What good is the thermodynamic limit? Daniel F. Styera) Department of Physics and Astronomy, Oberlin College, Oberlin, Ohio 44074 ͑Received 9 February 2000; accepted 27 August 2003͒ Statistical mechanics applies to large systems: technically, its results are exact only for infinitely large systems in ‘‘the thermodynamic limit.’’ The importance of this proviso is often minimized in undergraduate courses. This paper presents six paradoxes in statistical mechanics that can be resolved only by acknowledging the thermodynamic limit. For example, it demonstrates that the widely used microcanonical ‘‘thin phase space limit’’ must be taken after taking the thermodynamic limit. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1621028͔
Statistical mechanics is the study of matter in bulk. Kroemer,2 Baierlein,3 and Schroeder,4 only the last one men- Whereas undergraduate courses in subjects like classical or tions the thermodynamic limit at all. This is a pity, because quantum mechanics are loath to approach the three-body any student trained to ask questions and to delve into the problem, statistical mechanics courses routinely deal with meanings behind equations will find confusing situations the 6.02ϫ1023-body problem. How can one subject be so in statistical mechanics, and these confusions will vanish generous with particle number while others are so parsimo- only when the thermodynamic limit is invoked. This paper nious? introduces six such cases and shows how the resulting para- The answer has two facets: First, statistical mechanics doxes are resolved in the thermodynamic limit. Those desir- asks different questions from, say, classical mechanics. In- ing a more systematic treatment of the thermodynamic limit should consult Thompson,5 or Fisher,6 or the definitive for- stead of trying to trace all the particle trajectories for all time, 7 statistical mechanics is content to ask, for example, how the mal treatment by Ruelle. mean energy varies with temperature and pressure. Second, statistical mechanics turns the difficulty of bigness into a blessing by insisting on treating only very large systems, in I. THE LIMIT OF THIN PHASE SPACE which many of the details of system size fade into insignifi- Here is a story of statistical mechanics in the microcanoni- cance. The formal, mathematical term for this bigness con- cal ensemble: The system consists of N identical, classical dition is ‘‘the thermodynamic limit.’’ particles ͑perhaps interacting, perhaps independent͒ confined For example, in a fluid system specified by the tempera- to a container of volume V. The energy is known to have ture T, volume V, and particle number N, statistical mechan- some value between E and Eϩ⌬E. The volume of phase ics can calculate the Helmholtz free energy F(T,V,N), but space corresponding to this energy range is called usually finds that function interesting only in the limit that W(E,⌬E,V,N), and the entropy is given by V→ϱ and N→ϱ in such a way that their ratio ϭN/V ͑ ͒ϭ ͕ ͑ ⌬ ͒ 3N ͖ ͑ ͒ approaches a finite quantity. In this so-called thermodynamic S E,V,N kB ln W E, E,V,N /h N! , 1 limit, the free energy F itself grows to infinity, but the free where h is Planck’s constant. The use of an energy range ⌬E ϭ energy per particle f N(T, ) F(T,V,N)/N is expected to is, of course, nothing but a mathematical convenience. At the approach a finite value. end of any calculation, we will take the ‘‘thin phase space Alternatively, in the microcanonical ensemble, this fluid limit,’’ namely ⌬E→0. system would be specified by energy E, volume V, and par- Really? Why wait for the end of the calculation? In the