all practical purposes unmeasurable and the body can be treated classically. We can say that the quantum description of matter also converges to a classical description in the limit of large numbers of quantum particles. We call this "adequate" or statistical determinism. It is the apparent determinism we find behind Newton's laws of motion for macroscopic objects. The statistics of averaging over many independent quantum events then produces the "quantum to classical transition" for the same reason as the "law of large numbers" in probability theory. Both Bohr and Heisenberg suggested that just as relativistic effects can be ignored when the velocity is small compared to the velocity of light (v / c → 0), so quantum effects might be ignorable when Planck's quantum of action h → 0. But this is quite wrong, because h is a constant that never goes to zero. In the information interpretation, it is always a quantum world. The conditions needed for ignoring quantum indeterminacy are when the mass of the macroscopic "classical" object is large. Noting that the momentum p is the product of mass and velocity mv, Heisenberg's indeterminacy principle, Δp Δx > h, can be rewritten as Δv Δx > h / m. It is thus not when h is small, but when h / m is small enough, that errors in the position and momentum of macroscopic objects become smaller that can be measured.
Note that the macromolecules of biology are large enough to stabilize their information structures. DNA has been replicating its essential information for billions of years, resisting equilibrium despite the second law of thermodynamics
The creation of irreversible new information also marks the transition between the quantum world and the "adequately deterministic" classical world, because the information structure itself must be large enough (and stable enough) to be seen. The typical measurement apparatus is macroscopic, so the quantum of action h becomes small compared to the mass m and h / m approaches zero.
Decoherence theorists say that our failure to see quantum superpositions in the macroscopic world is the measurement problem
The information interpretation thus explains why quantum superpositions like Schrödinger's Cat are not seen in the macroscopic world. Stable new information structures in the dying cat reduce the quantum possibilities (and their potential interference effects) to a classical actuality. Upon opening the box and finding a dead cat, an autopsy will reveal that the time of death was observed/recorded. The cat is its own observer.
The "Possibilities Function"
The central element in quantum physics is the "wave function" ψ, with its mysterious waveparticle dual nature (sometimes a wave, sometimes a particle, etc.). We believe that teaching and understanding quantum mechanics would be much simpler if we called ψ the "possibilities function." It only looks like a wave in simple cases of lowdimensional coordinate space. But it always tells us the possibilities  the possible values of any observable, for example. Given the "possibilities function" ψ, quantum mechanics allows us to calculate the "probabilities" for each of the "possibilities." The calculation depends on the free choice of the experimenter as to which "observables" to look for. If the measurement apparatus can register n discrete values, ψ can be expanded in terms of a set of basis functions (eigenfunctions) appropriate for the chosen observable, say φ_{n}. The expansion is
ψ = ∑ c_{n} φ_{n}
When the absolute squares of the coefficients c_{n} are appropriately normalized to add up to 1, the probability P_{n} of observing an eigenvalue n is
P_{n} =  c_{n} ^{2} =  < ψ  φ_{n} >  ^{2}
These probabilities are confirmed statistically by repeated identical experiments that collect large numbers of results. Quantum mechanics is the most accurate physical theory in science, with measurements accurate to thirteen decimal places. In each individual experiment, generally just one of the possibilities becomes an actuality (some experiments leave the quantum system in a new superposition of multiple possibilities). In our information interpretation, a possibility is realized or actualized at the moment when information is created about the new state of the system. This new information requires that positive entropy be carried away from the local increase in negative entropy. Note that an "observer" would not be able to make a "measurement" unless there is new information to be "observed." Information must be (and is in all modern experimental systems) created and recorded before any observer looks at the results. An information approach can help philosophers to think more clearly about quantum physics. Instead of getting trapped in talk about mysterious "collapses of the wave function," "reductions of the wave packet," or the "projection postulate" (all important issues), the information interpretation proposes we simply say that one of the "possibilities" has become "actual." It is intuitively obvious that when one possibility becomes actual, all the others are annihilated, consigned to "nothingness," as JeanPaul Sartre put it. And because the other possibilities may have been extremely "distant" from the actuality, their instantanteous disappearances looked to Einstein to violate his principle of relativity, but they do not. Quantum theory lets us put quantitative values on the "probabilities" for each of the "possibilities." But this means that quantum theory is fundamentally statistical, meaning indeterministic and "random." It is not a question of our being ignorant about what is going on (an epistemological problem). What's happening is ontological chance, as Einstein first showed, but as he forever disliked. We can describe the "possibilities function" ψ as moving through space (at the speed of light, or even faster, as Einstein feared?), exploring all the possibilities for wherever the particle might be found. This too may be seen as a special kind of information. In the famous "twoslit experiment," the "possibilities function" travels everywhere, meaning that it passes through both slits, interfering with itself and thus changing the possibilities where the particle might be found. Metaphorically, it "knows" when both slits are open, even if our intuitive classical view imagines that the particle must go through only one. This changes the probabilities associated with each of the possibilities.
