What good is the thermodynamic limit?

5 Pages·0.169 MB·English

by  Styer D.F.

What good is the thermodynamic limit?
Daniel F. Styera)
DepartmentofPhysicsandAstronomy,OberlinCollege,Oberlin,Ohio44074
~Received 9 February 2000; accepted 27August 2003!
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
widelyusedmicrocanonical‘‘thinphasespacelimit’’mustbetakenaftertakingthethermodynamic
limit. © 2004AmericanAssociationofPhysicsTeachers.
@DOI:10.1119/1.1621028#
Statistical mechanics is the study of matter in bulk. Kroemer,2Baierlein,3andSchroeder,4onlythelastonemen-
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 6.0231023-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-
steadoftryingtotracealltheparticletrajectoriesforalltime,
mal treatment by Ruelle.7
statistical mechanics is content to ask, for example, how the
mean energy varies with temperature and pressure. Second,
statistical mechanics turns the difficulty of bigness into a
I. THE LIMIT OFTHIN PHASE SPACE
blessing by insisting on treating only very large systems, in
which many of the details of system size fade into insignifi-
Hereisastoryofstatisticalmechanicsinthemicrocanoni-
cance. The formal, mathematical term for this bigness con-
cal ensemble: The system consists of N identical, classical
dition is ‘‘the thermodynamic limit.’’ particles~perhapsinteracting,perhapsindependent!confined
For example, in a fluid system specified by the tempera-
to a container of volume V. The energy is known to have
tureT, volumeV, andparticlenumberN, statisticalmechan- some value between E and E1DE. 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,DE,V,N), and the entropy is given by
V!‘ and N!‘ in such a way that their ratio r5N/V
approachesafinitequantity.Inthisso-calledthermodynamic S~E,V,N!5kBln$W~E,DE,V,N!/h3NN!%, ~1!
limit, the free energy F itself grows to infinity, but the free whereh isPlanck’sconstant.TheuseofanenergyrangeDE
energy per particle fN(T,r)5F(T,V,N)/N is expected to is,ofcourse,nothingbutamathematicalconvenience.Atthe
approach a finite value. end of any calculation, we will take the ‘‘thin phase space
Alternatively, in the microcanonical ensemble, this fluid limit,’’namely DE!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 DE!0, the phase-space volume W vanishes, so
S(E,V,N), and one expects that a doubling of E, V, and N S5k ln(0)52‘. This holds true for any system, whether
B
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
bulksystemshavesomanyparticlesthattheycanbeconsid- 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
increases, the free energy density f 5F/N approaches the within the box, the position-space dimensions contribute a
N
limiting value f‘, and typically the difference between f1023 factor of VN to W. The total energy is
and f‘ is smaller than experimental error. 1
Undergraduate statistical mechanics texts usually have ~p2 1p2 1p2 1 1p2 1p2 1p2 !, ~2!
2m 1,x 1,y 1,z fl 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 tionimpliesthattheaccessibleregioninmomentumspaceis
25 Am.J.Phys.72~1!,January2004 http://aapt.org/ajp ©2004AmericanAssociationofPhysicsTeachers 25
S D S D
A
a spherical shell with inner radius 2mE and outer radius S 3 2pmev2/3N5/3 3
A2m(E1DE). The volume of a d-dimensional sphere is8 k 52Nln h2 2lnN!2ln 2N !
B FS D G
pd/2 de 3N/2
Vd~r!5~d/2!!rd, ~3! 1ln 11 e 21
S D
so the volume of this 3N-dimensional shell is 3 2pmev2/3
5 Nln
p3N/2 2 h2
@~2m~E1DE!!3N/22~2mE!3N/2#, ~4! S D FS D G
~3N/2!! 3 3 de 3N/2
1 NlnN5/32lnN!2ln N !1ln 11 21 .
whence 2 2 e
FS D G
~2pmE!3N/2 DE 3N/2 ~10!
W~E,DE,V,N!5VN 11 21 .
~3N/2!! E Now use Stirling’s approximation,
~5!
lnn!’nlnn2n for n@1, ~11!
~I have rearranged the expressions to make the dimensions
moreapparent:Thequantityinsquarebracketsisdimension- to simplify the expressions like ln(3N)! above. The middle
2
less.! three terms are approximately ~in an approximation that be-
The entropy follows immediately. It is comes exact as N!‘)
H J
S5k ln W 32NlnN5/32lnN!2ln~23N!!’25NlnN2NlnN1N
B N!h3N
HS D 2~3N!ln~3N!13N
2pmEV2/3 3N/2 1 2 2 2
5k ln
B h2 N!~3N/2!! 5N@523ln~3!#. ~12!
