《统计热力学》课程教学资源（课件讲稿）Insight into entropy

Insight into entropy Daniel F.Styer) Department of Physics.Oberlin College.Oberlin.Ohio 44074 (Received 20 September 1999;accepted 1 May 2000) What is the qualitative character of entropy?Several examples from statistical mechanics(including liquid crystal reentrant phases,two different lattice gas models,and the game of poker)demonstrate facets of this difficult question and point toward an answer.The common answer of "entropy as disorder'is regarded here as inadequate.An alternative but equally problematic analogy is "entropy as freedom.Neither simile is perfect,but if both are used cautiously and not too literally, then the combination provides considerable insight.2000 American Association of Physics Teachers. Of all the difficult concepts of classical physics-concepts found simply by counting:One counts the number W of mi- like acceleration,energy,electric field,and time-the most crostates that correspond to the given macrostate,and com- difficult is entropy.Even von Neumann'claimed that "no- putes the entropy of the macrostate through body really knows what entropy is anyway."It is concerning entropy that students are most likely to invert their familiar S=kg In W, (1) lament and say "'I can do the problems,but I just can't where k is Boltzmann's constant.Clearly,S is high for a understand the material." macrostate when many microstates correspond to that mac- The itative character of entropy has been discussed rostate,whereas it is low when few microstates correspond. widely, although often only in the restricted context of The entropy of a macrostate measures the number of ways in gases or even in the highly restricted context of noninteract- which a system can be different microscopically and yet still ing gases.The metaphoric images invoked for entropy in- be a member of the same macroscopic state.The corre- clude“disorder,”s“randomness,.”s“smoothness,'s“disper- sponding microstates are often called "accessible'for rea- sion,'and "homogeneity."In a posthumous fragment, sons relating to ergodic theory.Because ergodic theory is Gibbs4 mentionedentropy as mixed-up-ness.Images rarely discussed at the undergraduate level,I feel it best to such as these can be useful and important,but if taken too avoid that term.Other synonyms for "a microstate corre- literally they can confuse as well as enlighten,and when sponding to a macrostate''are "a microstate consistent with misused!5.16 they can result in simple error.Analogies and (or compatible with)a macrostate''or "a permissible mi- visualizations should be employed,but their limitations as crostate.''] well as their strengths must be kept firmly in mind. Note that it requires some skill and interpretation to trans- Section I of this paper serves to set the stage and fix the late this formal definition into an expression applicable to terminology by presenting the formal,mathematical defini- specific situations.For example,suppose the macrostate of tion of microcanonical entropy in statistical mechanics.(The the system is specified by an energy from E to E+AE.If the definition is extended to other ensembles in the Appendix. system in question is quantum mechanical with discrete en- Section II(Cautionary Tales")gives three examples of the ergy levels,then one must count not all quantal states with surprises nature provides when this definition is applied to mean energies in this range,nor all energy eigenstates with physical systems,and hence illustrates the difficulties in- energies in this range,but instead the number of elements of volved in seeking qualitative insight into entropy.This sur- an energy eigenbasis with eigenvalues in this range.(In other vey serves to frame the terms of debate and show why some words,one must properly count degenerate energy states.)If visualizations of entropy are not acceptable.Section IlI goes the system is a collection of N identical classical particles, to the heart of the matter by examining two versions of a whether interacting or not,then the macroscopic state may simple model system (the "lattice gas')in which the rela- be a gas,liquid,or solid,but in all cases W is the volume of tionship between microscopic configurations and macro- phase space corresponding to this energy range,divided by scopic thermodynamic states is particularly clear.Section IV (N!h).(In classical statistical mechanics,ho is an arbi- reinforces the ideas of Sec.III by applying them to the game trary constant with the dimensions of action.In quantal sta- of poker,and Sec.V draws conclusions.The ideal conclu- tistical mechanics,it takes on the value of Planck's constant. sion for this paper would be to resolve the difficulties raised The so-called delabeling factor,N!,reflects the fact that N! by producing a concise yet accurate visualization for entropy that appeals to both the gut and the intellect.I am not able to different phase space points correspond to the same physical system.These N!points all represent N particles at N given do this.But I am able to use a combination of mathematics and analogy to illuminate the character of entropy locations and with corresponding given velocities,and differ only in the labels affixed to the various particles.On a more pragmatic vein,if the factor of N!were absent,then the I.MATHEMATICAL DEFINITION OF ENTROPY resulting entropy would not be extensive.7) (MICROCANONCIAL ENSEMBLE) In statistical mechanics,many microstates may correspond IL.CAUTIONARY TALES to a single macrostate.(A macrostate is also called a "ther- modynamic state":for example,T,V,and N for the canoni- Before seeking qualitative insight into this formal defini- cal ensemble or E.I.and N for the microcanonical en- tion,we examine three situations that demonstrate just how semble.)In the microcanonical ensemble the entropy is hazardous our search for insight can be 1090 Am.J.Phys.68 (12),December 2000 http://ojps.aip.org/ajp/ 2000 American Association of Physics Teachers 1090

