On the dispute between Boltzmann and Gibbs entropy

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by  Pierfrancesco Buonsante

On the dispute between Boltzmann and Gibbs entropy
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Pierfrancesco Buonsante, Roberto Franzosi, Augusto Smerzi
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2 QSTAR & CNR - IstitutoNazionale di Ottica, Largo Enrico Fermi 2, I-50125 Firenze,
Italy.
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Abstract
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The validity of the concept of negative temperature has been recently chal-
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lenged by arguing that the Boltzmann entropy (that allows negative tempera-
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t tures)isinconsistentfromamathematicalandstatisticalpointofview,whereas
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s the Gibbs entropy (that does not admit negative temperatures) provides the
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a correctdefinitionforthemicrocanonicalentropy. Hereweprovethatthe Boltz-
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mannentropyis thermodynamicallyandmathematicallyconsistent. Analytical
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d results ontwo systems supporting negativetemperatures illustrate the scenario
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o wepropose. Inadditionwenumericallystudyalatticesystemtoshowthatneg-
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[ ativetemperatureequilibriumstatesareaccessibleandobeystandardstatistical

2 mechanics prediction.
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0 PACS numbers 05.20.-y,05.20.Gg, 05.30.-d,05.30.Ch
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0 Keywords: Statistical Mechanics, Microcanonical Ensemble
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6 1. Introduction
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v The concept of negative absolute temperature was1 invoked to explain the
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results ofexperiments with nuclear-spinsystems carriedout by Pound[2], Pur-
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a cellandPound[3]andRamseyandPound[4]. Shortly afterthese experiments,
Email address: [email protected] (RobertoFranzosi)
1This was not the first place where negative temperatures have been considered, in fact
[1] two years before proposed the existence of negative temperatures in order to explain the
formationoflargescalevorticesbyclusteringofsmallones inhydrodynamicsystems.
Preprint submitted toAnnals of Physics October25, 2016
Ramsey[5]discussedthethermodynamicimplicationsofnegativeabsolutetem-
perature and their meaning in statistical mechanics, thereby granting this con-
cept a well-grounded place in physics [6, 7].
Themicrocanonicalensemble,whichprovidesthestatisticaldescriptionofan
isolatedsystematequilibrium,isthemostappropriatevenuetodiscussnegative
temperatures. Inthisensemble,thethermodynamicquantities,liketemperature
andspecificheat,arederivedfromtheentropythroughsuitablethermodynamic
relations. Forinstance,theinversetemperatureisproportionaltothederivative
of the entropy with respect to the energy. In equilibrium statistical mechanics,
thereareatleasttwocommonlyaccepteddefinitionsofentropy: theBoltzmann
entropyisproportionaltothelogarithmofthenumberofmicrostatesinagiven
“energy shell”, whereas the Gibbs entropy is proportional to the logarithm of
the number of microstates up to a given energy. The debate as to which of
these definitions of entropy is the correctone has been going on for a long time
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17], although the generalconsensus is that they
arebasicallyinterchangeable. Infact,forstandardsystems2withalargenumber
of degrees of freedom they are practically equivalent [18]. Full equivalence is
obtained in the thermodynamic limit.
The existence of negative-temperature states provides a major bone of con-
tention in the debate. In fact, negative temperatures emerge in the Boltzmann
description whenever the number of microstates in a given energy shell is a
decreasing function of the relevant energy. On the contrary, the Gibbs tem-
perature can never be negative, since the number of microstates having energy
below a given value is always a non-decreasing function of such value. Thus,
systemsadmitting negative(Boltzmann)temperaturesprovideanidealcontext
to address the matter of the correct definition of entropy.
Recently [19, 20, 21] it was argued that for a broad class of physical sys-
tems,includingstandardclassicalHamiltoniansystems,onlytheGibbsentropy
2With“standardsystem”wemeanasystemwithunboundedenergyfromaboveforwhich
theenergygoestoinfinitywhenoneofthecanonical coordinatesgoes toinfinity.
2
yields a consistent thermodynamics, and that, consequently, negative tempera-
turesarenotachievablewithinastandardthermodynamicalframework. Inthis
respect, what is usually referred to Boltzmann temperature would not possess
the requiredpropertiesfor a temperature [19,22, 23,24, 20,25, 26]. These and
other relatedarguments [27, 28, 29] havebeen contended[30, 31, 32, 33, 34], in
what has become a lively debate.