Possibilities and Information Theory
It is of the deepest philosophical significance that information theory is based on the mathematics of probability. If all outcomes were certain, there would be no "surprises" in the universe. Information would be conserved and a universal constant, as some mathematicians mistakenly believe. Information philosophy requires the ontological uncertainty and probabilistic outcomes of modern quantum physics to produce new information. In Claude Shannon's theory of the communication of information, there must be multiple possible messages in order for information to be communicated. If there is but one possible message, there is no uncertainty, and no information can be communicated. In a universe describable by the classical Newtonian laws of motion, all the information needed to produce the next moment is contained in the positions, motions, and forces on the material particles. In a quantum world describable by the unitary evolution of the deterministic Schrödinger equation, nothing new ever happens, there is no new "outcome." Outcomes are added to standard quantum mechanics by the addition of the "projection postulate" or "collapse of the wave function" when the quantum system interacts with another system. Information is constant in a deterministic universe. There is "nothing new under the sun." The creation of new information is not possible without the random chance and uncertainty of quantum mechanics, plus the extraordinary temporal stability of quantum mechanical structures needed to store information once it is created. Without the extraordinary stability of quantized information structures over cosmological time scales, life and the universe we know would not be possible. That stability is the consequence of an underlying digital nature. Quantum mechanics reveals the architecture of the universe to be discrete rather than continuous, to be digital rather than analog. Digital information transfers are essentially perfect. All analog transfers are "lossy." It is Bohr's "correspondence principle" of quantum mechanics for large quantum numbers and the "law of large numbers" of statistics which ensure that macroscopic objects can normally average out microscopic uncertainties and probabilities to provide the statistical or "adequate" determinism that shows up in all our "Laws of Nature."
There is no separate classical world and no need for a quantumtoclassical transition. The quantum world becomes statistically deterministic when the mass of an object is such that h / m approaches zero.
We conclude, contrary to the views of Bohr and Heisenberg, that there is no need for a separate classical world. The classical laws of nature emerge statistically from quantum laws. Quantum laws, which are therefore universally applicable, converge in these two limits of large numbers to classical laws. There is no "transition" from the quantum world to a separate classical world. There is just one world, where quantum physics applies universally, but its mysterious properties, like interference, entanglement, and nonlocality, are normally invisible, averaged over, in the macroscopic world.
The problem for an informational interpretation of quantum mechanics is to explain exactly how these two convergences (large numbers of particles and large quantum numbers) allow continuous and apparently deterministic macroscopic information structures to emerge from the indeterministic and discontinuous microscopic quantum world. We must show how the determinism in the macroscopic world is only a statistical determinism or adequate determinism that results from "averaging over" the large number of independent quantum events happening in a macroscopic object. And even more important, we must show how the occasional magnification or amplification of microscopic quantum events leads to new macroscopic information that makes human beings the "authors of their lives", that makes them "cocreators of our universe," and that guarantees a genuinely open future with alternative possibilities, not in inaccessible "parallel universes" but in the one universe that we have.
Feynman's Path Integral, Diagrams, and Sum of Histories
In Richard Feynman's "path integral" formulation of quantum mechanics, we may have a way to help visualize our "possibilities" function.
Feynman proposed to reformulate quantum mechanics based on just three postulates:
The overall probability amplitude for a given process is the sum of the contributions over the space of all possible paths of the system in between the initial and final states. All probability amplitudes have equal weight but have varying phase of the complex action. Rapidly varying phase may significantly reduce the contribution along a path and paths may interfere with neighboring paths. The path integrals are described as the "sum of histories" of all paths through an infinite space–time. Local quantum field theory might restrict paths to lie within a finite causally closed region, with timelike separations inside a lightcone. A Feynman diagram is a graphical representation of one path's contribution to the probability amplitude. Feynman's path integral method gives the same results as quantum mechanics for Hamiltonians quadratic in the momentum. Feynman's amplitudes also obey the Schrödinger equation for the Hamiltonian corresponding to the action S. How do we interpret this as visualizing the "possibilities" in our information interpretation? We can compare the individual paths to the "virtual photons" that mediate the electromagnetic field in quantum field theory. The picture is then that an infinite number of virtual photons explore all the possibilities in the given situation. Large numbers of them go through both slits for example, and interfere with one another, preventing even a single real photon from landing on a null point in the interference pattern.
Information Creation without Observers
Information physics explores the quantum mechanical and thermodynamic properties of cosmic information structures, especially those that were created before the existence of human observers.