Insight into entropy Daniel F. Styera) Department of Physics, Oberlin College, Oberlin, Ohio 44074 ~Received 20 September 1999; accepted 1 May 2000! What is the qualitative character of entropy? Several examples from statistical mechanics ~including liquid crystal reentrant phases, two different lattice gas models, and the game of poker! demonstrate facets of this difficult question and point toward an answer. The common answer of ‘‘entropy as disorder’’ is regarded here as inadequate. An alternative but equally problematic analogy is ‘‘entropy as freedom.’’ Neither simile is perfect, but if both are used cautiously and not too literally, then the combination provides considerable insight. © 2000 American Association of Physics Teachers. Of all the difficult concepts of classical physics—concepts like acceleration, energy, electric field, and time—the most difficult is entropy. Even von Neumann1 claimed that ‘‘no￾body really knows what entropy is anyway.’’ It is concerning entropy that students are most likely to invert their familiar lament and say ‘‘I can do the problems, but I just can’t understand the material.’’ The qualitative character of entropy has been discussed widely,2–13 although often only in the restricted context of gases or even in the highly restricted context of noninteract￾ing gases. The metaphoric images invoked for entropy in￾clude ‘‘disorder,’’ ‘‘randomness,’’ ‘‘smoothness,’’ ‘‘disper￾sion,’’ and ‘‘homogeneity.’’ In a posthumous fragment, Gibbs14 mentioned ‘‘entropy as mixed-up-ness.’’ Images such as these can be useful and important, but if taken too literally they can confuse as well as enlighten, and when misused15,16 they can result in simple error. Analogies and visualizations should be employed, but their limitations as well as their strengths must be kept firmly in mind. Section I of this paper serves to set the stage and fix the terminology by presenting the formal, mathematical defini￾tion of microcanonical entropy in statistical mechanics. ~The definition is extended to other ensembles in the Appendix.! Section II ~‘‘Cautionary Tales’’! gives three examples of the surprises nature provides when this definition is applied to physical systems, and hence illustrates the difficulties in￾volved in seeking qualitative insight into entropy. This sur￾vey serves to frame the terms of debate and show why some visualizations of entropy are not acceptable. Section III goes to the heart of the matter by examining two versions of a simple model system ~the ‘‘lattice gas’’! in which the rela￾tionship between microscopic configurations and macro￾scopic thermodynamic states is particularly clear. Section IV reinforces the ideas of Sec. III by applying them to the game of poker, and Sec. V draws conclusions. The ideal conclu￾sion for this paper would be to resolve the difficulties raised by producing a concise yet accurate visualization for entropy that appeals to both the gut and the intellect. I am not able to do this. But I am able to use a combination of mathematics and analogy to illuminate the character of entropy. I. MATHEMATICAL DEFINITION OF ENTROPY „MICROCANONCIAL ENSEMBLE… In statistical mechanics, many microstates may correspond to a single macrostate. ~A macrostate is also called a ‘‘ther￾modynamic state’’: for example, T, V, and N for the canoni￾cal ensemble or E, V, and N for the microcanonical en￾semble.! In the microcanonical ensemble the entropy is found simply by counting: One counts the number W of mi￾crostates that correspond to the given macrostate, and com￾putes the entropy of the macrostate through S5kB ln W, ~1! where kB is Boltzmann’s constant. Clearly, S is high for a macrostate when many microstates correspond to that mac￾rostate, whereas it is low when few microstates correspond. The entropy of a macrostate measures the number of ways in which a system can be different microscopically and yet still be a member of the same macroscopic state. @The corre￾sponding microstates are often called ‘‘accessible’’ for rea￾sons relating to ergodic theory. Because ergodic theory is rarely discussed at the undergraduate level, I feel it best to avoid that term. Other synonyms for ‘‘a microstate corre￾sponding to a macrostate’’ are ‘‘a microstate consistent with ~or compatible with! a macrostate’’ or ‘‘a permissible mi￾crostate.’’# Note that it requires some skill and interpretation to trans￾late this formal definition into an expression applicable to specific situations. For example, suppose the macrostate of the system is specified by an energy from E to E1DE. If the system in question is quantum mechanical with discrete en￾ergy levels, then one must count not all quantal states with mean energies in this range, nor all energy eigenstates with energies in this range, but instead the number of elements of an energy eigenbasis with eigenvalues in this range. ~In other words, one must properly count degenerate energy states.! If the system is a collection of N identical classical particles, whether interacting or not, then the macroscopic state may be a gas, liquid, or solid, but in all cases W is the volume of phase space corresponding to this energy range, divided by (N!h0 3N). ~In classical statistical mechanics, h0 is an arbi￾trary constant with the dimensions of action. In quantal sta￾tistical mechanics, it takes on the value of Planck’s constant. The so-called delabeling factor, N!, reflects the fact that N! different phase space points correspond to the same physical system. These N! points all represent N particles at N given locations and with corresponding given velocities, and differ only in the labels affixed to the various particles. On a more pragmatic vein, if the factor of N! were absent, then the resulting entropy would not be extensive.17! II. CAUTIONARY TALES Before seeking qualitative insight into this formal defini￾tion, we examine three situations that demonstrate just how hazardous our search for insight can be. 1090 Am. J. Phys. 68 ~12!, December 2000 http://ojps.aip.org/ajp/ © 2000 American Association of Physics Teachers 1090

A.The monatomic ideal gas 80 isotropic The entropy of a pure classical monatomic ideal gas,as a function of energy E,volume V,and particle number N,is nematic given by the Sackur-Tetrode formula 3 4TmEV2B smectic S(E,V,N)=kRN In 2 3h6W丽 (2) 2 Let us examine this result qualitatively to see whether it agrees with our understanding of entropy as proportional to 0 0.10.20.3 0.40.5 the number of microstates that correspond to a given mac- composition rostate.If the volume is increased,then the formula states that the entropy S increases,which certainly seems reason- Fig.1.Phase diagram of a liquid crystal mixture.The variable"composi- able:If the volume goes up,then each particle has more tion''refers to the molecular weight ratio of 60CB to 80CB.Figure modi- places where it can be,so the entropy ought to increase.If fied from Refs.21,22. the energy E is increased,then S increases,which again seems reasonable:If there is more energy around,then there homogeneous bowl of meltwater,yet it is the ice slivers that will be more different ways to split it up and share it among the particles,so we expect the entropy to increase.But what have the lower entropy.The moral here is that the huge if the mass m of each particle increases?