ticle number N. The function of interest is now the entropy limit ⌬E→0, the phase-space volume W vanishes, so ϭ ϭϪϱ S(E,V,N), and one expects that a doubling of E, V, and N S kB ln(0) . This holds true for any system, whether will result in a doubling of S. However, we shall soon see gas, liquid, or solid. We have just completed all possible that this expectation—the expectation of extensivity—is not statistical mechanical calculations, and found that the result exact for any finite system. The expectation holds with in- is necessarily trivial! creasing accuracy for larger and larger systems, but is ex- Obviously something is wrong with the above analysis, actly true only in the thermodynamic limit. but what? Rather than resolve the paradox in the most gen- Why should anyone care about results in the thermody- eral case, we look to the simplest special case, namely the namic limit, when every real system is finite? Because real monatomic ideal gas. For this system, phase space consists bulk systems have so many particles that they can be consid- of 3N position-space dimensions and 3N momentum-space ered to exist in the thermodynamic limit. As the system size dimensions. Because the particles are restricted to positions ϭ within the box, the position-space dimensions contribute a increases, the free energy density f N F/N approaches the factor of VN to W. The total energy is limiting value f ϱ , and typically the difference between f 1023 and f ϱ is smaller than experimental error. 1 Undergraduate statistical mechanics texts usually have ͑p2 ϩp2 ϩp2 ϩ ϩp2 ϩp2 ϩp2 ͒, ͑2͒ 2m 1,x 1,y 1,z ¯ N,x N,y N,z little to say directly about the thermodynamic limit. Instead of an explicit mention, they vaguely invoke a ‘‘large sys- where m is the mass of each particle, so the energy restric- tem.’’ Of the four well-known texts by Reif,1 Kittel and tion implies that the accessible region in momentum space is
25 Am. J. Phys. 72 ͑1͒, January 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers 25 a spherical shell with inner radius ͱ2mE and outer radius S 3 2mev2/3N5/3 3 8 ϭ ͩ ͪ Ϫ Ϫ ͩ ͪ ͱ2m(Eϩ⌬E). The volume of a d-dimensional sphere is N ln 2 ln N! ln N ! kB 2 h 2 d/2 ␦ 3N/2 ͑ ͒ϭ d ͑ ͒ e Vd r r , 3 ϩlnͫͩ 1ϩ ͪ Ϫ1ͬ ͑d/2͒! e so the volume of this 3N-dimensional shell is 3 2mev2/3 ϭ N lnͩ ͪ 3N/2 2 h2 ͓͑ ͑ ϩ⌬ ͒͒3N/2Ϫ͑ ͒3N/2͔ ͑ ͒ ͑ ͒ 2m E E 2mE , 4 3N/2 ! 3 3 ␦e 3N/2 ϩ N ln N5/3Ϫln N!Ϫlnͩ Nͪ !ϩlnͫͩ 1ϩ ͪ Ϫ1ͬ. whence 2 2 e ͑2mE͒3N/2 ⌬E 3N/2 ͑10͒ W͑E,⌬E,V,N͒ϭVN ͫͩ 1ϩ ͪ Ϫ1ͬ. ͑3N/2͒! E Now use Stirling’s approximation, ͑5͒ ln n!Ϸn ln nϪn for nӷ1, ͑11͒ ͑I have rearranged the expressions to make the dimensions 3 more apparent: The quantity in square brackets is dimension- to simplify the expressions like ln( 2N)! above. The middle less.͒ three terms are approximately ͑in an approximation that be- The entropy follows immediately. It is comes exact as N→ϱ) W 3 5/3Ϫ Ϫ ͑ 3 ͒ Ϸ 5 Ϫ ϩ Sϭk lnͭ ͮ 2N ln N ln N! ln 2N ! 2N ln N N ln N N B N!h3N Ϫ͑ 3N͒ln͑ 3N͒ϩ 3N 2mEV2/3 3N/2 1 2 2 2 ϭ ͭͩ ͪ kB ln 2 ͒ ϭ ͓ 5Ϫ 3 ͑ 3 ͔͒ ͑ ͒ h N!͑3N/2 ! N 2 2 ln 2 . 12 ⌬E 3N/2 After this manipulation, we have ϫͫͩ 1ϩ ͪ Ϫ1ͬͮ , ͑6͒ E S 3 2mev2/3 5 3 3 Ϸ ͩ ͪ ϩ ͫ Ϫ ͩ ͪͬ N ln 2 N ln so kB 2 h 2 2 2 S 3 2mEV2/3 3 ␦e 3N/2 ϭ ͩ ͪ Ϫ Ϫ ͩ ͪ ϩ ͫͩ ϩ ͪ Ϫ ͬ ͑ ͒ N ln 2 ln N! ln N ! ln 1 1 . 