A key parameter is the amount of information per particle. When particles combine, the information per particle increases. Starting with quarks forming nucleons, nuclei combining with electrons to form atoms, atoms combining into molecules, macromolecules, crystal lattices, and other solid state structures, at every stage of growth two things are happening. Binding energy is being transferred away from the new composite structure, carrying away positive entropy. This positive entropy more than balances the negative entropy (or information) in the new structure, thus satisfying the second law of thermodynamics. But the important thing is the increasing information per particle, which allows the new information structure to approach classical behavior. Individual particles are no longer acting alone. Acting in concert allows them to average over quantum noise. They acquire new properties and capabilities that emerge from the component particles and are not reducible to the parts. Quantum noise destroys coherent actions of the lowerlevel parts of the new structure, preventing the lower level from exerting "bottonup" control. Information in the higherlevel structure allows the composite system to generate new behaviors that can exercise downward causal control. Some of these behaviors depend for their successful operation on the disruptive noise in the lower level. For example, a ribosome, assembling a polypeptide chain into a protein, depends on the chaotic and random motions of the transfer RNA molecules with amino acids attached to rapidly provide the next codon match. To understand information creation, information physics provides new insights into the puzzling "problem of measurement" and the mysterious "collapse of the wave function" in quantum mechanics. Information physics also probes deeply into the second law of thermodynamics to establish the irreversible increase of entropy on a quantum mechanical basis. "Information physics" provides a new "interpretation" of quantum mechanics. But it is is not an attempt to alter the basic assumptions of standard quantum mechanics, extending them to include "hidden variables," for example. It does reject the unfortunate idea that nothing happens to quantum systems without the intervention of an "observer."
Possibilities, Probabilities, and Actuality
1) The Wave Function. The central element in quantum physics is the "wave function" ψ, with its mysterious waveparticle dual nature (sometimes a wave, sometimes a particle, etc.). We believe that teaching and understanding quantum mechanics would be much simpler if we called ψ the "possibilities function." It only looks like a wave in simple cases of configuration space. But it always tells us the possibilities  the possible values of any observable, for example. 2. The Superposition Principle (and transformation theory). Given the "possibilities function" ψ, quantum mechanics allows us to calculate the "probabilities" for each of the "possibilities." The calculation will depend on the free choice of the experimenter as to which "observables" to look for, by designing an appropriate measurement apparatus. For example, if the measurement apparatus can register n discrete values, ψ can be expanded in terms of a set of n basis functions appropriate for the chosen observable, say φ_{n}. The expansion is then
ψ = ∑ c_{n} φ_{n},
and we say the system is in a "superposition" of these n states. When the absolute squares of the coefficients c_{n} are appropriately normalized to add up to 1, the probability P_{n} of observing value n is
P_{n} = c_{n}^{2} =  < ψ  φ_{n} >  ^{2}
These probabilities are confirmed statistically by repeated identical experiments that collect large numbers of results. Quantum mechanics is the most accurate physical theory in science, with measurements accurate to thirteen decimal places. 2. The Equation of Motion. The "possibilities function" ψ evolves in time according to the unitary Schrödinger equation of motion.
ih δψ_{a}/δt = H ψ_{a},
where H is the Hamiltonian.
3. The Projection Postulate ("Collapse" of the Wave Function).
4. The Axiom of Measurement (impossible without information creation). In each individual experiment, generally just one of the possibilities becomes an actuality (some experiments leave the quantum system in a new superposition of multiple possibilities). In our information interpretation, a possibility is realized or actualized when information is created about the new state of the system. This new information requires that more positive entropy be carried away than the local increase in negative entropy. Note that an "observer" would not be able to make a "measurement" unless there is new information to be "observed." Information can be (and usually is) created and recorded before any observer looks at the results. An information approach can help philosophers to think more clearly about quantum physics. Instead of getting trapped in talk about mysterious "collapses of the wave function," "reductions of the wave packet," or the "projection postulate" (all important issues), the information interpretation proposes we simply say that one of the "possibilities" has become "actual." It is intuitively obvious that when one possibility becomes actual, all the others are annihilated, consigned to "nothingness," as Jean PaulSartre put it. We can also say that quantum theory lets us put quantitative values on the "probabilities" for each of the "possibilities." But this means that quantum theory is statistical, meaning indeterministic and "random. It is not a question of our being ignorant about what is going on (an epistemological problem). What's happening is ontological chance. We can also say that the "possibilities function" ψ moves through space (at the speed of light , or even faster?), exploring all the possibilities for where the particle might be found. This too may be seen as a special kind of (potential?) information. In the famous "twoslit experiment," the "possibilities function" travels everywhere, meaning that it passes through both slits, interfering with itself and thus changing the possibilities where the particle might be found. Metaphorically, it "knows" when both slits are open, even if our intuitive classical view imagines the particle to go through only one slit. This changes the probabilities associated with each of the possibilities.
