Erratum: Insight into entropy (American Journal of Physics (2000) 68 (1090-1096) DOI: 10.1119/1.1287353);Insight into entropy

American Journal of Physics

Volumn 68, Issue 12, 2000, Pages 1090-1096

  Styer, Daniel F ^a  

^a OBERLIN COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034356604     PISSN: 00029505     EISSN: 19432909     Source Type: Journal    
DOI: 10.1119/5.0090247     Document Type: Erratum

References (45)

1. Oral remark by John von Neumann to Claude Shannon, recalled by Shannon. See page 354 of Myron Tribus, "Information theory and thermodynamics," in Heat Transfer, Thermodynamics, and Education: Boelter Anniversary Volume, edited by Harold A. Johnson (McGraw-Hill, New York, 1964), pp. 348-368.
2. Karl K. Darrow, "The concept of entropy," Am. J. Phys. 12, 183-196 (1944).
3. P. G. Wright, "Entropy and disorder," Contemp. Phys. 11, 581-588 (1970).
4. P. W. Atkins, The Second Law (Scientific American Books, New York, 1984).
5. Bernd Rodewald, "Entropy and homogeneity," Am. J. Phys. 58, 164-168 (1990).
6. Harvey S. Leff and Andrew F. Rex, Eds., Maxwell's Demon: Entropy, Information, Computing (Princeton University Press, Princeton, NJ, 1990).
7. Ralph Baierlein, "Entropy and the second law: A pedagogical alternative," Am. J. Phys. 62, 15-26 (1994).
8. Harvey S. Lett, "Thermodynamic entropy: The spreading and sharing of energy," Am. J. Phys. 64, 1261-1271 (1996).
9. Thomas A. Moore and Daniel V. Schroeder, "A different approach to introducing statistical mechanics," Am. J. Phys. 65, 26-36 (1997).
10. Vinay Ambegaokar and Aashish A. Clerk, "Entropy and time," Am. J. Phys. 67, 1068-1073 (1999).
11. J. Machta, "Entropy, information, and computation," Am. J. Phys. 67, 1074-1077 (1999).
12. Roger Balian, "Incomplete descriptions and relevant entropies," Am. J. Phys. 67, 1078-1090 (1999).
13. Harvey S. Leff, "What if entropy were dimensionless?" Am. J. Phys. 67, 1114-1122 (1999).
14. J. Willard Gibbs, Collected Works (Yale University Press, New Haven, CT, 1928), Vol. 1, p. 418.
15. John W. Patterson, "Thermodynamics and evolution," in Scientists Confront Creationism, edited by Laurie R. Godfrey (Norton, New York, 1983), pp. 99-116.
16. Duane T. Gish, Creation Scientists Answer their Critics [sic] (Institute for Creation Research, El Cajon, California, 1993). An Appendix contribution by D. R. Boylan seeks to split entropy into the usual entropy which is "due to random effects" and a different sort of entropy related to the "order or information in the system" (p. 429). An even greater error appears in Chap. 6 (on pp. 164 and 175) where Gish claims that scientists must show not that evolution is consistent with the second law of thermodynamics, but that evolution is necessary according to the second law of thermodynamics. The moon provides a counterexample.
17. Detailed discussion of this N! factor and the related "Gibbs paradox" can be found in David Hestenes, "Entropy and indistinguishability," Am. J. Phys. 38, 840-845 (1970); Barry M. Casper and Susan Freier, "'Gibbs paradox' paradox," ibid. 41, 509-511 (1973); and Peter D. Pešić, "The principle of identicality and the foundations of quantum theory. I. The Gibbs paradox," ibid. 59, 971-974 (1991).
18. Detailed discussion of this N! factor and the related "Gibbs paradox" can be found in David Hestenes, "Entropy and indistinguishability," Am. J. Phys. 38, 840-845 (1970); Barry M. Casper and Susan Freier, "'Gibbs paradox' paradox," ibid. 41, 509-511 (1973); and Peter D. Pešić, "The principle of identicality and the foundations of quantum theory. I. The Gibbs paradox," ibid. 59, 971-974 (1991).
19. Detailed discussion of this N! factor and the related "Gibbs paradox" can be found in David Hestenes, "Entropy and indistinguishability," Am. J. Phys. 38, 840-845 (1970); Barry M. Casper and Susan Freier, "'Gibbs paradox' paradox," ibid. 41, 509-511 (1973); and Peter D. Pešić, "The principle of identicality and the foundations of quantum theory. I. The Gibbs paradox," ibid. 59, 971-974 (1991).
20. Ar). The data in Ihsan Barin, Thermochemical Data of Pure Substances, 3rd ed. (VCH Publishers, New York, 1995), pp. 76 and 924, verify this prediction to 1.4% at 300 K, and to 90 parts per million at 2000 K.
