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 Lawrence Cahoone 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 Arthur Fine John Martin Fischer Frederic Fitch 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 Christoph Lehner 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 C.F. von Weizsäcker William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Scientists Michael Arbib Walter Baade Bernard Baars 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 Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Donald Campbell Anthony Cashmore Eric Chaisson Gregory Chaitin JeanPierre Changeux Arthur Holly Compton John Conway John Cramer E. P. Culverwell Olivier Darrigol 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 Steven Frautschi Edward Fredkin Lila Gatlin Michael Gazzaniga GianCarlo Ghirardi J. Willard Gibbs Nicolas Gisin Paul Glimcher Thomas Gold A. O. Gomes Brian Goodwin Joshua Greene Jacques Hadamard Mark Hadley 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 William R. Klemm 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 Warren McCulloch 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 Jürgen Renn/a> Juan Roederer Jerome Rothstein David Ruelle Tilman Sauer Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Claude Shannon David Shiang 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 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 Stephen Wolfram H. Dieter Zeh Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Presentations Biosemiotics Free Will Mental Causation James Symposium 
Irreversibility in Ideal, Classical, and Quantum Gases
Abstract
An ideal gas is defined as one in which the details of molecular collisions, with other molecules and with the walls of the container, can be ignored. Ideal molecules have no internal structure and no interaction with matter and fields outside the container. We define a classical gas as one described by classical dynamics including electrodynamics. Details of collisions are considered as are interactions with external electromagnetic and gravitational fields. Classical molecules have continuous internal degrees of freedom including vibration and rotation. A quantum gas adds electron spin, FermiDirac and BoseEinstein statistics for antisymmetric and symmetric particles, but most importantly it treats the kinetics of collisions  particle scattering  quantum mechanically. Quantum molecules have complex discrete internal structures. We ignore relativistic effects. Gases can be studied from three perspectives  thermodynamics, statistical mechanics, and kinetic theory (which follows the details of molecular interactions). They can also be treated classically or quantum mechanically. Boltzmann's original (1872) formulation of the Htheorem was based on the kinetics of molecular collisions, but it contained an unjustifiable assumption (Stosszahlansatz) about the absence of correlations of molecular velocities before and after the collisions (molecular chaos) that introduced probability, perhaps inadvertently, into his calculations.
f(r_{i}, r_{j}, p_{i}, p_{j}, t) = f(r_{i}, p_{i}, t) f(r_{j}, p_{j}, t) (1)
We will show that Boltzmann's original assumption of molecular chaos may be justified for quantum gases of particles with accessible internal energy states. With a combination of kinetic theory and probabilistic assumptions, Boltzmann derived the famous expression that he identified with Rudolf Clausius' thermodynamic entropy.
S = klogW (2)
Five years later, Boltzmann responded to a brief remark by his Vienna colleague Josef Loschmidt that a gas with the same molecular positions but opposite velocities should show an entropy decrease, since classical mechanics is time reversible. Boltzmann immediately agreed with Loschmidt and speculated that perhaps one could not prove the entropy increase with a purely mechanical analysis. In its place, he gave a new defense of the Htheorem (eq.2), using only the assumption of equiprobable microstates.
The entropy of a nonequilibrium macrostate is proportional to the logarithm of the number of microstates consistent with the macrostate description. Microstates compatible with thermal equilibrium outnumber nonequilibrium microstates by many orders of magnitude. Assuming that transitions between microstates are all equally likely (essentially the ergodic hypothesis as Ehrenfest named it), nonequilibrium macrostates quickly evolve to thermal equilibrium. Macrostates that depart from equilibrium contain information (negative entropy) that in principle can be observed and measured. This includes cases of states prepared by experimenters, including removing a barrier between unlike gases, or perfume in a bottle at time zero. In these cases, quantitatively calculable information in the initial state is completely lost at equilibrium. Increasing entropy can be equated to decreasing information. This is the core idea of statistical mechanics, whether the gas is ideal, classical, or quantum.
Ideal gas
For an ideal gas, for example hard spheres, Boltzmann's Htheorem has been shown to hold. Entropy will increase for arbitrary initial conditions.