13 kB 2 h 2 e ⌬E 3N/2 All the pieces grow linearly with N except for the right- ϩlnͫͩ 1ϩ ͪ Ϫ1ͬ. ͑7͒ ⌬ E most one—the piece related to E! In this rightmost piece, as N grows, the term (1ϩ␦e/e)3N/2 completely dominates 1 ⌬ → What happens when we take the limit E 0? As demanded as N→ϱ͑so long as ␦e/e is positive, no matter how small͒. by the paradox, the entropy approaches ln(0)ϭϪϱ. Thus The problem with this result for the entropy is that it at- ␦ 3N/2 ␦ 3N/2 tempts to hold for systems of any size. In justifying the defi- e e lnͫͩ 1ϩ ͪ Ϫ1ͬϷlnͩ 1ϩ ͪ nition of the entropy one relies upon the assumption of a e e ‘‘large’’ system, but in deriving the above expression we 3 ␦e never made use of that assumption. In fact, our expectation ϭ N lnͩ 1ϩ ͪ . ͑14͒ that we could let ⌬E→0 and obtain a sensible result holds 2 e only approximately for finite systems: the expectation holds This term is not only linear with N in the thermodynamic to higher and higher accuracy for larger and larger systems, limit, it also has a well-behaved limit ͑it vanishes͒ when but it holds exactly only for infinite systems, that is, for ␦ → systems in the thermodynamic limit. e 0! The thermodynamic limit, in the microcanonical case, This special case, the ideal gas, illustrates the general prin- consists of allowing the system’s particle number, volume, ciple that resolves our paradox: One must first take the ther- →ϱ energy, and energy spread all to grow without bound, but to modynamic limit N , and only then take the ‘‘thin phase do so in such a way that the intensive ratios remain finite. space’’ limit ␦eϵ⌬E/N→0. That is, we allow With this understanding in place, we find that the entropy is finite when ⌬E→0, that it does indeed grow linearly with →ϱ → → N in such a way that V/N v, E/N e, N, and that it is given by the well-known Sackur–Tetrode and ⌬E/N →␦e. ͑8͒ formula 2/3 In this limit we expect that the entropy will grow linearly 3 4mev 5 S͑E,V,N͒ϭk Nͫ lnͩ ͪ ϩ ͬ. ͑15͒ with system size, that is, B 2 3h2 2 S͑E,⌬E,V,N͒→Ns͑e,v,␦e͒. ͑9͒ II. EXTENSIVITY To prepare for taking the thermodynamic limit of Eq. ͑7͒, write V as vN, E as eN, and ⌬E as ␦eN, so that the only The expression ͑7͒ for the entropy of a finite system is not extensive variable is N. This results in extensive: If you double E, V, and N, you will not exactly
26 Am. J. Phys., Vol. 72, No. 1, January 2004 Daniel F. Styer 26 Table I. The entropy of a finite system is not extensive. ͑The ‘‘error’’ col- the two functions E(T,V,N) produced through these two umn lists the percentage difference between, for example, S20 and 2S10 .If ͒ very different procedures will be exactly the same! Surely, the entropy were extensive, all the entries in this column would be zero. this remarkable result requires explication. Rather than provide a detailed proof, undergraduate texts NSN 2SN/2 Error typically give plausibility arguments.9 This is the correct 10 52.99 pedagogical choice, because the detailed proofs ͑which hold 20 110.14 105.98 Ϫ3.8% even for interacting particles͒ are excruciatingly difficult.6,7 Ϫ 40 224.75 220.28 2.0% Yet any student trained to expect rigor will find this strategy inadequate. A reasonable approach is to discuss the equivalence of ensembles, refer to the plausibility arguments in the text, and double S. Only the Sackur–Tetrode formula ͑15͒, which then prove the result in the ideal gas case by having the holds in the thermodynamic limit, produces an exactly exten- student actually execute both procedures. ͓Suggested word- sive entropy. The lack of extensivity can be demonstrated ing for this problem is given in the appendix. Alternatively, through the following concrete illustration. Because we need the student could calculate the entropy S(E,V,N) using the a purely mathematical result, we ignore dimensions and ar- canonical ensemble, and compare it to the result previously ϭ ϭ ϭ ⌬ ϭ obtained using the microcanonical ensemble.