(Experimentally, number of microscopic degrees of freedom in the meltwater one could compare the absolute entropy of,say,argon and completely overshadow the minute number of macroscopic krypton under identical conditions.18)Our formula shows degrees of freedom in the jumbled ice slivers.But the analo- that entropy increases with mass,but is there any way to gies of entropy to "disorder"'or "smoothness''invite us to understand this qualitatively? ignore this moral and concentrate on the system's gross ap- pearance and nearly irrelevant macroscopic features. In fact,I can produce not just one but two qualitative arguments concemning the dependence of S on m.Unfortu- nately the two arguments give opposite results!The first re- C.Reentrant phases lies upon the fact that When the temperature falls at constant pressure,most pure materials pass from gas to liquid to solid.But the unusual (3) materials called "liquid crystals,"which consist of rodlike molecules,display a larger number of phases.For typical so for a given energy E,any individual particle may have a liquid crystals,the high-temperature liquid phase is isotropic. momentum ranging from 0 to v2mE.A larger mass implies meaning that the positions and the orientations of the mol- a wider range of possible momenta,which suggests more ecules are scattered about nearly at random.At lower tem- microstates and a greater entropy.The second argument re- peratures,the substance undergoes a transition into the so- lies upon the fact that called "nematic"phase,in which the molecules tend to orient in the same direction but in which positions are still (4) scattered.At still lower temperatures it passes into the "smectic''phase,in which the molecules orient in the same direction and their positions tend to fall into planes.Finally, so for a given energy E,any individual particle may have a at even lower temperatures,the molecules freeze into a con- speed ranging from 0 to v2E/m.A larger mass implies a ventional solid.The story told so far reinforces the picture of narrowed range of possible speeds,which suggests fewer "entropy as disorder,with lower-temperature(hence lower microstates and a smaller entropy.The moral is simple: entropy)phases showing more and more qualitative order. Qualitative arguments can backfire! But not all liquid crystals behave in exactly this fashion. One material called "hexyloxycyanobiphenyl'or 60CB' B.Freezing water passes from isotropic liquid to nematic to smectic and then back to nematic again as the temperature is lowered.2122 The It is common to hear entropy associated with"disorder," first transition suggests that the nematic phase is "less or- s“smoothness,.”ors“homogeneity.”How do these associa- derly''than the smectic phase,while the second transition tions stand up to the simple situation of a bowl of liquid suggests the opposite! water placed into a freezer?Initially the water is smooth and One might argue that the lower-temperature nematic homogeneous.As its temperature falls,the sample remains phase-the so-called "reentrant nematic"-is somehow homogeneous until the freezing point is reached.At the qualitatively different in character from the higher- freezing temperature the sample is an inhomogeneous mix- temperature nematic,but the experiments summarized in Fig. ture of ice and liquid water until all the liquid freezes.Then 1 demonstrate that this is not the case.These experiments the sample is homogeneous again as the temperature contin- involve a similar liquid crystal material called "octyloxy- ues to fall.Thus the sample has passed from homogeneous to cyanobiphenyl'or 8OCB'which has no smectic phase at inhomogeneous to homogeneous,yet all the while its entropy all.Adding a bit of 8OCB into a sample of 6OCB reduces has decreased. the temperature range over which the smectic phase exists. Suppose the ice is then cracked out of its bowl to make Adding a bit more reduces that range further.Finally,addi- slivers,which are placed back into the bowl and allowed to tion of enough 80CB makes the smectic phase disappear rest at room temperature until they melt.The jumble of altogether.The implication of Fig.I is clear:there is no irregular ice slivers certainly seems disordered relative to the qualitative difference between the usual nematic and the re- 1091 Am.J.Phys.,Vol.68,No.12,December 2000 Daniel F.Styer 1091

A. The monatomic ideal gas The entropy of a pure classical monatomic ideal gas, as a function of energy E, volume V, and particle number N, is given by the Sackur–Tetrode formula S~E,V,N!5kBNF 3 2 lnS 4pmEV2/3 3h0 2 N5/3 D 1 5 2G . ~2! Let us examine this result qualitatively to see whether it agrees with our understanding of entropy as proportional to the number of microstates that correspond to a given mac￾rostate. If the volume V is increased, then the formula states that the entropy S increases, which certainly seems reason￾able: If the volume goes up, then each particle has more places where it can be, so the entropy ought to increase. If the energy E is increased, then S increases, which again seems reasonable: If there is more energy around, then there will be more different ways to split it up and share it among the particles, so we expect the entropy to increase. But what if the mass m of each particle increases? ~Experimentally, one could compare the absolute entropy of, say, argon and krypton under identical conditions.18! Our formula shows that entropy increases with mass, but is there any way to understand this qualitatively? In fact, I can produce not just one but two qualitative arguments concerning the dependence of S on m. Unfortu￾nately the two arguments give opposite results! The first re￾lies upon the fact that E5 1 2m ( i pi 2 , ~3! so for a given energy E, any individual particle may have a momentum ranging from 0 to A2mE. A larger mass implies a wider range of possible momenta, which suggests more microstates and a greater entropy. The second argument re￾lies upon the fact that E5 m 2 ( i vi 2 , ~4! so for a given energy E, any individual particle may have a speed ranging from 0 to A2E/m. A larger mass implies a narrowed range of possible speeds, which suggests fewer microstates and a smaller entropy. The moral19 is simple: Qualitative arguments can backfire! B. Freezing water It is common to hear entropy associated with ‘‘disorder,’’ ‘‘smoothness,’’ or ‘‘homogeneity.’’ How do these associa￾tions stand up to the simple situation of a bowl of liquid water placed into a freezer? Initially the water is smooth and homogeneous. As its temperature falls, the sample remains homogeneous until the freezing point is reached. At the freezing temperature the sample is an inhomogeneous mix￾ture of ice and liquid water until all the liquid freezes. Then the sample is homogeneous again as the temperature contin￾ues to fall. Thus the sample has passed from homogeneous to inhomogeneous to homogeneous, yet all the while its entropy has decreased. Suppose the ice is then cracked out of its bowl to make slivers, which are placed back into the bowl and allowed to rest at room temperature until they melt.9 The jumble of irregular ice slivers certainly seems disordered relative to the homogeneous bowl of meltwater, yet it is the ice slivers that have the lower entropy. The moral here is that the huge number of microscopic degrees of freedom in the meltwater completely overshadow the minute number of macroscopic degrees of freedom in the jumbled ice slivers. But the analo￾gies of entropy to ‘‘disorder’’ or ‘‘smoothness’’ invite us to ignore this moral and concentrate on the system’s gross ap￾pearance and nearly irrelevant macroscopic features. C. Reentrant phases When the temperature falls at constant pressure, most pure materials pass from gas to liquid to solid. But the unusual materials called ‘‘liquid crystals,’’ which consist of rodlike molecules, display a larger number of phases.20 For typical liquid crystals, the high-temperature liquid phase is isotropic, meaning that the positions and the orientations of the mol￾ecules are scattered about nearly at random. At lower tem￾peratures, the substance undergoes a transition into the so￾called ‘‘nematic’’ phase, in which the molecules tend to orient in the same direction but in which positions are still scattered. At still lower temperatures it passes into the ‘‘smectic’’ phase, in which the molecules orient in the same direction and their positions tend to fall into planes. Finally, at even lower temperatures, the molecules freeze into a con￾ventional solid. The story told so far reinforces the picture of ‘‘entropy as disorder,’’ with lower-temperature ~hence lower entropy! phases showing more and more qualitative order. But not all liquid crystals behave in exactly this fashion. One material called ‘‘hexyloxycyanobiphenyl’’ or ‘‘6OCB’’ passes from isotropic liquid to nematic to smectic and then back to nematic again as the temperature is lowered.21,22 The first transition suggests that the nematic phase is ‘‘less or￾derly’’ than the smectic phase, while the second transition suggests the opposite! One might argue that the lower-temperature nematic phase—the so-called ‘‘reentrant nematic’’—is somehow qualitatively different in character from the higher￾temperature nematic, but the experiments summarized in Fig. 1 demonstrate that this is not the case. These experiments involve a similar liquid crystal material called ‘‘octyloxy￾cyanobiphenyl’’ or ‘‘8OCB’’ which has no smectic phase at all. Adding a bit of 8OCB into a sample of 6OCB reduces the temperature range over which the smectic phase exists. Adding a bit more reduces that range further. Finally, addi￾tion of enough 8OCB makes the smectic phase disappear altogether. The implication of Fig. 1 is clear: there is no qualitative difference between the usual nematic and the re￾Fig. 1. Phase diagram of a liquid crystal mixture. The variable ‘‘composi￾tion’’ refers to the molecular weight ratio of 6OCB to 8OCB. Figure modi- fied from Refs. 21, 22. 1091 Am. J. Phys., Vol. 68, No. 12, December 2000 Daniel F. Styer 1091

entrant nematic phases-you can move continuously from one to the other in the temperature-composition phase dia- gram. The implication of reentrant phases23 for entropy is pro- found:Under some conditions the nematic phase has more entropy than the smectic phase and under other conditions less,while in all cases the nematic is qualitatively less or- dered. Reentrant phases are also encountered in type-II superconductors.24 For example,when the temperature is lowered at constant magnetic field,some superconductors pass from the "vortex liquid"phase to the "vortex glass" phase and then back to the vortex liquid''phase.The usual and reentrant instances of the vortex liquid are the same phase,as can be demonstrated by making them merge by changing both the temperature and the magnetic field. A third example of a reentrant phase appears in the phase diagram of the mixture of water and nicotine.25 For a wide range of mixing ratios,this mixture is a homogeneous solu- tion at high temperatures,segregates into water-rich and nicotine-rich phases at moderate temperatures,yet becomes homogeneous again at low temperatures.At mixing ratios closer to pure water or pure nicotine,the mixture is homo- geneous at all temperatures.Thus the high-temperature and Fig.2.A lattice gas configuration generated by the program Toss/. reentrant homogeneous phases are in fact the same phase. An obvious question at this point is "Why do reentrant phases exist?"The answer is "I don't know."I don't know classes,two pools,of microstates,and that I used my two why water has the phases that it does either.Nature displays computer programs to select one member from each of those an extraordinary variety of phase behaviors,26 all of which two pools.The selection was done at random,so the selected come about through maximizing the entropy of an isolated configuration should be considered typical.Thus my ques- system,and none of which can (at present)be calculated tion should have been "Which configuration was drawn from first principles.We can only grope at why some phase from the larger pool?,orWhich configuration is typical diagrams are relatively simple and others are mind- of the larger class?".Given this corrected question,you numbingly complex.Such gropings are exciting and impor- might want to go back and look at the configurations again. tant,but they do little to illuminate the qualitative character I have asked these questions of a number of people (both of entropy.That illumination comes instead through examin- students and professionals)and most of them guess that Fig. ing a simpler-rather than a more complex-model. 3 is typical of the class with larger entropy.That configura- tion is smoother,less clumpy.They look at the configuration in Fig.2 and see patterns,which suggests some orderly pro- IIL.ENTROPY AND THE LATTICE GAS The three examples above should caution us about relying on qualitative arguments concerning entropy.Here is another situation2to challenge your intuition:Figures 2 and 3 show 。 two configurations of 132=169 squares tossed down on an area that has 35x35=1225 empty spaces,each of which could hold a square.This system,called the "lattice gas model,"lacks some familiar features (e.g.,energy and tem- perature),yet it is nevertheless a legitimate and well-studied thermodynamic system.28 The two configurations shown were produced by two different computer programs(Toss/ and Toss)that used different rules to position the squares2 (The rules will be presented in due course;for the moment I shall reveal only that both rules employ random numbers.) Which configuration do you think has the greater entropy? Be sure to look at the configurations,ponder,and make a guess(no matter how ill-informed)before reading on. Before analyzing these pictures,I must first confess that my question was very misleading.I asked "Which configu- ration has the greater entropy?,"but entropy is not defined in terms of a single configuration (a single microstate).Instead, entropy is defined for a macrostate,and is related to the number of microstates that the system can take on and still be classified in that same macrostate.Instead of asking the question I did,I should have pointed out that I had two Fig.3.A lattice gas configuration generated by the program Toss2. 1092 Am.J.Phys.,Vol.68,No.12,December 2000 Daniel F.Styer 1092