The Axioms of an Information Quantum Mechanics
Information physics accepts the principle of superposition (that arbitrary quantum states can be represented as linear combinations of system eigenstates) and the projection postulate (that interactions between quantum systems can project the two system states  or "collapse" them  into eigenstates of the separate systems or the combined entangled system). But we replace the Copenhagen axiom of "measurement" (as involving a "measurer" or "observer") by considering instead the increased information that can sometimes occur when systems interact. If this increased information is recorded for long enough to be seen by an observer, then we can call it a measurement. The appearance of a particle at a particular spot on a photographic plate, the firing of a Geiger counter recording a particle event, or a ribosome adding an amino acid to a polypeptide chain may all be considered to be "measurements." Just as the traditional axiom of measurement says that measurements of observables (quantities that commute with the Hamiltonian) can only yield eigenvalues, information physics accepts that the quantities of information added correspond to these eigenvalues. What these measurements (or simply interactions without "measurements") have in common is the appearance of a microscopic particle in a particular "observable" state. But whether or not "observed," quantum particles are always interacting with other particles, and some of these interactions increase the perparticle information as the particles build up information structures. In classical mechanics, the material universe is thought to be made up of tiny particles whose motions are completely determined by forces that act between the particles, forces such as gravitation, electromagnetic attractions and repulsions, nuclear and weak forces, etc. The equations that describe those motions, Newton's laws of motion, were for many centuries thought to be perfect and sufficient to predict the future of any mechanical system. They provided support for many philosophical ideas about determinism.
The Schrödinger Equation of Motion
In quantum mechanics, Newton's laws of motion, especially in the Hamiltonian formulation of those laws, are replaced by Erwin Schrodinger's wave equation, which describes the time evolution of a probability amplitude wave ψ_{a},
ih δψ_{a}/δt = H ψ_{a},
where H is the Hamiltonian.
The probability amplitude looks like a wave and the Schrödinger equation is a wave equation. But the wave is an abstract quantity whose absolute square is interpreted as the probability of finding a quantum particle somewhere. It is distinctly not the particle itself, whose exact position is unknowable while the quantum system is evolving deterministically. It is the probability amplitude wave that interferes with itself. Particles, as such, never interfere (although they may collide). And while probabilities can be distributed throughout space, some here, some there, parts of a particle are never found. It is the whole quantum particle that is always found. Information is conserved (a constant of the motion) during the unitary and deterministic time evolution of the wave function according to the Schrödinger equation. Information can only be created (or destroyed) during an interaction between quantum systems. It is because the system goes from a superposition of states to one of the possible states that the information changes. In what Wolfgang Pauli called a "measurement of the second kind," a system prepared in an eigenstate will, when measured again, be found with certainty, in the same state. There is no information gained in this case.
Clarifying WaveParticle Duality
In the information interpretation of quantum mechanics, the time evolution of the wave function is not interpreted as a motion of the particle. What is evolving, what is moving, is the probability, more fundamentally, the possibilities, of finding the particle in different places, or the expectation value of various physical quantities as a function of the time.
An electron is not both a wave and a particle, think "eitheror"
There is no logical contradiction in the use of the deterministic Schrödinger wave equation to predict values of the (indeterministic) probability amplitude wave functions at future times. The Schrödinger equation is a universal equation (and quantum mechanics is universally true), but there is no universal wave function. Each wave function is particular to specific quantum systems, sometimes to just a single particle. And when a particle is actually found, the potential (probability) of finding it elsewhere simply vanishes instantaneously.
first a wave of possibilities, then an actual particle. As Max Born put it, the equations of motion for the probability waves are deterministic and continuous, but the motion of a particle itself is indeterministic, discontinuous, and probabilistic. John von Neumann was bothered by the fact that the projection postulate or wavefunction collapse (his Process 1) was not a formal part of quantum mechanics. Dozens, if not hundreds, of physicists have attempted to produce a formal theory to describe the collapse process and make it predictable. But this is to misunderstand the irreducible chance nature of the wavefunction collapse. Long before Werner Heisenberg formulated his indeterminacy principle, it was Albert Einstein who discovered the fundamental acausal and discontinuous nature of quantum mechanics. He postulated "light quanta" in 1905, more than twenty years before quantum mechanics was developed. And in 1916, his explanation of light emission and absorption by atoms showed that the process is essentially a chance process. All attempts to predict a random process are denials of the indeterministic nature of the quantum world. Such efforts often assume that probability itself can be understood as epistemic, the consequence of human ignorance about underlying, as yet undiscovered, deterministic laws. Historically, both scientists and philosophers have had what William James called an "antipathy to chance." The information interpretation says that if deterministic underlying laws existed, there would be no new information creation. There would be only one possible future. But we know that the universe has been creating new information from the time of the Big Bang. Information physics adds this cosmological information creation to Einstein's discovery of microscopic quantum chance to give us a consistent picture of a quantum world that appears adequately determined at the macroscopic level. But why then did Einstein object all his life to the indeterminism ("God does not play dice," etc.) that was his own fundamental discovery?