͔ The student bitrarily select kB 1, E 3N, V 8N, E 0.3N, and 2m/h2ϭ1. Evaluating expression ͑7͒ for selected values of will find that the results of the two different procedures are N produces the results in Table I. Clearly, this expression for indeed identical, but only in the thermodynamic limit! the entropy is not extensive. ͑The Appendix presents a prob- lem that can help underscore this point for students.͒ IV. PHASE TRANSITIONS Iced tea, boiling water, and other aspects of two-phase III. EQUIVALENCE OF ENSEMBLES coexistence are familiar features of daily life. Yet we will soon see that phase transitions do not exist at all in finite The microcanonical and canonical ensembles are concep- systems! They appear only in the thermodynamic limit. tually quite distinct. The microcanonical ensemble— A phase transition is marked by a singularity ͑usually a characterized by E, V, and N—is a collection of microstates discontinuity͒ in the entropy function S(T). How can such a Ϫ ⌫ ⌫ with energies H(⌫) ranging from E to Eϩ⌬E, and with singularity appear? The Boltzmann factor e H( )/kBT is an equal probability of encountering any such microstate. The analytic function of T except at Tϭ0. For TϾ0 the partition canonical ensemble—characterized by T, V, and N—is a function, collection of microstates ⌫ with any possible energy, and Ϫ ⌫ with probability Z͑T͒ϭ ͚ e H( )/kBT, ͑19͒ Ϫ ⌫ microstates ⌫ e H( )/kBT ͑16͒ Z͑T,V,N͒ is a sum of positive, analytic functions of T, so it is a posi- tive, analytic function of T. The free energy, of encountering microstate ⌫. ͑Here Z is the partition func- ͑ ͒ϭϪ ͑ ͑ ͒͒ ͑ ͒ tion.͒ F T kBT ln Z T , 20 In addition, the microcanonical and canonical ensembles will have a singularity whenever Zϭ0, but Z is never equal are operationally quite distinct. For example, described be- to zero, so F(T) is likewise analytic for TϾ0. The entropy, low are the two procedures for finding E(T,V,N). Fץ Using the microcanonical ensemble, the procedure is to S͑T͒ϭϪ , ͑21͒ Tץ find the number of accessible microstates ⍀(E,V,N), ͑2͒ 1͒͑ calculate the thermodynamic entropy function S(E,V,N) ϭ ⍀ ͑ ͒ is of course analytic again. Thus there is no mechanism to kB ln( ), 3 calculate the temperature through the thermo- produce a phase transition except at zero temperature. dynamic relation The analysis of the above paragraph is absolutely correct S for finite systems. Phase transitions arise when an additionalץ 1 ϭ ͩ ͪ , ͑17͒ .E mathematical step is introduced: the thermodynamic limitץ T͑E,V,N͒ V,N For any finite system, the curve of S as a function of T might and finally ͑4͒ solve this expression for E to produce the be very steep, but it is never discontinuous. The ‘‘growth of desired function E(T,V,N). a phase transition’’ as N approaches infinity is insightfully In contrast, using the canonical ensemble one must ͑1͒ discussed in the lectures by Fisher.10 A specific example, find the partition function Z(T,V,N), ͑2͒ calculate the ther- with graphs, is given in Ref. 11. ϭϪ modynamic free energy function F(T,V,N) kBT ln(Z), and then ͑3͒ use the Gibbs–Helmholtz relation V. DENSITY OF LEVELS IN k-SPACE ln Z ץ F/T͒͑ץ E͑T,V,N͒ϭ ͩ ͪ ϭϪ ͩ ͪ ͑18͒ Sooner or later, every statistical mechanics text introduces ץ T͒/1͑ץ V,N V,N the energy eigenstates of a single independent particle in a to produce the desired function E(T,V,N). cube of volume VϭL3, usually with periodic boundary Despite these vast conceptual and operational differences, conditions.12 These so-called levels are characterized by the principle of ‘‘equivalence of ensembles’’ guarantees that wave vectors13