entrant nematic phases—you can move continuously from one to the other in the temperature–composition phase dia￾gram. The implication of reentrant phases23 for entropy is pro￾found: Under some conditions the nematic phase has more entropy than the smectic phase and under other conditions less, while in all cases the nematic is qualitatively less or￾dered. Reentrant phases are also encountered in type-II superconductors.24 For example, when the temperature is lowered at constant magnetic field, some superconductors pass from the ‘‘vortex liquid’’ phase to the ‘‘vortex glass’’ phase and then back to the ‘‘vortex liquid’’ phase. The usual and reentrant instances of the vortex liquid are the same phase, as can be demonstrated by making them merge by changing both the temperature and the magnetic field. A third example of a reentrant phase appears in the phase diagram of the mixture of water and nicotine.25 For a wide range of mixing ratios, this mixture is a homogeneous solu￾tion at high temperatures, segregates into water-rich and nicotine-rich phases at moderate temperatures, yet becomes homogeneous again at low temperatures. At mixing ratios closer to pure water or pure nicotine, the mixture is homo￾geneous at all temperatures. Thus the high-temperature and reentrant homogeneous phases are in fact the same phase. An obvious question at this point is ‘‘Why do reentrant phases exist?’’ The answer is ‘‘I don’t know.’’ I don’t know why water has the phases that it does either. Nature displays an extraordinary variety of phase behaviors,26 all of which come about through maximizing the entropy of an isolated system, and none of which can ~at present! be calculated from first principles. We can only grope at why some phase diagrams are relatively simple and others are mind￾numbingly complex. Such gropings are exciting and impor￾tant, but they do little to illuminate the qualitative character of entropy. That illumination comes instead through examin￾ing a simpler—rather than a more complex—model. III. ENTROPY AND THE LATTICE GAS The three examples above should caution us about relying on qualitative arguments concerning entropy. Here is another situation27 to challenge your intuition: Figures 2 and 3 show two configurations of 1325169 squares tossed down on an area that has 3533551225 empty spaces, each of which could hold a square. This system, called the ‘‘lattice gas model,’’ lacks some familiar features ~e.g., energy and tem￾perature!, yet it is nevertheless a legitimate and well-studied thermodynamic system.28 The two configurations shown were produced by two different computer programs ~Toss1 and Toss2! that used different rules to position the squares.29 ~The rules will be presented in due course; for the moment I shall reveal only that both rules employ random numbers.! Which configuration do you think has the greater entropy? Be sure to look at the configurations, ponder, and make a guess ~no matter how ill-informed! before reading on. Before analyzing these pictures, I must first confess that my question was very misleading. I asked ‘‘Which configu￾ration has the greater entropy?,’’ but entropy is not defined in terms of a single configuration ~a single microstate!. Instead, entropy is defined for a macrostate, and is related to the number of microstates that the system can take on and still be classified in that same macrostate. Instead of asking the question I did, I should have pointed out that I had two classes, two pools, of microstates, and that I used my two computer programs to select one member from each of those two pools. The selection was done at random, so the selected configuration should be considered typical. Thus my ques￾tion should have been ‘‘Which configuration was drawn from the larger pool?’’, or ‘‘Which configuration is typical of the larger class?’’. Given this corrected question, you might want to go back and look at the configurations again. I have asked these questions of a number of people ~both students and professionals! and most of them guess that Fig. 3 is typical of the class with larger entropy. That configura￾tion is smoother, less clumpy. They look at the configuration in Fig. 2 and see patterns, which suggests some orderly pro￾Fig. 2. A lattice gas configuration generated by the program Toss1. Fig. 3. A lattice gas configuration generated by the program Toss2. 1092 Am. J. Phys., Vol. 68, No. 12, December 2000 Daniel F. Styer 1092

cess for producing that configuration,while the smoothness of the lower configuration suggests a random,disorderly construction process. Having posed the central question,let me end the suspense and tell you how the computer programs produced the two configurations.Figure 2 was produced by tossing 169 =(13)2squares down at random onto an area with 35x35 =1225 locations,subject only to the rule that two squares could not fall on the same location.Figure 3 was produced in exactly the same manner except that there was an additional ● rule,namely that two squares could not fall on adjacent lo- cations either.Thus Fig.2 was drawn from the pool of all patterns with 169 squares,while Fig.3 was drawn from the much smaller pool of patterns with 169 squares and with no two squares adjacent.The configuration of Fig.2 is typical of the class with more configurations and hence greater entropy.30 Look again at Fig.3.You will notice that there are no squares immediately adjacent to any given square.This "nearest neighbor exclusion''rule acts to spread out the squares,giving rise to the smooth appearance that tricks so many into guessing that the configuration of Fig.3 is typical of a class with high entropy. Now look again at Fig.2.You will notice holes and Fig.4.An orderly lattice gas configuration that is a member of both the large class and the small class of configurations. clumps of squares,the inhomogeneities that lead many to guess that it is typical of a small class.But in fact one should expect a random configuration to have holes-only a very exceptional configuration is perfectly smooth.31 This in- because this hand is a member of an enormous class of not- particularly-valuable poker hands.But the probability of be- volves the distinction between a typical configuration and an ing dealt this hand is exactly the same as the probability of average configuration.Typical configurations have holes: being dealt the roval flush of hearts.The reason that one some have holes in the upper right,some in the middle left, hand is memorable and the other is not has nothing to do some in the very center.Because the holes fall in various with the rarity of that particular hand;it has everything to do locations,the average configuration-the one produced by with the size of the class of which that hand is a member. adding all the configurations and dividing by the number of This powerful and graphic illustration of the importance of configurations-is smooth.The average configuration is ac- class rather than of individual configuration may be called tually atypical.