Probability Amplitude Waves Are Not "Fields" like Electromagnetic or Gravitational Fields
In classical electrodynamics, electromagnetic radiation (light, radio) is known to have wave properties such as interference. When the crest of one wave in the electromagnetic field meets the trough of another, the two waves cancel one another. This field is a ponderable object. A disturbance of the field at one place is propagated to other parts of the field at the velocity of light. Einstein's gravitational field in general relativity is similar. Matter moving at one point produces changes in the field which propagate to other parts of space, again at the velocity of light, in order to be relativistically invariant. The probability amplitude wave function ψ of quantum mechanics is not such a field, however. Although Einstein sometimes referred to it as a "ghost field," and Louis de Broglie later developed a "pilot wave" theory in which waves guide the motion of material particles along paths perpendicular to the wave fronts, the wave function ψ has no ponderable substance. When a particle is detected at a given point, the probability of its being found elsewhere goes immediately to zero. Probability does not propagate at some speed to other places, reducing the probability of the particle subsequently being found there. Once located, the particle cannot be elsewhere. And that is instantaneously true. Einstein, who spent his life searching for a "unified field theory," found this very hard to accept. He thought that instantaneous changes in the "ghost field" must violate his special theory of relativity by traveling faster than the speed of light. He described this as "nonlocal reality," the idea that something at one point (the probability of finding a particle there) could move to be somewhere else faster than light speed. In our information interpretation, nothing that is material or energy is transferred when the probability "collapses," it is only immaterial information that changes instantly. Animation of a probability amplitude wave function ψ collapsing  click to restart
Whereas the abstract probability amplitude moves continuously and deterministically throughout space, the concrete particle, for all we know, may move discontinuously and indeterministically to a particular point in space.
Who Really Discovered Quantum Indeterminism and Discontinuity?
In 1900, Max Planck made the assumption that radiation is produced by resonant oscillators whose energy is quantized and proportional to the frequency of radiation ν.
E = hν
For Planck, the proportionality constant h is a "quantum of action," a heuristic mathematical device that allowed him to apply Ludwig Boltzmann's work on the statistical mechanics and kinetic theory of gases to the radiation field. (Boltzmann had shown in the 1870's that the increase in entropy (the second law) could be explained if gases were made up of enormous numbers of particles his HTheorem). Planck applied Boltzmann's statistics of many particles to radiation and derived the distribution of radiation at different frequencies (or wavelengths) just as James Clerk Maxwell and Boltzmann had derived the distribution of velocities (or energies) of the gas particles. Note the mathematical similarity of Planck's radiation distribution law (photons) and the MaxwellBoltzmann velocity distribution (molecules). Both curves have a power law increase on one side to a maximum and an exponential decrease on the other side of the maximum. The molecular velocity curves cross one another because the total number of molecules is the same. With increasing temperature T, the number of photons increases at all wavelengths. But Planck did not actually believe that radiation came in discrete particles, at least until a dozen years later. In the meantime, Albert Einstein's 1905 paper on the photoelectric effect showed that light came in discrete particles he called "light quanta," subsequently called "photons," by analogy to electrons. Planck was not happy about the idea of light particles, because his use of Boltzmann's statistics implied that chance was real. Boltzmann himself had qualms about the reality of chance. Although Einstein also did not like the idea of chancy statistics, he did believe that energy came in packages of discrete "quanta." It was Einstein, not Planck, who quantized mechanics and electrodynamics. Nevertheless, it was for the introduction of the quantum of action h that Planck was awarded the Nobel prize in 1918. Meanwhile, in 1916, after completing his work on the general theory of relativity, Einstein returned to thinking about a quantum theory for the interaction of radiation and matter (N.B., ten years before quantum mechanics).
In 1924, Louis de Broglie argued that if photons, with their known wavelike properties, could be described as particles, electrons as particles might show wavelike properties with a wavelength λ inversely proportional to their momentum p = m_{e}v.
p = h/2πλ
Experiments confirmed de Broglie's assumption and led Erwin Schrödinger to derive a "wave equation" to describe the motion of de Broglie's waves. Schrödinger's equation replaces the classical Newton equations of motion. Note that Schrödinger's equation describes the motion of only the wave aspect, not the particle aspect, and as such it implies interference. Note also that the Schrödinger equation is as fully deterministic an equation of motion as Newton's equations. Schrödinger attempted to interpret his "wave function" for the electron as a probability density for electrical charge, but charge density would be positive everywhere and unable to interfere with itself. Max Born shocked the world of physics by suggesting that the absolute values of the wave function ψ squared (ψ^{2}) could be interpreted as the probability of finding the electron in various position and momentum states  if a measurement is made. This allows the probability amplitude ψ to interfere with itself, producing highly nonintuitive phenomena such as the twoslit experiment. Despite the probability amplitude going through two slits and interfering with itself, experimenters never find parts of electrons. They always are found whole. In 1932 John von Neumann explained that two fundamentally different processes are going on in quantum mechanics.
Von Neumann claimed there is another major difference between these two processes. Process 1 is thermodynamically irreversible. Process 2 is reversible. This confirms the fundamental connection between quantum mechanics and thermodynamics that is explainable by information physics. Information physics establishes that process 1 may create information. Process 2 is information preserving.
Collapse of the Wave Function
Physicists calculate the deterministic evolution of the Schrödinger wave function in time as systems interact or collide. At some point, they make the ad hoc assumption that the wave function "collapses." This produces a set of probabilities of finding the resulting combined system in its various eigenstates.