21. The moral of the paradox is given in the body of this paper. The resolution of the paradox is both deeper and more subtle: It hinges on the fact that the proper home of statistical mechanics is phase space, not configuration space, because Liouville's theorem implies conservation of volume in phase space, not in configuration space. See Ludwig Boltzmann, Vorlesungen über Gastheorie (J. A. Barth, Leipzig, 1896-98), Part II, Chaps. III and VII [translated into English by Stephen G. Brush: Lectures on Gas Theory (University of California Press, Berkeley, 1964)]; J. Willard Gibbs, Elementary Principles in Statistical Mechanics (C. Scribner's Sons, New York, 1902), p. 3; and Richard C. Tolman, The Principles of Statistical Mechanics (Oxford University Press, Oxford, U.K., 1938), pp. 45, 51-52.
22. The moral of the paradox is given in the body of this paper. The resolution of the paradox is both deeper and more subtle: It hinges on the fact that the proper home of statistical mechanics is phase space, not configuration space, because Liouville's theorem implies conservation of volume in phase space, not in configuration space. See Ludwig Boltzmann, Vorlesungen über Gastheorie (J. A. Barth, Leipzig, 1896-98), Part II, Chaps. III and VII [translated into English by Stephen G. Brush: Lectures on Gas Theory (University of California Press, Berkeley, 1964)]; J. Willard Gibbs, Elementary Principles in Statistical Mechanics (C. Scribner's Sons, New York, 1902), p. 3; and Richard C. Tolman, The Principles of Statistical Mechanics (Oxford University Press, Oxford, U.K., 1938), pp. 45, 51-52.
23. The moral of the paradox is given in the body of this paper. The resolution of the paradox is both deeper and more subtle: It hinges on the fact that the proper home of statistical mechanics is phase space, not configuration space, because Liouville's theorem implies conservation of volume in phase space, not in configuration space. See Ludwig Boltzmann, Vorlesungen über Gastheorie (J. A. Barth, Leipzig, 1896-98), Part II, Chaps. III and VII [translated into English by Stephen G. Brush: Lectures on Gas Theory (University of California Press, Berkeley, 1964)]; J. Willard Gibbs, Elementary Principles in Statistical Mechanics (C. Scribner's Sons, New York, 1902), p. 3; and Richard C. Tolman, The Principles of Statistical Mechanics (Oxford University Press, Oxford, U.K., 1938), pp. 45, 51-52.
24. The moral of the paradox is given in the body of this paper. The resolution of the paradox is both deeper and more subtle: It hinges on the fact that the proper home of statistical mechanics is phase space, not configuration space, because Liouville's theorem implies conservation of volume in phase space, not in configuration space. See Ludwig Boltzmann, Vorlesungen über Gastheorie (J. A. Barth, Leipzig, 1896-98), Part II, Chaps. III and VII [translated into English by Stephen G. Brush: Lectures on Gas Theory (University of California Press, Berkeley, 1964)]; J. Willard Gibbs, Elementary Principles in Statistical Mechanics (C. Scribner's Sons, New York, 1902), p. 3; and Richard C. Tolman, The Principles of Statistical Mechanics (Oxford University Press, Oxford, U.K., 1938), pp. 45, 51-52.
25. P. G. de Gennes, The Physics of Liquid Crystals (Oxford University Press, London, 1974).
26. J. David Lister and Robert J. Birgeneau, "Liquid crystal phases and phase transitions," Phys. Today 35, 26-33 (May 1982).
27. D. Guillon, P. E. Cladis, and J. Stamatoff, "X-ray study and microscopic study of the reentrant nematic phase," Phys. Rev. Lett. 41, 1598-1601 (1978).
28. The phenomenon of reentrance in general, and particularly in liquid crystals, is reviewed in section 6 of Shri Singh, "Phase transitions in liquid crystals," Phys. Rep. 324, 107-269 (2000).
29. Y. Yeshurun, A. P. Malozemoff, and A. Shaulov, "Magnetic relaxation in high-temperature superconductors," Rev. Mod. Phys. 68, 911-949 (1996). See page 916 and Fig. 4b on page 915.
30. G. W. Castellan, Physical Chemistry, 2nd ed. (Addison-Wesley, Reading, MA, 1971), p. 330.
31. 2O ice." Science 281, 809-812 (1998).
32. 2O ice." Science 281, 809-812 (1998).
33. 2O ice." Science 281, 809-812 (1998).
34. 2O ice." Science 281, 809-812 (1998).
