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Philosophers
Mortimer Adler Rogers Albritton Alexander of Aphrodisias G.E.M.Anscombe Anselm Thomas Aquinas Aristotle David Armstrong Augustine J.L.Austin A.J.Ayer Alexander Bain Mark Balaguer William Belsham Henri Bergson Isaiah Berlin Bernard Berofsky Susanne Bobzien Emil du Bois-Reymond George Boole Émile Boutroux F.H.Bradley C.D.Broad C.A.Campbell Joseph Keim Campbell Carneades Ernst Cassirer Roderick Chisholm Chrysippus Cicero Randolph Clarke Samuel Clarke Anthony Collins Diodorus Cronus Donald Davidson Democritus Daniel Dennett René Descartes Richard Double Fred Dretske 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 Carl Ginet Nicholas St. John Green H.Paul Grice Ian Hacking Ishtiyaque Haji Stuart Hampshire W.F.R.Hardie R.M.Hare Georg W.F. Hegel Martin Heidegger R.E.Hobart Thomas Hobbes David Hodgson Shadsworth Hodgson Ted Honderich Pamela Huby David Hume Ferenc Huoranszki William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan William King Christine Korsgaard Keith Lehrer Gottfried Leibniz Leucippus Michael Levin C.I.Lewis David Lewis Peter Lipton John Locke Michael Lockwood John R. Lucas Lucretius James Martineau Hugh McCann Colin McGinn Michael McKenna Paul E. Meehl Alfred Mele John Stuart Mill Dickinson Miller G.E.Moore Thomas Nagel Friedrich Nietzsche P.H.Nowell-Smith Robert Nozick William of Ockham Timothy O'Connor David F. Pears Charles Sanders Peirce Derk Pereboom Steven Pinker Plato Karl Popper H.A.Prichard Hilary Putnam Willard van Orman Quine Frank Ramsey Ayn Rand Thomas Reid Charles Renouvier Nicholas Rescher C.W.Rietdijk Josiah Royce Bertrand Russell Paul Russell Gilbert Ryle T.M.Scanlon Moritz Schlick Arthur Schopenhauer John Searle Wilfrid Sellars Henry Sidgwick Walter Sinnott-Armstrong J.J.C.Smart Saul Smilansky Michael Smith L. Susan Stebbing George F. Stout Galen Strawson Peter Strawson Eleonore Stump Richard Taylor Kevin Timpe 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 Alfred North Whitehead David Widerker David Wiggins Bernard Williams Ludwig Wittgenstein Susan Wolf Scientists Michael Arbib Bernard Baars John S. Bell Charles Bennett Margaret Boden David Bohm Neils Bohr Ludwig Boltzmann Emile Borel Max Born Leon Brillouin Stephen Brush Henry Thomas Buckle Donald Campbell Anthony Cashmore Eric Chaisson Jean-Pierre Changeux Arthur Holly Compton John Conway E. H. Culverwell Charles Darwin Abraham de Moivre Paul Dirac John Eccles Arthur Stanley Eddington Paul Ehrenfest Albert Einstein Richard Feynman Joseph Fourier Michael Gazzaniga GianCarlo Ghirardi Nicolas Gisin Thomas Gold A.O.Gomes Joshua Greene Jacques Hadamard Patrick Haggard Augustin Hamon Sam Harris Martin Heisenberg Werner Heisenberg William Stanley Jevons Pascual Jordan Simon Kochen Stephen Kosslyn Rolf Landauer Alfred Landé Pierre-Simon Laplace David Layzer Benjamin Libet Josef Loschmidt Ernst Mach Henry Margenau James Clerk Maxwell Ernst Mayr Jacques Monod Roger Penrose Steven Pinker Max Planck Henri Poincaré Adolphe Quételet Jerome Rothstein Erwin Schrödinger Claude Shannon Herbert Simon Dean Keith Simonton B. F. Skinner Henry Stapp Antoine Suarez Leo Szilard William Thomson (Kelvin) John von Neumann Daniel Wegner Steven Weinberg Norbert Wiener Eugene Wigner E. O. Wilson Ernst Zermelo |
The Problem of Microscopic Reversibility
Loschmidt's Paradox
In 1876, Josef Loschmidt criticized his younger colleague Ludwig Boltzmann's 1866 attempt to derive the increasing entropy required by the second law of thermodynamics from basic classical dynamics.
Loschmidt's criticism was based on the simple idea that the laws of classical dynamics are time reversible. Consequently, if we just turned the time around, the system should lead to decreasing entropy.
This is the intimate connection between time and the second law of thermodynamics that Arthur Stanley Eddington later called the Arrow of Time.
Microscopic time reversibility is one of the foundational assumptions of both classical mechanics and quantum mechanics. A careful quantum analysis shows that reversibility fails even in the most ideal conditions - the case of two particles in collision.
Microscopic irreversibility provides a new justification for Ludwig Boltzmann's assumption of "molecular disorder" and strengthens his H-Theorem.
In quantum mechanics, microscopic time reversibility is taken to mean that the deterministic linear Schrödinger equation is time reversible.
A careful quantum analysis shows that ideal reversibility fails
even in the simplest conditions - the case of two particles in collision.
When they collide, even structureless particles should not be treated as individual particles with single-particle wave functions, but as a single system with a two-particle wave function, because they are now entangled.
Treating two atoms as a temporary molecule means we must use molecular, rather than atomic, wave functions. The quantum description of the molecule now transforms the six independent degrees of freedom into three for the molecule's center of mass and three more that describe vibrational and rotational quantum states.
The possibility of quantum transitions between closely spaced
vibrational and rotational energy levels in the "quasi-molecule' introduces uncertainty, which could be different for the hypothetical perfectly reversed path.
Even assuming the practical impossibility of a perfect classical time reversal, in which we simply turn the two particles around, quantum physics requires two measurements to locate the two particles, followed by two state preparations to send them in the opposite direction, preserving the momenta.
Heisenberg indeterminacy puts calculable limits on the accuracy with which perfect reversed paths could be achieved.
Let us assume this impossible task can be completed, and it sends the two particles into the reverse collision paths. But on the return path, there is only a finite probability that a "sum over histories" calculation will produce the same (or reversed) quantum transitions between vibrational and rotational states that occurred in the first collision.
Thus a quantum description of a two-particle collision establishes the microscopic irreversibility that Boltzmann sometimes described as his assumption of "molecular disorder." In his second (1872) derivation of the H-theorem, Boltzmann used a statistical approach and the molecular disorder assumption to get away from the time reversibility assumptions of classical dynamics.
Zermelo's paradox was a later criticism of Ludwig Boltzmann's attempt to derive the increasing entropy required by the second law of thermodynamics. It also involves time. Assuming infinite available time, a finite universe with fixed matter, energy, and information will at some point return to any given earlier state.
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