In the present manuscript, at first we focus on the class of systems whose
canonical (or, possibly, grand canonical) ensemble is equivalent to the micro-
canonical ensemble. Thus we implicitly exclude non-extensive systems, and
systems at the first-order phase transitions. We show that such equivalence
can be rigorouslysatisfied only if the thermodynamics of the latter ensemble is
derived by the Boltzmann entropy. For such systems we show that the Boltz-
manntemperature providesaconsistentdescriptionwiththoseofthe canonical
and grandcanonical ensembles. Therefore we conclude that, also in the case of
isolated systems for which a comparisonbetween different statistical ensembles
is not possible, the Boltzmann entropy provides the correct description.
Later, we focus on a general system and we prove that the Boltzmann entropy
isthermostatisticallyconsistentanddoesnotviolateanyfundamentalcondition
for the microcanonical entropy.
The outline of the paper is the following. In Sec. 2 we summarize the
essential features of systems in which negative Boltzmann temperatures are
expected. In Sec. 3 we consider an isolated Hamiltonian system and, under
the hypothesis of ergodicity, we show that only for the Boltzmann entropy all
the thermodynamic quantities can be measured as time averages (along the
dynamics) of suitable functions. In Sec. 4 we prove that, for systems whose
canonicalandmicrocanonicalensemblesare equivalent,the thermodynamically
consistentdefinitionforthe temperatureis the onederivedwiththe Boltzmann
entropy. Furthermore we show that, in the thermodynamic limit, the Gibbs
and Boltzmann temperatures do coincide when the latter is positive whereas
theinverseGibbstemperatureisidenticallynullinthe regionwhereBoltzmann
provides negative values for the temperature. In Sec. 5 we recall the critique
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of consistency of Boltzmann entropy raised recently in literature and, in Sec.
6 we prove that the Boltzmann entropy is consistent from a mathematical and
thermodynamical point of view. In section 7, we give two examples of systems
supportingnegativeBoltzmanntemperaturesforwhichthe grand-canonical(or
canonical)ensembleandthemicrocanonicaldescriptiongivenbytheBoltzmann
entropy do agree on the whole parameter space and on the complete range of
values of the energy-density. We show that the equipartition theorem fails for
system with negative Boltzmann temperatures in Sec. 8. In Sec. 9 we show
throughnumericalsimulationsona specific system,thatnegativetemperatures
are accessible. We show that, irrespective of the sign of the temperature, a
large microcanonical lattice acts as a thermostat for a small grand canonical
sublattice, and this confirms the ensemble equivalence. Furthermore, we have
shownthat, irrespective of the sign of the temperature, two isolated systems at
equilibrium at different inverse temperatures, reach an equilibrium state at an
intermediate inverse temperature, after that they are brought in contact.
2. Negative temperatures
The microcanonicalensemble describes the equilibrium properties of aniso-
lated system, that is to say in which energy, and possibly further quantities,
are conserved. Within the microcanonical description, all the thermodynamic
quantities are derivedfrom the entropy,for instance the inverse temperature of
the system is defined as
1 ∂s
β = , (1)
k ∂ǫ
B
wherek istheBoltzmannconstantands(ǫ)istheentropydensitycorrespond-
B
ing to a given energy density ǫ. The two alternative definitions for the entropy
usedinequilibriumstatisticalmechanicsareascribedtoBoltzmannandGibbs3.
According to Boltzmann’s definition
s (ǫ)=L−1k ln(ω(ǫ)∆), (2)
B B
3WerefertoRef. [35]forhistoricaldetails.
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where ω(ǫ) is the density of microstates at a fixed value energy density ǫ and,
possibly,ata fixed value ofthe additionalconservedquantities, ∆ is a constant
withthe same dimensionasǫ, andL is the number ofdegreesoffreedomin the
system. The Gibbs entropy is
s (ǫ)=L−1k lnΩ(ǫ), (3)
G B
where Ω(ǫ) is the number of microstates with energy density less than or equal
to ǫ and, possibly, at a fixed value of the additional conserved quantities. It is
knownthat inthe thermodynamiclimit these twodefinitions ofentropyleadto
equivalentthermodynamic results in“standard”systems [36]. So far,these two
entropy definitions have been used in an alternative (interchangeable) way in
statistical mechanics, by resorting to the most suitable form depending on the
specificproblemconsidered. Thesetwoentropydefinitionsareconnectedbythe
relation between ω and Ω
∂Ω
ω(ǫ)= (ǫ), (4)
∂ǫ
Since ∂Ω/∂ǫ 0, Gibbs’ temperatures are not negative and consequently the
≥
two entropies have incompatible outcomes if applied to systems that admit
negative Boltzmann temperatures.
Anecessary(althoughnotsufficient)conditionforasystemforhavingnega-
tivetemperaturesistheboundednessoftheenergy(asinthecaseofnuclear-spin
systemsdiscussedby Poundet al), inthis casealocalmaximuminside the sys-
tem’s density energy interval of the Boltzmann entropy s (ǫ) is not forbidden
B
and, both positive and negative Boltzmann temperatures are possible.