Although the collapse appears to be a random and ad hoc addition to the deterministic formalism of the Schrödinger equation, It is very important to note that the experimental accuracy of quantum mechanical predictions is unparalleled in physics, providing the ultimate justification for this theoretical kluge.
Moreover, without wave functions collapsing, no new information can come into the universe. Nothing unpredicatable would ever emerge. Determinism is "informationpreserving." All the information we have today would have to have already existed in the original fireball.
The "Problem" of Measurement
The irreducibly random process of wave function "collapse," when one of the possibilities becomes an actuality in a quantum measurement, is not a part of the mathematical formalism describing the time evolution of the wave function (the perfectly deterministic Schrödinger equation of motion).
Collapse is an ad hoc addition to the theory, a heuristic description that enables a method of calculation to predict the probabilities of what will happen when an observer makes a measurement. The ad hoc nature of this "collapse" of the wavefunction (or the "reduction" of the wavepacket) is the original problem of measurement. Decoherence theorists today describe the problem of measurement as the absence of macroscopic superpositions of states, for example, the famous Schrödinger's Cat. Our information interpretation clarifies both of these "measurement problems." Many physicists (especially those who, following Albert Einstein and Erwin Schrödinger, would prefer a deterministic theory) think that the collapse should be included in the formalism, to make it a predictable part of quantum theory. Ironically, it was Einstein, many years before Heisenberg's 1927 announcement of his "uncertainty principle," who found that electromagnetic radiation is not continuous and deterministic, but discontinuous, discrete, indeterministic, and acausal. He found it a "weakness in the theory," but quantum events like the emission of a photon or decay of a radioactive nucleus happen by chance (Zufall). We cannot say exactly when, or where, or in which direction a particle will go. We can only give statistical probabilities for such events. This Einstein was first to see and understand, but it was something he could never accept as the complete picture of quantum reality. Both he and Schrödinger (even Max Planck) denied the reality of the quantum and "quantum jumps." In many standard discussions of quantum mechanics, and most every popular treatment, it is said that we need the consciousness of a physicist to collapse the wave function. Eugene Wigner and John Wheeler sometimes describe the observer as making up the "mind of the universe." Von Neumann contributed a lot to this confusion by claiming that the location of the Heisenberg "cut" (Schnitt) between the microscopic system and macroscopic measurement can be put anywhere, including inside an observer's brain or mind. The information interpretation of quantum mechanics removes the observer from the moment of the Heisenbergvon Neumann "cut." This cut is when information is created about the interaction of the quantum system with another system (which may be a measurement apparatus). And the moment that irreversible information enters the universe is the moment of the transition from the quantum world to the classical world that the decoherence theories attempt to explain. An "observable" information structure (with mass m) has reduced the probability amplitudes to probabilities. Interference between the different possibilities disappears. The fundamental particles are still quantum, but averages over them cancel out and indeterminacy in their positions and velocities Δx Δv approaches zero as h / m approaches zero. Measurement requires the interaction of something macroscopic, assumed to be large and adequately determined. In physics experiments, this is the observing apparatus. But in general, measurement does not require a conscious observer. It does require information creation or there will be nothing to observe. In our discussion of Schrödinger's Cat, the cat can be its own observer.
Thermodynamics
The second law of thermodynamics says that the entropy (or disorder) of a closed physical system increases until it reaches a maximum, the state of thermodynamic equilibrium. It requires that the entropy of the universe is now and has always been increasing. (The first law is that energy is conserved.)
This established fact of increasing entropy has led many scientists and philosophers to assume that the universe we have is running down. They think that means the universe began in a very high state of information, since the second law requires that any organization or order is susceptible to decay. The information that remains today, in their view, has always been here. This fits nicely with the idea of a deterministic universe. There is nothing new under the sun. Physical determinism is "informationpreserving."
But the universe is not a closed system. It is in a dynamic state of expansion that is moving away from thermodynamic equilibrium faster than entropic processes can keep up. The maximum possible entropy is increasing much faster than the actual increase in entropy. The difference between the maximum possible entropy and the actual entropy is potential information.
Creation of information structures means that in parts of the universe the local entropy is actually going down. Reduction of entropy locally is always accompanied by radiation of entropy away from the local structures to distant parts of the universe, into the night sky for example. Since the total entropy in the universe always increases, the amount of entropy radiated away always exceeds (often by many times) the local reduction in entropy, which mathematically equals the increase in information.
"Ergodic" Processes
We will describe processes that create information structures, reducing the entropy locally, as "ergodic." This is a new use for a term from statistical mechanics that describes a hypothetical property of classical mechanical gases. See the Ergodic Hypothesis. Ergodic processes (in our new sense of the word) are those that appear to resist the second law of thermodynamics because of a local increase in information or "negative entropy" (Erwin Schrödinger's term). But any local decrease in entropy is more than compensated for by increases elsewhere, satisfying the second law. Normal entropyincreasing processes we will call "entropic". Encoding new information requires the equivalent of a quantum measurement  each new bit of information produces a local decrease in entropy but requires that at least one bit (generally much much more) of entropy be radiated or conducted away. Without violating the inviolable second law of thermodynamics overall, ergodic processes reduce the entropy locally, producing those pockets of cosmos and negative entropy (order and informationrich structures) that are the principal objects in the universe and in life on earth.
