"out"'(p=0)if they do not.If the number of correspond- Mechanics (Oxford University Press,Oxford,U.K.,1938),pp.45,51-52 ing microstates is W,then the nonvanishing probabilities are 2p.G.de Gennes,The Physics of Liquid Crystals (Oxford University Press, London.1974). p=1/W,and the general entropy definition Eq.(Al)be- 2J.David Lister and Robert J.Birgeneau,"Liquid crystal phases and phase comes transitions,"Phys.Today 35,26-33(May 1982). 2D.Guillon,P.E.Cladis,and J.Stamatoff,"X-ray study and microscopic S=kg In W, (A3) study of the reentrant nematic phase,"Phys.Rev.Lett.41,1598-1601 which is Eq.(1)of Sec.I. (1978) 2The phenomenon of reentrance in general,and particularly in liquid crys- tals,is reviewed in section 6 of Shri Singh,"Phase transitions in liquid aElectronic mail:Dan.Styer@oberlin.edu crystals,''Phys.Rep.324.107-269(2000). Oral remark by John von Neumann to Claude Shannon,recalled by Shan- 24Y.Yeshurun,A.P.Malozemoff,and A.Shaulov,"Magnetic relaxation in non.See page 354 of Myron Tribus,"Information theory and thermody- high-temperature superconductors,"'Rev.Mod.Phys.68,911-949 namics,'in Heat Transfer,Thermodynamics.and Education:Boelter An- (1996).See page 916 and Fig.4b on page 915. niversary Volume,edited by Harold A.Johnson (McGraw-Hill,New York, 25G.W.Castellan,Physical Chemistry,2nd ed.(Addison-Wesley,Reading, 1964),Pp.348-368 MA,1971),p.330 2Karl K.Darrow,The concept of entropy,"Am.J.Phys.12,183-196 (1944). 2See,for example,Max Hansen and Kurt Anderko,Constimtion of Binary 3P.G.Wright,"Entropy and disorder,"Contemp.Phys.11,581-588 Alloys (McGraw-Hill,New York,1958);David A.Young,Phase Dia- (1970). grams of the Elements (University of California Press,Berkeley,1991). "P.W.Atkins,The Second Law (Scientific American Books,New York, Robert J.Birgeneau,"Novel magnetic phenomena and high-temperature 1984). superconductivity in lamellar copper oxides,"Am.J.Phys.58,28-40 (1990)(Fig.4);C.Lobban,J.L.Finney,and W.F.Kuhs,"The structure 5Bernd Rodewald,"Entropy and homogeneity,Am.J.Phys.58,164-168 of a new phase of ice,"'Nature (London)391,268-270 (1998);and (1990). I-Ming Chou,J.G.Blank,A.F.Gohcharov,H.-k.Mao,and R.J.Hemley, PHarvey S.Leff and Andrew F.Rex,Eds.,Maxwell's Demon:Entropy, Information.Computing (Princeton University Press,Princeton,NJ,1990). "In situ observations of a high-pressure phase of H2O ice,"'Science 281, Ralph Baierlein,"Entropy and the second law:A pedagogical alterna- 809-812(1998). tive,”Am.J.Phys.62,15-26(1994). 27The argument of this section was invented by Edward M.Purcell and is Harvey S.Leff,"Thermodynamic entropy:The spreading and sharing of summarized in Stephen Jay Gould,Bully for Brontosaurus (W.W.Norton, energy,”Am.J.Phys.64,1261-1271(1996). New York,1991),pp.265-268,260-261 Thomas A.Moore and Daniel V.Schroeder,"A different approach to See,for example,H.Eugene Stanley,Introduction to Phase Transitions introducing statistical mechanics,"'Am.J.Phys.65,26-36(1997). and Critical Phenomena (Oxford University Press,New York,1971):D. Vinay Ambegaokar and Aashish A.Clerk,"Entropy and time,"Am.J. C.Radulescu and D.F.Styer,"The Dobrushin-Shlosman phase unique- ness criterion and applications to hard squares,''J.Stat.Phys.49,281- Phys.67,1068-1073(1999) J.Machta,Entropy,information,and computation,Am.J.Phys.67. 295(1987);and Ref.2. 1074-1077(1999). 29These computer programs,which work under MS-DOS,are available for Roger Balian,Incomplete descriptions and relevant entropies,Am.J. free downloading through http://www.oberlin.edu/physics/dstyer/. Phys.67,1078-1090(1999) The model of Fig.2 is called the "ideal lattice gas,while the nearest- Harvey S.Leff,"What if entropy were dimensionless?"Am.J.Phys.67, neighbor-excluding model of Fig.3 is called the "hard-square lattice 1114-1122(1999). gas.''(These are just two of the infinite number of varieties of the lattice 4J.Willard Gibbs,Collected Works (Yale University Press,New Haven, gas model.)Although the entropy of the hard-square lattice gas is clearly CT,1928),Vol.1,p.418 less than that of the corresponding ideal lattice gas,it is difficult to calcu- John W.Patterson,Thermodynamics and evolution,in Scientists Con- late the exact entropy for either model.Such values can be found (to high front Creationism,edited by Laurie R.Godfrey (Norton,New York, accuracy)by extrapolating power series expansions in the activity =de- 1983),pp.99-116 tails and results are given in D.S.Gaunt and M.E.Fisher,"Hard-sphere Duane T.Gish,Creation Scientists Answer their Critics [sic](Institute for lattice gases.I.Plane-square