(Analogy:A typical person is not of average s“the poker paradox.' height.A typical person is somewhat taller or somewhat shorter than average,and very few people are exactly of average height.Any clothing manufacturer that produced V.CONCLUSION only shirts of average size would quickly go bankrupt.)The It is often said that entropy measures the disorder of a presence of holes or clumps,therefore,need not be an indi- system.This qualitative concept has at least three failings: cation of a pattern or of a design.However,we humans tend First,it is vague.There is no precise definition of disorder. to find patterns wherever we look.even when no design is Some find the abstract paintings of Jackson Pollock to be present.In just this way the ancient Greeks looked into the disorderly;others find them pregnant with structure.Second, nighttime sky,with stars sprinkled about at random,and saw the animals,gods,and heroes that became our constellations. it uses an emotionally charged word.Most of us have feel- ings about disorder (either for it or against it),and the anal- ogy encourages us to transfer that like or dislike from disor- der,where our feelings are appropriate,to entropy,where IV.ENTROPY AND POKER they are not.The most important failing,however,is that the An excellent illustration of the nature of entropy is given analogy between entropy and disorder invites us to think about a single configuration rather than a class of configura- by the card game poker.There are many possible hands in poker,some valuable and most less so.For example,the tions.In the lattice gas model there are many "orderly' hand configurations (such as the checkerboard pattern of Fig.4) that are members of both classes.There are many other or- A,K9,Q9,J,10 derly'configurations (such as the solid block pattern of Fig is an example of a royal flush,the most powerful hand in 5)that are members only of the larger (higher entropy!) poker.There are only four royal flushes(the royal flush of class.32.33 The poker hand hearts,of diamonds,of spades,and of clubs)and any poker 2,4◆69,8◆，10* player who has ever been dealt a royal flush will remember it for the rest of his or her life. is very orderly,but a member of a very large class of nearly worthless poker hands. By contrast.no one can remember whether he or she has been dealt the hand Given the clear need for an intuition concerning entropy, and the appealing but unsatisfactory character of the simile 4◆，3◆，J,2◆.7◆ 'entropy as disorder,"what is to be done?I suggest an 1093 Am.J.Phys.,Vol.68,No.12,December 2000 Daniel F.Styer 1093

cess for producing that configuration, while the smoothness of the lower configuration suggests a random, disorderly construction process. Having posed the central question, let me end the suspense and tell you how the computer programs produced the two configurations. Figure 2 was produced by tossing 169 5(13)2 squares down at random onto an area with 35335 51225 locations, subject only to the rule that two squares could not fall on the same location. Figure 3 was produced in exactly the same manner except that there was an additional rule, namely that two squares could not fall on adjacent lo￾cations either. Thus Fig. 2 was drawn from the pool of all patterns with 169 squares, while Fig. 3 was drawn from the much smaller pool of patterns with 169 squares and with no two squares adjacent. The configuration of Fig. 2 is typical of the class with more configurations and hence greater entropy.30 Look again at Fig. 3. You will notice that there are no squares immediately adjacent to any given square. This ‘‘nearest neighbor exclusion’’ rule acts to spread out the squares, giving rise to the smooth appearance that tricks so many into guessing that the configuration of Fig. 3 is typical of a class with high entropy. Now look again at Fig. 2. You will notice holes and clumps of squares, the inhomogeneities that lead many to guess that it is typical of a small class. But in fact one should expect a random configuration to have holes—only a very exceptional configuration is perfectly smooth.31 This in￾volves the distinction between a typical configuration and an average configuration. Typical configurations have holes: some have holes in the upper right, some in the middle left, some in the very center. Because the holes fall in various locations, the average configuration—the one produced by adding all the configurations and dividing by the number of configurations—is smooth. The average configuration is ac￾tually atypical. ~Analogy: A typical person is not of average height. A typical person is somewhat taller or somewhat shorter than average, and very few people are exactly of average height. Any clothing manufacturer that produced only shirts of average size would quickly go bankrupt.! The presence of holes or clumps, therefore, need not be an indi￾cation of a pattern or of a design. However, we humans tend to find patterns wherever we look, even when no design is present. In just this way the ancient Greeks looked into the nighttime sky, with stars sprinkled about at random, and saw the animals, gods, and heroes that became our constellations. IV. ENTROPY AND POKER An excellent illustration of the nature of entropy is given by the card game poker. There are many possible hands in poker, some valuable and most less so. For example, the hand Ak,Kk,Qk,Jk,10k is an example of a royal flush, the most powerful hand in poker. There are only four royal flushes ~the royal flush of hearts, of diamonds, of spades, and of clubs! and any poker player who has ever been dealt a royal flush will remember it for the rest of his or her life. By contrast, no one can remember whether he or she has been dealt the hand 4l,3l,Jk,2;,7l because this hand is a member of an enormous class of not￾particularly-valuable poker hands. But the probability of be￾ing dealt this hand is exactly the same as the probability of being dealt the royal flush of hearts. The reason that one hand is memorable and the other is not has nothing to do with the rarity of that particular hand; it has everything to do with the size of the class of which that hand is a member. This powerful and graphic illustration of the importance of class rather than of individual configuration may be called ‘‘the poker paradox.’’ V. CONCLUSION It is often said that entropy measures the disorder of a system. This qualitative concept has at least three failings: First, it is vague. There is no precise definition of disorder. Some find the abstract paintings of Jackson Pollock to be disorderly; others find them pregnant with structure. Second, it uses an emotionally charged word. Most of us have feel￾ings about disorder ~either for it or against it!, and the anal￾ogy encourages us to transfer that like or dislike from disor￾der, where our feelings are appropriate, to entropy, where they are not. The most important failing, however, is that the analogy between entropy and disorder invites us to think about a single configuration rather than a class of configura￾tions. In the lattice gas model there are many ‘‘orderly’’ configurations ~such as the checkerboard pattern of Fig. 4! that are members of both classes. There are many other ‘‘or￾derly’’ configurations ~such as the solid block pattern of Fig. 5! that are members only of the larger ~higher entropy!! class.32,33 The poker hand 2',4l,6k,8;,10' is very orderly, but a member of a very large class of nearly worthless poker hands. Given the clear need for an intuition concerning entropy, and the appealing but unsatisfactory character of the simile ‘‘entropy as disorder,’’ what is to be done? I suggest an Fig. 4. An orderly lattice gas configuration that is a member of both the large class and the small class of configurations. 1093 Am. J. Phys., Vol. 68, No. 12, December 2000 Daniel F. Styer 1093

microscopically disordered''directs attention avay from glib emotional baggage and toward the perspective of"more en- tropy means more microstates. "The unexamined life is not worth living,"claims Socrates.Similarly,the unexamined analogy is as likely to mislead as to edify.Using two different analogies.which superficially appear in opposition,can help insure that both ■■■■■■■■■■■■ ■■■■■■■■■■■■■ analogies are used critically and thoughtfully. ■■■■■■■■■■■ ACKNOWLEDGMENTS I appreciate detailed and probing comments from Profes- ■■■ sors Norman Craig,Harvey Gould,Daniel Schroeder,and ■■■■■■■■■■■■ ■■■■■■■■■■■■ Jan Tobochnik.A conscientious and persistent referee posed ■■■ excellent challenges that led to improvements in this paper. APPENDIX:GENERAL MATHEMATICAL DEFINITION OF ENTROPY The formal statistical-mechanical definition of entropy, valid for any equilibrium ensemble,is Fig.5.An orderly lattice gas configuration that is a member of only the S=-kB∑PmInpm, (A1) large (high entropy!)class of configurations. where kg is Boltzmann's constant,the sum runs over all the microstates m of the system,and pm is the probability that microstate m is occupied in the given ensemble.34 The en- additional simile,namely"entropy as freedom,"which is to tropy is a function of the system's macrostate(also called its be used not by itself but only in conjunction with"entropy "thermodynamic state'':for example,T,V,and N for the as disorder.' canonical ensemble or E.V.and N for the microcanonical "Freedom"'means a range of possible actions,while "'en- ensemble).The microstate probabilities p are of course also tropy'means a range of possible microstates.If only one functions of that macrostate microstate corresponds to a certain macrostate,then the sys- Note that the function -Inx is monotonically decreasing tem has no freedom to choose its microstate-and it has zero and that entropy.If you are free to manage your own bedroom,then you may keep it either neat or messy,just as high entropy S=-kB(nPm）where∑Pm=l (A2) macrostates encompass both orderly and disorderly mi- crostates.The entropy gives the number of ways that the These three facts together give us a good picture of the rela- constituents of a system can be arranged and still be a mem- tion between the occupation probabilities pm and the en- ber of the club (or class).If the class entropy is high,then there are a number of different ways to satisfy the class tropy.The largest value of pm corresponds to the smallest membership criteria.If the class entropy is low,then that value of-Inpm,and because p is large this small value is class is very demanding-very restrictive-about which mi- weighed heavily when forming the average -(Inp) crostates it will admit as members.In short,the advantage of Clearly,a small entropy comes from having a few mi- the "'entropy as freedom''analogy is that it focuses attention crostates with high occupation probabilities while all the rest on the variety of microstates corresponding to a macrostate have small occupation probabilities.(Indeed,the minimum whereas the "entropy as disorder''analogy invites focus on value S=0 comes whenever one microstate has pm=1 and a single microstate. all the rest have pm=0.)Just as clearly,a large entropy While"entropy as freedom''has these benefits,it also has comes when the occupation probability is nearly equally two of the drawbacks of "entropy as disorder."First,the shared among the several possible microstates.(Indeed,us- term"freedom''is laden with even more emotional baggage ing Lagrange multipliers it is a trivial matter to show that the than the term "disorder."'Second,it is even more vague; maximum value of S comes when all the occupation prob- political movements from the far right through the center to abilities are equal.)In short,the entropy of a given mac- the extreme left all characterize themselves as "freedom rostate measures the extent to which the occupation prob- fighters."Is there any way to reap the benefits of this anal- abilities pm are "spread around'among the various ogy without sinking into the mire of drawbacks? microstates corresponding to the macrostate. For maximum pedagogical advantage,I suggest using The significance of the "spread around''quality is most both of these analogies.The emotions and vaguenesses at- clearly seen in the microcanonical ensemble,as described in tached to freedom''are very different from those attached Sec.I.(In an undergraduate course,I feel that the general to"'disorder,"so using them together tends to cancel out the argument of this Appendix should not be used,and only the emotion.A simple sentence like "For macrostates of high microcanonical-specific argument of Sec.I presented.)In entropy,the system has the freedom to chose one of a large this ensemble,microstates are either "in'(pm=aconstant) number of microstates,and the bulk of such microstates are if they correspond to the given macrostate or else they're 1094 Am.J.Phys.,Vol.68,No.12,December 2000 Daniel F.Styer 1094