Entropy and Classical Mechanics
Ludwig Boltzmann attempted in the 1870's to prove Rudolf Clausius' second law of thermodynamics, namely that the entropy of a closed system always increases to a maximum and then remains in thermal equilibrium. Clausius predicted that the universe would end with a "heat death" because of the second law.
Boltzmann formulated a mathematical quantity H for a system of n ideal gas particles, showing that it had the property δΗ/δτ ≤ 0, that H always decreased with time. He identified his H as the opposite of Rudolf Clausius' entropy S. In 1850 Clausius had formulated the second law of thermodynamics. In 1857 he showed that for a typical gas like air at standard temperatures and pressures, the gas particles spend most of their time traveling in straight lines between collisions with the wall of a containing vessel or with other gas particles. He defined the "mean free path" of a particle between collisions. Clausius and essentially all physicists since have assumed that gas particles can be treated as structureless "billiard balls" undergoing "elastic" collisions. Elastic means no motion energy is lost to internal friction. Shortly after Clausius first defined the entropy mathematically and named it in 1865, James Clerk Maxwell determined the distribution of velocities of gas particles (Clausius for simplicity had assumed that all particles moved at the average speed ^{1}/_{2}mv^{2} = ^{3}/_{2}kT). Maxwell's derivation was very simple. He assumed the velocities in the x, y, and z directions were independent. [more...] Boltzmann improved on Maxwell's statistical derivation by equating the number of particles entering a given range of velocities and positions to the number leaving the same volume in 6ndimensional phase space. This is a necessary state for the gas to be in equilibrium. Boltzmann then used Newtonian physics to get the same result as Maxwell, which is thus called the MaxwellBoltzmann distribution. Boltzmann's first derivation of his Htheorem (1872) was based on the same classical mechanical analysis he had used to derive Maxwell's distribution function. It was an analytical mathematical consequence of Newton's laws of motion applied to the particles of a gas. But it ran into immediate objections. The objection is the hypothetical and counterfactual idea of time reversibility. If time were reversed, the entropy would simply decrease. Since the fundamental Newtonian equations of motion are time reversible, this appears to be a paradox. How could the irreversibile increase of the macroscopic entropy result from microscopic physical laws that are time reversible?Lord Kelvin (William Thomson) was the first to point out the time asymmetry in macroscopic processes, but the criticism of Boltzmann's Htheorem is associated with his lifelong friend Joseph Loschmidt. Boltzmann immediately agreed with Loschmidt that the possibility of decreasing entropy could not be ruled out if the classical motion paths were reversed. Boltzmann then reformulated his Htheorem (1877). He analyzed a gas into "microstates" of the individual gas particle positions and velocities. For any "macrostate" consistent with certain macroscopic variables like volume, pressure, and temperature, there could be many microstates corresponding to different locations and speeds for the individual particles. Any individual microstate of the system was intrinsically as probable as any other specific microstate, he said. But the number of microstates consistent with the disorderly or uniform distribution in the equilibrium case of maximum entropy simply overwhelms the number of microstates consistent with an orderly initial distribution. About twenty years later, Boltzmann's revised argument that entropy statistically increased ran into another criticism, this time not so counterfactual. This is the recurrence objection. Given enough time, any system could return to its starting state, which implies that the entropy must at some point decrease. These reversibility and recurrence objections are still prominent in the physics literature.The recurrence idea has a long intellectual history. Ancient Babylonian astronomers thought the known planets would, given enough time, return to any given position and thus begin again what they called a "great cycle," estimated by some at 36,000 years. Their belief in an astrological determinism suggested that all events in the world would also recur. Friedrich Nietzsche made this idea famous in the nineteenth century, at the same time as Boltzmann's hypothesis was being debated, as the "eternal return" in his Also Sprach Zarathustra. The recurrence objection was first noted in the early 1890's by French mathematician and physicist Henri Poincaré. He had found an analytic solution to the threebody problem and noted that the configuration of three bodies returns arbitrarily close to the initial conditions after calculable times. Even for a handful of planets, the recurrence time is longer than the age of the universe, if the positions are specified precisely enough. Poincaré then proposed that the presumed "heat death" of the universe predicted by the second law of thermodynamics could be avoided by "a little patience." Another mathematician, Ernst Zermelo, a young colleague of Max Planck in Berlin, is more famous for this recurrence paradox. Boltzmann accepted the recurrence criticism. He calculated the extremely small probability that entropy would decrease noticeably, even for gas with a very small number of particles (1000). He showed the time associated with such an event was 10^{1010} years. But the objections in principle to his work continued, especially from those who thought the atomic hypothesis was wrong. It is very important to understand that both Maxwell's original derivation of the velocities distribution and Boltzmann's Htheorem showing an entropy increase are only statistical or probabilistic arguments. Boltzmann's work was done twenty years before atoms were established as real and fifty years before the theory of quantum mechanics established that at the microscopic level all interactions of matter and energy are fundamentally and irreducibly statistical and probabilistic.