27 Am. J. Phys., Vol. 72, No. 1, January 2004 Daniel F. Styer 27 2 2000. During the 13 years that Professor Romer edited this kϭ ͑n ,n ,n ͒, where n ϭ0,Ϯ1,Ϯ2,... . ͑22͒ journal, he has often suggested improvements in my L x y z i papers—ranging from deep physics issues to spelling And sooner or later, in the process of executing sums over corrections—but he never allowed me to acknowledge these these levels, every text replaces these sums with volume in- suggestions. ‘‘I’m just doing my job,’’ he would say, and tegrals in k-space. then use his power as editor to excise any sentence of ac- It is clear that a sum is not an integral, so why can we get knowledgment that I had included. Now that Professor away with this replacement? Because the distance between Romer is no longer editor, I am at last able to acknowledge adjacent ‘‘allowed wavevectors’’ is 2/L. In the thermody- his help not only in this paper, but also in all the other papers namic limit, L→ϱ so the allowed wave vectors become that I have published in AJP. In grateful acknowledgment I densely packed in k space. In all but the rarest situations ͑see dedicate this paper to him. the next section͒ this limit permits the sums to be replaced by integrals. APPENDIX VI. BOSE CONDENSATION Here are two problems that can be assigned to students to The argument for Bose condensation can be outlined as help drive home the ideas presented in this paper. follows:14 The chemical potential (T,V,N) is determined ͑1͒ The extensivity of entropy: For the classical monatomic by demanding that ideal gas, plot the entropy as a function of particle number using both the ‘‘finite size’’ form ͑7͒ and the Sackur–Tetrode 1 ϭ ͑ ͒ form ͑15͒. All other things being equal, is the thermody- N ͚ (⑀ Ϫ)/k T , 23 r e r B Ϫ1 namic limit approached more rapidly for atoms of high mass ⑀ or for atoms of low mass? where the one-particle level r has energy eigenvalue r .To ͑2͒ Equivalence of canonical and microcanonical en- carry out this process, it is convenient to approximate this sembles: For the classical monatomic ideal gas, the canonical sum by an integral which, with full consideration for degen- partition function is eracy, turns out to be N 3N/2 V 2 mkBT ͱ2m3 ϱ ͱE Z͑T,V,N͒ϭ ͩ ͪ . ͑A1͒ ϭ ͫ ͬ ͵ E ͑ ͒ N! h2 N V 2 3 (EϪ)/k T d . 24 2 ប 0 e B Ϫ1 Carry out the procedure described in Sec. III to show that the The integral on the right cannot be evaluated in closed form, energy calculated in the canonical ensemble is but one can establish an upper bound, namely ϭ 3 ͑ ͒ E 2NkBT. A2 2mk T 3/2 Ͻ ͫ B ͬ ͑ ͒ ͑ ͒ In addition, start with the ‘‘finite size’’ microcanonical en- N V 2 2.612... . 25 h tropy equation ͑7͒ and carry out the microcanonical proce- For a system with a given N and V, this condition will be dure described in Sec. III to find violated when the temperature is low enough. This violation 3 ␦e ␦e (3N/2)Ϫ1 ␦e (3N/2) Ϫ1 marks the Bose condensation. Eϭ Nk Tͭ1Ϫ ͩ 1ϩ ͪ ͫͩ 1ϩ ͪ Ϫ1ͬ ͮ . 2 B e e e Anyone approaching this argument critically will find it ͑ ͒ extremely suspect. ‘‘You approximated a sum by an integral, A3 and when that approximation proved untenable, you should Under what conditions are these two expressions for E iden- have gone back to evaluate the sum more accurately. Instead, tical? you threw your hands into the air and invented a phase tran- sition!’’ Indeed, arguments15 similar to those of Sec. IV show a͒Electronic mail: [email protected] rigorously that the exact summation ͑22͒ cannot admit a 1F. Reif, Fundamentals of Statistical and Thermal Physics ͑McGraw-Hill, phase transition, suggesting that the so-called ‘‘Bose conden- New York, 1965͒. 