35. 2O ice." Science 281, 809-812 (1998).
36. The argument of this section was invented by Edward M. Purcell and is summarized in Stephen Jay Gould, Bully for Brontosaurus (W. W. Norton, New York, 1991), pp. 265-268, 260-261.
37. See, for example, H. Eugene Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971); D. C. Radulescu and D. F. Styer, "The Dobrushin-Shlosman phase uniqueness criterion and applications to hard squares," J. Stat. Phys. 49, 281-295 (1987); and Ref. 2.
38. See, for example, H. Eugene Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971); D. C. Radulescu and D. F. Styer, "The Dobrushin-Shlosman phase uniqueness criterion and applications to hard squares," J. Stat. Phys. 49, 281-295 (1987); and Ref. 2.
39. These computer programs, which work under MS-DOS, are available for free downloading through http://www.oberlin.edu/physics/dstyer/.
40. The model of Fig. 2 is called the "ideal lattice gas," while the nearest-neighbor-excluding model of Fig. 3 is called the "hard-square lattice gas." (These are just two of the infinite number of varieties of the lattice gas model.) Although the entropy of the hard-square lattice gas is clearly less than that of the corresponding ideal lattice gas, it is difficult to calculate the exact entropy for either model. Such values can be found (to high accuracy) by extrapolating power series expansions in the activity z: details and results are given in D. S. Gaunt and M. E. Fisher, "Hard-sphere lattice gases. I. Plane-square lattice," J. Chem. Phys. 43, 2840-2863 (1965) and R. J. Baxter, I. G. Enting, and S. K. Tsang, "Hard-square lattice gas," J. Stat. Phys. 22, 465-489 (1980).
41. The model of Fig. 2 is called the "ideal lattice gas," while the nearest-neighbor-excluding model of Fig. 3 is called the "hard-square lattice gas." (These are just two of the infinite number of varieties of the lattice gas model.) Although the entropy of the hard-square lattice gas is clearly less than that of the corresponding ideal lattice gas, it is difficult to calculate the exact entropy for either model. Such values can be found (to high accuracy) by extrapolating power series expansions in the activity z: details and results are given in D. S. Gaunt and M. E. Fisher, "Hard-sphere lattice gases. I. Plane-square lattice," J. Chem. Phys. 43, 2840-2863 (1965) and R. J. Baxter, I. G. Enting, and S. K. Tsang, "Hard-square lattice gas," J. Stat. Phys. 22, 465-489 (1980).
42. 10×2×40=41.96. The actual numbers for the six illustrations are 28, 5, 5, 4, 4, and 0. I am confident that the sites in these illustrations were not occupied at random, but rather to give the impression of uniformity.
43. Someone might raise the objection: "Yes, but how many configurations would you have to draw from the pool, on average, before you obtained exactly the special configuration of Fig. 5?" The answer is, "Precisely the same number that you would need to draw, on average, before you obtained exactly the special configuration of Fig. 2." These two configurations are equally special and equally rare.
44. In this connection it is worth observing that in the canonical ensemble (where all microstates are "accessible") the microstate most likely to be occupied is the ground state, and that this is true at any positive temperature no matter how high. The ground state energy is not the most probable energy, nor is the ground state typical, yet the ground state is the most probable microstate. In specific, even at a temperature of 1 000 000 K, a sample of helium is more likely to be in a particular crystalline microstate than in any particular plasma microstate. However, there are so many more plasma than crystalline microstates that (in the thermodynamic limit) the sample occupies a plasma macrostate with probability 1.
45. The definition of entropy in Eq. (1) is the best starting place for teaching about entropy, but it holds only for the microcanonical ensemble. The definition Eq. (A1) is harder to understand but is also more general, applying to any ensemble. The two definitions are logically equivalent. See, for example, Richard E. Wilde and Surjit Singh, Statistical Mechanics: Fundamentals and Modern Applications (Wiley, New York, 1998), Sec. 1.6.1.