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 2␲mev2/3N5/3 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 2␲mev2/3 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 ͑2␲mE͒3N/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 2␲mEV2/3 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 2␲mev2/3 5 3 3 Ϸ ͩ ͪ ϩ ͫ Ϫ ͩ ͪͬ N ln 2 N ln so kB 2 h 2 2 2 S 3 2␲mEV2/3 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.