additional simile, namely ‘‘entropy as freedom,’’ which is to be used not by itself but only in conjunction with ‘‘entropy as disorder.’’ ‘‘Freedom’’ means a range of possible actions, while ‘‘en￾tropy’’ means a range of possible microstates. If only one microstate corresponds to a certain macrostate, then the sys￾tem has no freedom to choose its microstate—and it has zero entropy. If you are free to manage your own bedroom, then you may keep it either neat or messy, just as high entropy macrostates encompass both orderly and disorderly mi￾crostates. The entropy gives the number of ways that the constituents of a system can be arranged and still be a mem￾ber of the club ~or class!. If the class entropy is high, then there are a number of different ways to satisfy the class membership criteria. If the class entropy is low, then that class is very demanding—very restrictive—about which mi￾crostates it will admit as members. In short, the advantage of the ‘‘entropy as freedom’’ analogy is that it focuses attention on the variety of microstates corresponding to a macrostate whereas the ‘‘entropy as disorder’’ analogy invites focus on a single microstate. While ‘‘entropy as freedom’’ has these benefits, it also has two of the drawbacks of ‘‘entropy as disorder.’’ First, the term ‘‘freedom’’ is laden with even more emotional baggage than the term ‘‘disorder.’’ Second, it is even more vague; political movements from the far right through the center to the extreme left all characterize themselves as ‘‘freedom fighters.’’ Is there any way to reap the benefits of this anal￾ogy without sinking into the mire of drawbacks? For maximum pedagogical advantage, I suggest using both of these analogies. The emotions and vaguenesses at￾tached to ‘‘freedom’’ are very different from those attached to ‘‘disorder,’’ so using them together tends to cancel out the emotion. A simple sentence like ‘‘For macrostates of high entropy, the system has the freedom to chose one of a large number of microstates, and the bulk of such microstates are microscopically disordered’’ directs attention away from glib emotional baggage and toward the perspective of ‘‘more en￾tropy means more microstates.’’ ‘‘The unexamined life is not worth living,’’ claims Socrates. Similarly, the unexamined analogy is as likely to mislead as to edify. Using two different analogies, which superficially appear in opposition, can help insure that both analogies are used critically and thoughtfully. ACKNOWLEDGMENTS I appreciate detailed and probing comments from Profes￾sors Norman Craig, Harvey Gould, Daniel Schroeder, and Jan Tobochnik. A conscientious and persistent referee posed excellent challenges that led to improvements in this paper. APPENDIX: GENERAL MATHEMATICAL DEFINITION OF ENTROPY The formal statistical-mechanical definition of entropy, valid for any equilibrium ensemble, is S52kB( m pm ln pm , ~A1! where kB is Boltzmann’s constant, the sum runs over all the microstates m of the system, and pm is the probability that microstate m is occupied in the given ensemble.34 The en￾tropy is a function of the system’s macrostate ~also called its ‘‘thermodynamic state’’: for example, T, V, and N for the canonical ensemble or E, V, and N for the microcanonical ensemble!. The microstate probabilities pm are of course also functions of that macrostate. Note that the function 2ln x is monotonically decreasing and that S52kB^ln pm& where ( m pm51. ~A2! These three facts together give us a good picture of the rela￾tion between the occupation probabilities pm and the en￾tropy. The largest value of pm corresponds to the smallest value of 2ln pm , and because pm is large this small value is weighed heavily when forming the average 2^ln pm&. Clearly, a small entropy comes from having a few mi￾crostates with high occupation probabilities while all the rest have small occupation probabilities. ~Indeed, the minimum value S50 comes whenever one microstate has pm51 and all the rest have pm50.! Just as clearly, a large entropy comes when the occupation probability is nearly equally shared among the several possible microstates. ~Indeed, us￾ing Lagrange multipliers it is a trivial matter to show that the maximum value of S comes when all the occupation prob￾abilities are equal.! In short, the entropy of a given mac￾rostate measures the extent to which the occupation prob￾abilities pm are ‘‘spread around’’ among the various microstates corresponding to the macrostate. The significance of the ‘‘spread around’’ quality is most clearly seen in the microcanonical ensemble, as described in Sec. I. ~In an undergraduate course, I feel that the general argument of this Appendix should not be used, and only the microcanonical-specific argument of Sec. I presented.! In this ensemble, microstates are either ‘‘in’’ (pm5a constant) if they correspond to the given macrostate or else they’re Fig. 5. An orderly lattice gas configuration that is a member of only the large ~high entropy!! class of configurations. 1094 Am. J. Phys., Vol. 68, No. 12, December 2000 Daniel F. Styer 1094

"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

probable microstate.In specific,even at a temperature of 1 000 000 K.a about entropy,but it holds only for the microcanonical ensemble.The sample of helium is more likely to be in a particular crystalline microstate definition Eq.(Al)is harder to understand but is also more general,ap- than in any particular plasma microstate.However,there are so many plying to any ensemble.The two definitions are logically equivalent.See, more plasma than crystalline microstates that(in the thermodynamic limit) for example,Richard E.Wilde and Surjit Singh,Statistical Mechanics: the sample occupies a plasma macrostate with probability 1. Fundamentals and Modern Applications (Wiley,New York,1998).Sec. The definition of entropy in Eq.(1)is the best starting place for teaching 1.6.1. COMPUTERS IN THE CLASSROOM I've paid my dues in physics and astronomy,and in those basic sciences,computers have nothing to do with learning. No computer can help someone understand the meaning of a wave function,angular momen- tum,or the relativistic-twins paradox.Software can simulate these on a glass screen,but these simplifications depend on someone else's understanding,which may be quite limited. Up and down the line,computer programs feed us someone else's logic,instead of encouraging us to develop our own.When confronted by a quandary,we're fed someone else's rubric rather than creating our own assaults on the problem. Clifford Stoll,Silicon Snake Oil (Doubleday,New York,1995),p.121. 1096 Am.J.Phys.,Vol.68,No.12,December 2000 Daniel F.Styer 1096

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. 34The 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, ap￾plying 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. COMPUTERS IN THE CLASSROOM I’ve paid my dues in physics and astronomy, and in those basic sciences, computers have nothing to do with learning. No computer can help someone understand the meaning of a wave function, angular momen￾tum, or the relativistic-twins paradox. Software can simulate these on a glass screen, but these simplifications depend on someone else’s understanding, which may be quite limited. Up and down the line, computer programs feed us someone else’s logic, instead of encouraging us to develop our own. When confronted by a quandary, we’re fed someone else’s rubric rather than creating our own assaults on the problem. Clifford Stoll, Silicon Snake Oil ~Doubleday, New York, 1995!, p. 121. 1096 Am. J. Phys., Vol. 68, No. 12, December 2000 Daniel F. Styer 1096