Hamiltonians with bounded energies can also be characterized by the exis-
tence of more than one first integral of motion and, for this reason, in addition
to the statistical mechanics of systems with one first integral, we will consider
also the case of systems with more then one first integrals. Within the lat-
terclassforinstance therearemodels usuallyemployedfordescribingultracold
atoms. Thepossibilityofobservingnegativetemperaturestatesinultracoldsys-
tems,hasbeentheoreticallypredictedbysomeauthorswithdifferentapproaches
[37,38,39]and,theexperimentalevidenceoftheexistenceofstatesformotional
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degreesoffreedomofabosonicgasatnegative(Boltzmann)temperatures,have
been achieved a few years ago by Braun et al. [40]. The interpretation of such
experimentalresultshasbeencontestedin[19]. Successively[20,21]ithasbeen
argued that for a broad class of systems –that includes all “standard classical
Hamiltonian systems”– only the Gibbs entropy satisfies all three thermody-
namic laws exactly. These papers have engendered a glowing debate between
supportersofthe Gibbs entropy[23, 24,25, 26,22,20, 28,29, 21,34]andthose
consideringcorrecttheBoltzmannentropy[31,32,30,41,42,43,44,45,34,46].
3. Dynamics and statistical mechanics for classical systems
Let us consider first a generic classical many-particle system described by
an autonomous Hamiltonian H(x ,...,x ), in which the energy is the sole first
1 L
integral of motion. The Boltzmann entropy density s (ǫ) in this case is given
B
by
s (ǫ)=L−1k ln dLxδ(Lǫ H(x)), (5)
B B
−
Z
whereas the one of Gibbs is
s (ǫ)=L−1k ln dLxΘ(Lǫ H(x)), (6)
G B
−
Z
where δ is the Dirac function and Θ is the Heaviside function.
As a consequence of the conservation of energy, the system dynamics takes
placeonenergy-levelsets. FromLiouvilletheoremitdescendsthatthemeasure
oftheEuclideanvolumeispreservedbythedynamicsandthisinducesameasure
µ conserved on each energy level set Σ of energy density ǫ which is given by
ǫ
[47, 48]
dΣ
dµ= , (7)
H
k∇ k
where dΣ is the Euclidean measure induced on Σ and is the Euclidean
ǫ
k·k
norm.
This means that, under the hypothesis of ergodicity, the averages of each
dynamical observable Φ of the system can be equivalently measured along the
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dynamics as
1 τ
Φ = lim dtΦ(t), (8)
h i τ→∞τ
Z0
or as average on the hypersurface Σ according to
ǫ
dµΦ
Φ = Σǫ . (9)
h i dµ
R Σǫ
Now, it is reasonable to expect that tRhe temperature, the specific heat, and
the other thermodynamic observables could be measured as time averages of
suitable observables Φ along the dynamics, in a way analogous to Eq. (8).
Consequently, when ergodicity holds, the measures of these quantities have to
be derived from averages upon the energy level sets Σ , according to Eq. (9).
ǫ
Furthermore, temperature, specific heat and other thermodynamic quantities
depend on derivatives of the microcanonical entropy with respect to energy.
Therefore, they are computed by means of a functional of the form (9) if and
onlyif themicrocanonicalentropyisdefined`alaBoltzmann. Thisfactisproven
by Rugh [49] in the case of many-particle systems for which the Hamiltonian is
the only conserved quantity, and in Ref. [50] and Ref. [51] for the case of two
and k N conserved quantities, respectively.
∈
For instance, in the simpler case studied in Ref. [49] e.g., it results
s =L−1k ln dµ,
B B
ZΣǫ
and from the definition (1) in the case of Boltzmann we obtain 4
dµ ( H/ H 2)
β = Σǫ ∇· ∇ k∇ k , (10)
B
dµ
R Σǫ
where β = 1/(k T )5. In the case oRf s the expression for the inverse tem-
B B B G
perature is
dµ
β = Σǫ , (11)
G dLx
RMǫ
R
4TheFederer-Laurencederivationformula[52,53]is∂k( ψdΣ)/∂ǫk =Lk Ak(ψ)dΣ,
Σǫ Σǫ
R R
whereA(•)=1/k ▽ H k ▽(▽H/k▽Hk•).
5HigherderivativesofsBrespecttoǫarecomputedbymeanstheFederer-Laurenceformula
[52,53]thatleadstoaveragessimilartotheoneinEq. (9).
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where M = x RL H(x) Lǫ , that cannot be expressed in the form (9).