FS D GJ 2 2 2
DE 3N/2 After this manipulation, we have
3 11 21 , ~6! S D F S DG
E S 3 2pmev2/3 5 3 3
’ Nln 1N 2 ln
so k 2 h2 2 2 2
S D S D B FS D G
S 3 2pmEV2/3 3 de 3N/2
5 Nln 2lnN!2ln N ! 1ln 11 21 . ~13!
k 2 h2 2 e
B FS D G
DE 3N/2 All the pieces grow linearly with N except for the right-
1ln 11 21 . ~7! most one—the piece related to DE! In this rightmost piece,
E
as N grows, the term (11de/e)3N/2 completely dominates 1
WhathappenswhenwetakethelimitDE!0?Asdemanded asN!‘ ~solongasde/e ispositive,nomatterhowsmall!.
by the paradox, the entropy approaches ln(0)52‘. Thus
FS D G S D
The problem with this result for the entropy is that it at-
de 3N/2 de 3N/2
temptstoholdforsystemsofanysize.Injustifyingthedefi- ln 11 21 ’ln 11
nition of the entropy one relies upon the assumption of a e e
S D
‘‘large’’ system, but in deriving the above expression we
3 de
never made use of that assumption. In fact, our expectation 5 Nln 11 . ~14!
that we could let DE!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 vanishes! when
but it holds exactly only for infinite systems, that is, for
de!0!
systems in the thermodynamic limit.
Thisspecialcase,theidealgas,illustratesthegeneralprin-
The thermodynamic limit, in the microcanonical case,
ciple that resolves our paradox: One must first take the ther-
consists of allowing the system’s particle number, volume,
modynamic limit N!‘, and only then take the ‘‘thin phase
energy, and energy spread all to grow without bound, but to
do so in such a way that the intensive ratios remain finite. space’’limit de[DE/N!0.
That is, we allow With this understanding in place, we find that the entropy
isfinitewhenDE!0,thatitdoesindeedgrowlinearlywith
N!‘ insuchawaythatV/N!v, E/N!e,
N, and that it is given by the well-known Sackur–Tetrode
and DE/N!de. ~8! formula F S D G
In this limit we expect that the entropy will grow linearly 3 4pmev2/3 5
S~E,V,N!5k N ln 1 . ~15!
with system size, that is, B 2 3h2 2
S~E,DE,V,N!!Ns~e,v,de!. ~9!
II. EXTENSIVITY
To prepare for taking the thermodynamic limit of Eq. ~7!,
write V as vN, E as eN, and DE as deN, so that the only Theexpression~7!fortheentropyofafinitesystemisnot
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,January2004 DanielF.Styer 26
TableI. Theentropyofafinitesystemisnotextensive.~The‘‘error’’col- the two functions E(T,V,N) produced through these two
umnliststhepercentagedifferencebetween,forexample,S20and2S10. If very different procedures will be exactly the same! Surely,
theentropywereextensive,alltheentriesinthiscolumnwouldbezero.!
this remarkable result requires explication.
Rather than provide a detailed proof, undergraduate texts
N S 2S Error
N N/2 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 23.8% even for interacting particles! are excruciatingly difficult.6,7
40 224.75 220.28 22.0% Yet any student trained to expect rigor will find this strategy
inadequate.
A reasonable approach is to discuss the equivalence of
ensembles,refertotheplausibilityargumentsinthetext,and
double S. Only the Sackur–Tetrode formula ~15!, which then prove the result in the ideal gas case by having the
holdsinthethermodynamiclimit,producesanexactlyexten- 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,
throughthefollowingconcreteillustration.Becauseweneed 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
bitrarily select k 51, E53N, V58N, DE50.3N, and obtained using the microcanonical ensemble.# The student
B
2pm/h251.Evaluatingexpression~7!forselectedvaluesof will find that the results of the two different procedures are
indeed identical, but only in the thermodynamic limit!
N producestheresultsinTableI.Clearly,thisexpressionfor
theentropyisnotextensive.~TheAppendixpresentsaprob-
lem that can help underscore this point for students.!
IV. PHASE TRANSITIONS
Iced tea, boiling water, and other aspects of two-phase
III. EQUIVALENCE OFENSEMBLES 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
characterizedbyE, V, andN—isacollectionofmicrostates discontinuity!intheentropyfunctionS(T). Howcansucha
G with energies H(G) ranging from E to E1DE, and with singularity appear? The Boltzmann factor e2H(G)/kBT is an
equal probability of encountering any such microstate. The analyticfunctionofT exceptatT50.ForT.0 thepartition
canonical ensemble—characterized by T, V, and N—is a function,
collection of microstates G with any possible energy, and
(
with probability Z~T!5 e2H(G)/kBT, ~19!
e2H(G)/kBT microstates G
~16!