lattice,"J.Chem.Phys.43,2840-2863 Creation Research,El Cajon,California,1993).An Appendix contribution (1965)and R.J.Baxter,I.G.Enting,and S.K.Tsang,"Hard-square by D.R.Boylan seeks to split entropy into the usual entropy which is lattice gas,J.Stat.Phys.22,465-489(1980). "due to random effects'and a different sort of entropy related to the 3 Reference 4 promotes the idea that entropy is a measure of homogeneity. "order or information in the system"(p.429).An even greater error (This despite the everyday observation of two-phase coexistence.)To but- appears in Chap.6(on pp.164 and 175)where Gish claims that scientists tress this argument,the book presents six illustrations (on pp.54,72,74 must show not that evolution is consistent with the second law of thermo- 75,and 77)of"equilibrium lattice gas configurations."Each configura- dynamics,but that evolution is necessary according to the second law of tion has 100 occupied sites on a 40x40 grid.If the occupied sites had thermodynamics.The moon provides a counterexample. been selected at random,then the probability of any site being occupied Detailed discussion of this N!factor and the related"Gibbs paradox''can would be 100/1600,and the probability of any given pair of sites both be found in David Hestenes,"Entropy and indistinguishability,"'Am.J. being occupied would be 1/(16)-.The array contains 2X39X39 adjacent Phys.38,840-845 (1970);Barry M.Casper and Susan Freier,Gibbs site pairs,so the expected number of occupied adjacent pairs would be paradox'paradox,"ibid.41,509-511 (1973);and Peter D.Pesic,"The 2(39/16)=11.88.The actual numbers of occupied nearest-neighbor pairs principle of identicality and the foundations of quantum theory.I.The in the six illustrations are 0,7,3,7,4,and 3.A similar calculation shows Gibbs paradox,"ibid.59,971-974 (1991). that the expected number of empty rows or columns in a randomly occu- IsThe Sackur-Tetrode formula (2) predicts that SK-SA pied array is(15/16)10x2x40=41.96.The actual numbers for the six =(3/2)kgN In(mgr/mAr).The data in Ihsan Barin,Thermochemical Data illustrations are 28,5,5,4,4,and 0.I am confident that the sites in these of Pure Substances,3rd ed.(VCH Publishers,New York,1995),pp.76 illustrations were not occupied at random,but rather to give the impression and 924,verify this prediction to 1.4%at 300 K,and to 90 parts per of uniformity million at 2000 K. Someone might raise the objection:"Yes,but how many configurations The moral of the paradox is given in the body of this paper.The resolution would you have to draw from the pool,on average,before you obtained of the paradox is both deeper and more subtle:It hinges on the fact that the exactly the special configuration of Fig.5?The answer is,"Precisely the proper home of statistical mechanics is phase space,not configuration same number that you would need to draw,on average,before you ob- space,because Liouville's theorem implies conservation of volume in tained exactly the special configuration of Fig.2."These two configura- phase space,not in configuration space.See Ludwig Boltzmann,Vorlesun- tions are equally special and equally rare. gen iiber Gastheorie (J.A.Barth,Leipzig,1896-98),Part II,Chaps.III 33In this connection it is worth observing that in the canonical ensemble and VII [translated into English by Stephen G.Brush:Lectures on Gas (where all microstates are "accessible")the microstate most likely to be Theory (University of California Press,Berkeley,1964)],J.Willard Gibbs, occupied is the ground state,and that this is true at amy positive tempera- Elementary Principles in Statistical Mechanics (C.Scribner's Sons,New ture,no matter how high.The ground state energy is not the most probable York,1902),p.3;and Richard C.Tolman,The Principles of Statistical energy,nor is the ground state typical,yet the ground state is the most 1095 Am.J.Phys.,Vol.68,No.12,December 2000 Daniel F.Styer 1095‘‘out’’ (pm50) if they do not. If the number of correspond￾ing microstates is W, then the nonvanishing probabilities are pm51/W, and the general entropy definition Eq. ~A1! be￾comes S5kB ln W, ~A3! which is Eq. ~1! of Sec. I. a! Electronic mail: Dan.Styer@oberlin.edu 1 Oral remark by John von Neumann to Claude Shannon, recalled by Shan￾non. See page 354 of Myron Tribus, ‘‘Information theory and thermody￾namics,’’ in Heat Transfer, Thermodynamics, and Education: Boelter An￾niversary 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 alterna￾tive,’’ Am. J. Phys. 62, 15–26 ~1994!. 8 Harvey S. Leff, ‘‘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!. 