2 ͑ sation’’ is nothing but a failure of the approximation with no Charles Kittel and Herbert Kroemer, Thermal Physics, 2nd ed. W. H. Freeman, New York, 1980͒. physical significance. 3Ralph Baierlein, Thermal Physics ͑Cambridge University Press, Cam- The resolution of this paradox would require more pages bridge, UK, 1999͒. 15,16 than are in this paper. Suffice it to say that when the 4Daniel V. Schroeder, An Introduction to Thermal Physics ͑Addison- thermodynamic limit is taken with exquisite care ͑and it is so Wesley, San Francisco, 2000͒. taken in the papers cited͒, then the crossover behavior at the 5Colin J. Thompson, Classical Equilibrium Statistical Mechanics ͑Claren- don, Oxford, UK, 1988͒, Chap. 3. Bose condensation point is no mere breakdown in an ap- 6 ͑ Michael E. Fisher, ‘‘The free energy of a macroscopic system,’’ Arch. proximation, but a true physical effect. The thermodynamic Ration. Mech. Anal. 17, 377–410 ͑1964͒. limit also plays an interesting, although less central, role in 7David Ruelle, Statistical Mechanics: Rigorous Results ͑W. A. Benjamin, 17 Fermi–Dirac statistics. Also of interest is the Bose conden- Reading, MA, 1969͒. sation of particles that are independent but not free.18͒ 8Standard Mathematical Tables and Formulae, edited by Daniel Zwillinger ͑CRC Press, Boca Raton, FL, 1996͒,Eq.͑4.18.2͒. 9See, for example, Ref. 1, pp. 205–206 and 219–232. ACKNOWLEDGMENTS 10Michael E. Fisher, ‘‘The nature of critical points,’’ Lectures in Theoretical ͑ ͒ A referee made numerous suggestions and posed excellent Physics University of Colorado Press, Boulder, Colorado, 1965 , Vol. 7, Part C, pp. 1–159. See particularly Secs. 12 and 13. questions that improved the quality of this paper. Professor 11Arthur E. Ferdinand and Michael E. Fisher, ‘‘Bounded and inhomoge- Robert Romer, then editor of the American Journal of Phys- neous Ising models. I. Specific-heat anomaly of a finite lattice,’’ Phys. Rev. ics, suggested that I write a paper on this topic back in May 185, 832–846 ͑1969͒.
28 Am. J. Phys., Vol. 72, No. 1, January 2004 Daniel F. Styer 28 12See Ref. 1, pp. 353–360; Ref. 2, pp. 72–73; Ref. 3, pp. 75–80; or Ref. 4, 16G. Scharf, ‘‘On Bose–Einstein condensation,’’ Am. J. Phys. 61,843–845 pp. 251–255. ͑1993͒; William J. Mullin, ‘‘The loop-gas approach to Bose–Einstein con- 13 Professor W. Bruce Richards of the Oberlin College Physics Department densation for trapped particles,’’ ibid. 68, 120–128 ͑2000͒. calls these ‘‘allowed wavevectors’’ the ‘‘special ks.’’ 17 ¨ 14 K. Schonhammer, ‘‘Thermodynamics and occupation numbers of a Fermi For details, see Ref. 2, pp. 199–206; Ref. 3, pp. 199–209; or Ref. 4, pp. gas in the canonical ensemble,’’ Am. J. Phys. 68, 1032–1037 ͑2000͒. 315–319. 18Martin Ligare, ‘‘Numerical analysis of Bose–Einstein condensation in a 15Robert M. Ziff, George E. Uhlenbeck, and Mark Kac, ‘‘The ideal Bose– Einstein gas, revisited,’’ Phys. Rep. 32, 169–248 ͑1977͒, especially pp. three-dimensional harmonic oscillator potential,’’ Am. J. Phys. 66,185– ͑ ͒ 172–173. 190 1998 .
Bottle Bursting Apparatus. This apparatus at Denison University was bought from the Central Scientific Company of Chicago ca. 1905. In the 1927 catalogue it is listed at $3.25, including two extra 175 cc bottles. It demonstrates the incompressibility of water. The bottle is filled completely with water, and the top is clamped down firmly. The piston rod is then pushed down, highly incompressible water pushes against the sides of the glass bottle, and the bottle breaks. Today, safety precautions would be taken when doing the demonstration! ͑Photograph and notes by Thomas B. Greenslade, Jr., Kenyon College͒
29 Am. J. Phys., Vol. 72, No. 1, January 2004 Daniel F. Styer 29