Z~T,V,N! 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 G. ~Here Z is the partition func-
tion.! F~T!52kBTln~Z~T!!, ~20!
In addition, the microcanonical and canonical ensembles will have a singularity whenever Z50, but Z is never equal
are operationally quite distinct. For example, described be- to zero, so F(T) is likewise analytic for T.0. The entropy,
low are the two procedures for finding E(T,V,N).
Using the microcanonical ensemble, the procedure is to ]F
~1!findthenumberofaccessiblemicrostatesV(E,V,N), ~2! S~T!52]T, ~21!
calculate the thermodynamic entropy function S(E,V,N)
is of course analytic again. Thus there is no mechanism to
5k ln(V),~3!calculatethetemperaturethroughthethermo-
B produce a phase transition except at zero temperature.
dynamic relation
S D The analysis of the above paragraph is absolutely correct
1 ]S for finite systems. Phase transitions arise when an additional
T~E,V,N!5 ]E , ~17! mathematical step is introduced: the thermodynamic limit.
V,N Foranyfinitesystem,thecurveofS asafunctionofT might
and finally ~4! 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 ~1! discussed in the lectures by Fisher.10 A specific example,
find the partition function Z(T,V,N), ~2! calculate the ther- with graphs, is given in Ref. 11.
modynamic free energy function F(T,V,N)52k Tln(Z),
B
and then ~3! use the Gibbs–Helmholtz relation
S D S D V. DENSITYOFLEVELS IN k-SPACE
]~F/T! ]lnZ
E~T,V,N!5 52 ~18!
]~1/T! ]b Soonerorlater,everystatisticalmechanicstextintroduces
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 V5L3, 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,January2004 DanielF.Styer 27
2p 2000. During the 13 years that Professor Romer edited this
k5 ~n ,n ,n !, where n 50,61,62,.... ~22! 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-
Itisclearthatasumisnotanintegral,sowhycanweget 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 2p/L. In the thermody- hishelpnotonlyinthispaper,butalsoinalltheotherpapers
namic limit, L!‘ so the allowed wave vectors become that I have published in AJP. In grateful acknowledgment I
denselypackedink space.Inallbuttherarestsituations~see dedicate this paper to him.
thenextsection!thislimitpermitsthesumstobereplacedby
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 m(T,V,N) is determined ~1!Theextensivityofentropy:Fortheclassicalmonatomic
by demanding that ideal gas, plot the entropy as a function of particle number
usingboththe‘‘finitesize’’form~7!andtheSackur–Tetrode
N5( 1 , ~23! form ~15!. All other things being equal, is the thermody-
r e(er2m)/kBT21 namiclimitapproachedmorerapidlyforatomsofhighmass
or for atoms of low mass?
where the one-particle level r has energy eigenvalue er. To ~2! Equivalence of canonical and microcanonical en-
carry out this process, it is convenient to approximate this
sembles:Fortheclassicalmonatomicidealgas,thecanonical
sum by an integral which, with full consideration for degen-
partition function is
eracy, turns out to be S D
FA2m3GE‘ AE Z~T,V,N!5VN 2pmkBT 3N/2. ~A1!
N5V dE. ~24! N! h2
2p2\3 0 e(E2m)/kBT21
CarryouttheproceduredescribedinSec.IIItoshowthatthe
Theintegralontherightcannotbeevaluatedinclosedform, energy calculated in the canonical ensemble is
but one can establish an upper bound, namely
F G E53Nk T. ~A2!
2pmk T 3/2 2 B
N,V B ~2.612...!. ~25! In addition, start with the ‘‘finite size’’ microcanonical en-
h2 tropy equation ~7! and carry out the microcanonical proce-
For a system with a given N and V, this condition will be dure describHed in SeSc. III toDfind FS D G J
violated when the temperature is low enough.This violation 3 de de (3N/2)21 de (3N/2) 21
marks the Bose condensation. E5 Nk T 12 11 11 21 .
2 B e e e
Anyone approaching this argument critically will find it
~A3!
extremelysuspect.‘‘Youapproximatedasumbyanintegral,
and when that approximation proved untenable, you should UnderwhatconditionsarethesetwoexpressionsforE iden-
havegonebacktoevaluatethesummoreaccurately.Instead, tical?
you threw your hands into the air and invented a phasetran-
sition!’’Indeed,arguments15similartothoseofSec.IVshow a!Electronicmail:[email protected]
rigorously that the exact summation ~22! cannot admit a 1F.Reif,FundamentalsofStatisticalandThermalPhysics~McGraw-Hill,
phasetransition,suggestingthattheso-called‘‘Boseconden- NewYork,1965!.