However, both the reversibility objection and the recurrence objection are still valid. Given special initial conditions corresponding to the highly improbable "timereversed" velocities, the entropy will decrease. PierreSimon Laplace had postulated a super intelligence who can predict the future, given perfect knowledge of the positions and velocities of material particles together with their force laws. But given the essential impossibility of preparing an initial state for a test of this or Loschmidt's hypothesis, Boltzmann thought it not a practical objection. Extreme sensitivity to initial conditions was appreciated by James Clerk Maxwell as early as 1865 [ref?], when he noted the occurrence of singular points in hydrodynamical flows and argued that something like them in the mind might allow living creatures to escape from strict determinism. Modern computer simulations of ideal gases confirm Maxwell's and Boltzmann's intuitions, showing that miniscule errors in original positions lead very quickly to randomness in the distributions. "Chaos theory" is the deterministic mathematical formalism that describes the dynamics of physical systems near singular points in their motions where infinitesimal differences in position or velocity lead to exponentially large differences at later times. It does not involve quantum uncertainty. Zermelo's recurrence objection was based on Poincare's studies of the nbody problem, which indicated that recurrence would be "quasiperiodic." Poincare also discovered exceptional nonrecurrent paths, which have been shown to be infinite in number yet with zero probability, a "set of measure zero" in modern terminology ^{1}.
Conclusions for an Ideal Gas
Classical gas
Boltzmann was as aware of the ideal gas approximations as any modern scientist. He was the first to include external forces (a gravitational field). He attempted kinetic calculations for polyatomic molecules. And he knew that collisions with real container walls are likely to be inelastic and nonspecular, adding randomness to the time evolution of the gas. Boltzmann also knew that the exactness of deterministic laws of classical mechanics themselves went "beyond experience." No observational evidence justified their perfection. Newton himself doubted analytical perfection, in part for theological reasons that such perfection limited God's powers. Loschmidt's reversibility objection forced Boltzmann to see that some configurations of position and velocity might lead to to a reduction in entropy, but Boltzmann appreciated the practical impossibility of preparing such a state. He thought fluctuations of local reductions in the entropy would occur. But he argued they would be shortlived in a nonideal gas, given the effects of random perturbations from external forces. In modern times, calculations have shown that even tiny amounts of matter at stellar distances can alter the trajectories of classical particles. A gram of matter at the distance of Sirius can cause a particle to miss a predicted collision after as few as 50 collisions. [Berry?] Faced with Zermelo's recurrence objection, Boltzmann calculated the recurrence time for even a small number of particles and showed that it exceeded by many orders of magnitude the likely age of the universe. For all practical purposes, recurrence to a prepared state (with all particles in half the container volume for example) was impossible.
Conclusions for a Classical Gas
Quantum gas
Just as Hamilton's classical equations of motion are time reversible, Erwin Schrodinger's reformulation of them as the equation of motion for the probability amplitude ψ(r, t) of a quantum state is time reversible. If H is the Hamiltonian,
^{ih}/_{2π} ^{δψ(r, t)}/_{δτ} = Ηψ(r, t). (3)
Quantum statistical mechanics embodies the equiprobability assumptions about phasespace volumes (the ergodic hypothesis in both Boltzmann and Gibbs formulations) into the Fermi Golden Rule and Master Equation, which says that transition probabilities from microstate i to microstate j are equal to the reverse transition from j to i. The matrix element P_{ij} is the complex conjugate of P_{ji}.
Quantum kinetic theory treats the collisions of gas molecules as problems in quantum scattering. In this case, inelastic collisions could include Raman scattering that changes the internal quantum states of the colliding molecules. Moreover, colliding atoms could combine to form molecules, emitting the binding energy as radiation. In these cases, the resulting dissociation at a later time, however short, would mean that information has been created and destroyed. As a consequence, the kind of quantum coherence needed for time reversibility of the Schrodinger equation is lost. Boltzmann's original guess that velocities of particles after collisions are random would be justified.
Conclusions for a Quantum Gas
Final Conclusions
Deterministic chaos for a classical gas and the assumption of equal probabilities for forward and reverse transitions between microstates (Fermi's Golden Rule) for a quantum gas are more than adequate to account for entropy increase and irreversibility in the normal time scales of terrestrial physics.
But the addition of molecular collisions analyzed as quantum scattering processes that create and destroy information, with temporary and local entropy decreases (fluctuations), strengthens the Htheorem by vitiating Loschmidt's reversibility objection. In addition, it replaces Zermelo's recurrence objection with the unavoidable but essentially vanishingly small probability that any prepared system should return to its initial conditions after a finite and physically significant time. For Teachers
For Scholars
Notes:1. Brush, Stephen, Kinetic Theory of Gases, vol.2, Irreversible Processes, Pergamon, Oxford, 1966. p.17