10Vinay Ambegaokar and Aashish A. Clerk, ‘‘Entropy and time,’’ Am. J. Phys. 67, 1068–1073 ~1999!. 11J. Machta, ‘‘Entropy, information, and computation,’’ Am. J. Phys. 67, 1074–1077 ~1999!. 12Roger Balian, ‘‘Incomplete descriptions and relevant entropies,’’ Am. J. Phys. 67, 1078–1090 ~1999!. 13Harvey S. Leff, ‘‘What if entropy were dimensionless?’’ Am. J. Phys. 67, 1114–1122 ~1999!. 14J. Willard Gibbs, Collected Works ~Yale University Press, New Haven, CT, 1928!, Vol. 1, p. 418. 15John W. Patterson, ‘‘Thermodynamics and evolution,’’ in Scientists Con￾front Creationism, edited by Laurie R. Godfrey ~Norton, New York, 1983!, pp. 99–116. 16Duane 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 thermo￾dynamics, but that evolution is necessary according to the second law of thermodynamics. The moon provides a counterexample. 17Detailed 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. Pesˇic´, ‘‘The principle of identicality and the foundations of quantum theory. I. The Gibbs paradox,’’ ibid. 59, 971–974 ~1991!. 18The Sackur-Tetrode formula ~2! predicts that SKr2SAr 5(3/2)kBN ln(mKr /mAr). 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. 19The 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, Vorlesun￾gen u¨ber 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. 20P. G. de Gennes, The Physics of Liquid Crystals ~Oxford University Press, London, 1974!. 21J. David Lister and Robert J. Birgeneau, ‘‘Liquid crystal phases and phase transitions,’’ Phys. Today 35, 26–33 ~May 1982!. 22D. 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!. 23The phenomenon of reentrance in general, and particularly in liquid crys￾tals, is reviewed in section 6 of Shri Singh, ‘‘Phase transitions in liquid crystals,’’ Phys. Rep. 324, 107–269 ~2000!. 24Y. 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. 25G. W. Castellan, Physical Chemistry, 2nd ed. ~Addison-Wesley, Reading, MA, 1971!, p. 330. 26See, for example, Max Hansen and Kurt Anderko, Constitution of Binary Alloys ~McGraw-Hill, New York, 1958!; David A. Young, Phase Dia￾grams of the Elements ~University of California Press, Berkeley, 1991!; Robert J. Birgeneau, ‘‘Novel magnetic phenomena and high-temperature superconductivity in lamellar copper oxides,’’ Am. J. Phys. 58, 28–40 ~1990! ~Fig. 4!; C. Lobban, J. L. Finney, and W. F. Kuhs, ‘‘The structure of a new phase of ice,’’ Nature ~London! 391, 268–270 ~1998!; and I-Ming Chou, J. G. Blank, A. F. Gohcharov, H.-k. Mao, and R. J. Hemley, ‘‘In situ observations of a high-pressure phase of H2O ice,’’ Science 281, 809–812 ~1998!. 27The 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. 28See, 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 unique￾ness criterion and applications to hard squares,’’ J. Stat. Phys. 49, 281– 295 ~1987!; and Ref. 2. 29These computer programs, which work under MS-DOS, are available for free downloading through http://www.oberlin.edu/physics/dstyer/. 30The 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 calcu￾late the exact entropy for either model. Such values can be found ~to high accuracy! by extrapolating power series expansions in the activity z: de￾tails 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!. 31Reference 4 promotes the idea that entropy is a measure of homogeneity. ~This despite the everyday observation of two-phase coexistence.! To but￾tress this argument, the book presents six illustrations ~on pp. 54, 72, 74, 75, and 77! of ‘‘equilibrium lattice gas configurations.’’ Each configura￾tion has 100 occupied sites on a 40340 grid. If the occupied sites had been selected at random, then the probability of any site being occupied would be 100/1600, and the probability of any given pair of sites both being occupied would be 1/(16)2. The array contains 2339339 adjacent site pairs, so the expected number of occupied adjacent pairs would be 2(39/16)2511.88. The actual numbers of occupied nearest-neighbor pairs in the six illustrations are 0, 7, 3, 7, 4, and 3. A similar calculation shows that the expected number of empty rows or columns in a randomly occu￾pied array is (15/16)1032340541.96. 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. 32Someone 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 ob￾tained exactly the special configuration of Fig. 2.’’ These two configura￾tions are equally special and equally rare. 33In 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 tempera￾ture, 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 1095 Am. J. Phys., Vol. 68, No. 12, December 2000 Daniel F. Styer 1095