Entropy and Quantum Mechanics
A quantum mechanical analysis of the microscopic collisions of gas particles (these are usually molecules  or atoms in a noble gas) can provide revised analyses for the two problems of reversibility and recurrence. Note this requires more than quantum statistical mechanics. It needs the quantum kinetic theory of collisions in gases.
There are great differences between Ideal, Classical, and Quantum Gases. Boltzmann assumed that collisions would result in random distributions of velocities and positions so that all the possible configurations would be realized in proportion to their number. He called this "molecular chaos." But if the path of a system of n particles in 6ndimensional phase space should be closed and repeat itself after a short and finite time during which the system occupies only a small fraction of the possible states, Boltzmann's assumptions would be wrong. What is needed is for collisions to completely randomize the directions of particles after collisions, and this is just what the quantum theory of collisions can provide. Randomization of directions is the norm in some quantum phenomena, for example the absorption and reemission of photons by atoms as well as Raman scattering of photons. In the deterministic evolution of the Schrödinger equation, just as in the classical path evolution of the Hamiltonian equations of motion, the time can be reversed and all the coherent information in the wave function will describe a particle that goes back exactly the way it came before the collision. But if when two particles collide the internal structure of one or both of the particles is changed, and particularly if the two particles form a temporary larger molecule (even a quasimolecule in an unbound state), then the separating atoms or molecules lose the coherent wave functions that would be needed to allow time reversal back along the original path. During the collision, one particle can transfer energy from one of its internal quantum states to the other particle. At room temperature, this will typically be a transition between rotational states that are populated. Another possibility is an exchange of energy with the background thermal radiation, which at room temperatures peaks at the frequencies of molecular rotational energy level differences. Such a quantum event can be analyzed by assuming a shortlived quasimolecule is formed (the energy levels for such an unbound system are a continuum of, so that almost any photon can cause a change of rotational state of the quasimolecule. A short time later, the quasimolecule dissociates into the two original particles but in different energy states. We can describe the overall process as a quasimeasurement, because there is temporary information present about the new structure. This information is lost as the particles separate in random directions (consistent with conservation of energy, momentum, and angular momentum). The decoherence associated with this quasimeasurement means that if the postcollision wave functions were to be time reversed, the reverse collision would be very unlikely to send the particles back along their incoming trajectories. Boltzmann's assumption of random occupancy of possible configurations is no longer necessary. Randomness in the form of "molecular chaos" is assured by quantum mechanics. The result is a statistical picture that shows that entropy would normally increase even if time could be reversed. This does not rule out the kind of departures from equilibrium that occur in small groups of particles as in Brownian motion, which Boltzmann anticipated long before Brown's experiments and Einstein's explanation. These fluctuations can be described as forming shortlived information structures, brief and localized regions of negative entropy, that get destroyed in subsequent interactions. Nor does it change the remote possibility of a recurrence of any particular initial microstate of the system. But it does prove that Poincaré was wrong about such a recurrence being periodic. Periodicity depends on the dynamical paths of particles being classical, deterministic, and thus time reversible. Since quantum mechanical paths are fundamentally indeterministic, recurrences are simply statistically improbable departures from equilibrium, like the fluctuations that cause Brownian motion.
Entropy is Lost Information
Entropy increase can be easily understood as the loss of information as a system moves from an initially ordered state to a final disordered state. Although the physical dimensions of thermodynamic entropy (joules/ºK) are not the same as (dimensionless) mathematical information, apart from units they share the same famous formula.
S = ∑ p_{i} ln p_{i}To see this very simply, let's consider the wellknown example of a bottle of perfume in the corner of a room. We can represent the room as a grid of 64 squares. Suppose the air is filled with molecules moving randomly at room temperature (blue circles). In the lower left corner the perfume molecules will be released when we open the bottle (when we start the demonstration).
What is the quantity of information we have about the perfume molecules? We know their location in the lower left square, a bit less than 1/64th of the container. The quantity of information is determined by the minimum number of yes/no questions it takes to locate them. The best questions are those that split the locations evenly (a binary tree). For example:
If the room had no air, the perfume would rapidly reach an equilibrium state, since the molecular velocity at room temperature is about 400 meters/second. Collisions with air molecules prevent the perfume from dissipating quickly. This lets us see the approach to equilibrium. When the perfume has diffused to onesixteenth of the room, the entropy will have risen 2 bits for each molecule, to onequarter of the room, four bits, etc. Let's look at a computer visualization of the equilibration process in a new window.
Ergodic Processes Increase Information
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