2Charles Kittel and Herbert Kroemer, Thermal Physics, 2nd ed. ~W. H.
sation’’isnothingbutafailureoftheapproximationwithno
Freeman,NewYork,1980!.
physical significance.
3Ralph Baierlein, Thermal Physics ~Cambridge University Press, Cam-
The resolution of this paradox would require more pages bridge,UK,1999!.
than are in this paper.15,16 Suffice it to say that when the 4Daniel V. Schroeder, An Introduction to Thermal Physics ~Addison-
thermodynamiclimitistakenwithexquisitecare~anditisso Wesley,SanFrancisco,2000!.
taken in the papers cited!, then the crossover behavior at the 5ColinJ.Thompson,ClassicalEquilibriumStatisticalMechanics~Claren-
Bose condensation point is no mere breakdown in an ap- don,Oxford,UK,1988!,Chap.3.
proximation, but a true physical effect. ~The thermodynamic 6Michael E. Fisher, ‘‘The free energy of a macroscopic system,’’Arch.
Ration.Mech.Anal.17,377–410~1964!.
limit also plays an interesting, although less central, role in 7David Ruelle, Statistical Mechanics: Rigorous Results ~W.A. Benjamin,
Fermi–Diracstatistics.17AlsoofinterestistheBoseconden- Reading,MA,1969!.
sation of particles that are independent but not free.18! 8StandardMathematicalTablesandFormulae,editedbyDanielZwillinger
~CRCPress,BocaRaton,FL,1996!,Eq.~4.18.2!.
9See,forexample,Ref.1,pp.205–206and219–232.
ACKNOWLEDGMENTS
10MichaelE.Fisher,‘‘Thenatureofcriticalpoints,’’LecturesinTheoretical
Physics~UniversityofColoradoPress,Boulder,Colorado,1965!,Vol.7,
Arefereemadenumeroussuggestionsandposedexcellent
PartC,pp.1–159.SeeparticularlySecs.12and13.
questions that improved the quality of this paper. Professor
11Arthur E. Ferdinand and Michael E. Fisher, ‘‘Bounded and inhomoge-
Robert Romer, then editor of theAmerican Journal of Phys-
neousIsingmodels.I.Specific-heatanomalyofafinitelattice,’’Phys.Rev.
ics, suggested that I write a paper on this topic back in May 185,832–846~1969!.
28 Am.J.Phys.,Vol.72,No.1,January2004 DanielF.Styer 28
12SeeRef.1,pp.353–360;Ref.2,pp.72–73;Ref.3,pp.75–80;orRef.4, 16G.Scharf,‘‘OnBose–Einsteincondensation,’’Am.J.Phys.61,843–845
pp.251–255. ~1993!;WilliamJ.Mullin,‘‘Theloop-gasapproachtoBose–Einsteincon-
13ProfessorW.BruceRichardsoftheOberlinCollegePhysicsDepartment densationfortrappedparticles,’’ibid.68,120–128~2000!.
callsthese‘‘allowedwavevectors’’the‘‘specialks.’’ 17K.Scho¨nhammer,‘‘ThermodynamicsandoccupationnumbersofaFermi
14Fordetails,seeRef.2,pp.199–206;Ref.3,pp.199–209;orRef.4,pp. gasinthecanonicalensemble,’’Am.J.Phys.68,1032–1037~2000!.
315–319.
18Martin Ligare, ‘‘Numerical analysis of Bose–Einstein condensation in a
15RobertM.Ziff,GeorgeE.Uhlenbeck,andMarkKac,‘‘TheidealBose–
Einstein gas, revisited,’’Phys. Rep. 32, 169–248 ~1977!, especially pp. three-dimensional harmonic oscillator potential,’’Am. J. Phys. 66, 185–
172–173. 190~1998!.
Bottle BurstingApparatus. This apparatus at Denison University was bought from the Central Scientific Company of Chicago ca. 1905. In the 1927
catalogueitislistedat$3.25,includingtwoextra175ccbottles.Itdemonstratestheincompressibilityofwater.Thebottleisfilledcompletelywithwater,and
thetopisclampeddownfirmly.Thepistonrodisthenpusheddown,highlyincompressiblewaterpushesagainstthesidesoftheglassbottle,andthebottle
breaks.Today,safetyprecautionswouldbetakenwhendoingthedemonstration!~PhotographandnotesbyThomasB.Greenslade,Jr.,KenyonCollege!
29 Am.J.Phys.,Vol.72,No.1,January2004 DanielF.Styer 29