ǫ
{ ∈ | ≤ }
Inother words,by adopting the Gibbs entropydefinitionwhenergodicityholds
true, one has to trust in the very singular fact that time averages of thermo-
dynamic quantities taken along the dynamics (and then on the energy level set
Σ ) coincide with the averages of quantities taken on a set that includes all the
ǫ
energy levels below to the one on which the dynamics takes place, analogously
to Eq. (11) of the inverse temperature6.
In the case of β , one could suggest that the Gibbs temperature can be
G
measuredasamicrocanonicalaveragebyresortingtotheequipartitiontheorem.
Nevertheless,as we will show in Sec. 8, for instance in the case of systems with
negative Boltzmann temperatures, the “standard” equipartition theorem fails.
This is a first signal of inconsistency for the Gibbs entropy.
It is worth emphasizing that in the case of k > 1 conserved quantities, a
geometric structure similar to the one of Eq. (10) keeps on to be valid. In fact,
inRef. [50]ithasbeenconsideredthe casek =2bystudyingageneralclassical
autonomous many-body Hamiltonian system, whose coordinates and canonical
momenta are indicated with x RL, and for which V(x) is a further conserved
∈
quantity in involution with H. For such a system, the motion takes place on
the manifolds =Σ V , where V = (x) RL V(x)=Lu are subsets of
ǫ u u
M ∩ { ∈ | }
RL where V is constant. In Ref. [50] it is shown that
s =L−1k ln dLxδ(H(x) Lǫ)δ(V(x) Lu)=
B B
− −
Z
dτ
L−1k ln , (12)
B
W
ZM
where dτ is the volume form of , and
M
1/2
L 2
∂H ∂V ∂H ∂V
W = . (13)
"µ<ν=1(cid:18)∂xµ∂xν − ∂xν ∂xµ(cid:19) #
X
Furthermore, in [50] it is derived the generalization of (10) that gives the mi-
6Inadditiontothecaseoftheinversetemperature,thesamescenarioholdforthechemical
potential.
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crocanonicalinverse temperature for these systems, it results
dτΦ
β = M 2 , (14)
B
dτ
R M
where the complicated functional is nowR
W nξ (nV )(nξ)
Φ (x)= ·∇ nV , (15)
2
H nξ ∇ W − W ·
∇ · (cid:20) (cid:18) (cid:19) (cid:21)
that is given in terms of the unitary vectors nH = H/ H and nV =
∇ k∇ k
V/ V through the vector ξ = nH (nH nV)nV, from which is defined
∇ k∇ k − ·
the unitary vector nξ = ξ/ ξ that appears in Eq. (15). Remarkably, by ex-
k k
changing H and V in expression (15) the functional so obtained allows to mea-
sure the chemical potential of the system. This fact shows a further “esthetic
advantage” of the Boltzmann entropy: it leads to expressions formally identical
independently from the number of conserved quantities.
4. Comparison between statistical ensembles
Inastatisticaldescriptionofamany-bodysystem,temperaturehasadiffer-
ent meaning depending on the statistical ensemble. In the canonical ensemble
and in the grand-canonical one, (inverse) temperature is just a Lagrangian pa-
rameter that is introduced in order to fix the mean energy. On the contrary,
in the microcanonical ensemble the temperature is a quantity derived from the
entropy density s, according to the relation T = (∂s/∂ǫ)−1. Therefore it is
clear that T(ǫ) will depend on the entropy definition assumed within the mi-
crocanonical statistical description. The main point here is that the meaning
of temperature cannot be reduced to the issue of the coherent definition inside
to microcanonicalensemble, at leastif there is equivalence of ensembles. In the
latter case, one expects that temperature, or more in general thermodynamics,
defined for a system by two microscopic models, for instance canonical and mi-
crocanonical, coincide in the thermodynamic limit and they coincide with the
experimentally known thermodynamics of such system [54]. This amounts to
requiring that the thermodynamics of a large isolated (microcanonical) system
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and the thermodynamics of a “small” (even if big enough) subsystem of it co-
incide. In fact, in the thermodynamic limit, the complement of the subsystem,
acts on it as a thermostat and the subsystem is well described in the canonical
ensemble. The problem of equivalence of ensembles is only incompletely solved
[55],forinstanceitisknownthatsystemswithlong-rangeinteractioncanviolate
this equivalence. In fact, for this class of systems the energy is not extensive:
a system cannot be divided into independent macroscopic parts at variance of
the case ofthe short-rangeinteraction. Inthe followingwe show thatif there is
equivalence between statistical ensembles, Helmholtz free energy density is the
LegendretransformofBoltzmannentropydensityandviceversa. Consequently,
thermodynamics derived for a systems by Boltzmann entropy and by canonical
partitionfunction rigorouslycoincide in the thermodynamic limit. We consider
this as a strong evidence supporting the legitimacy of the Boltzmann entropy.
Letus now discuss abouta case wherethere is notequivalence betweencanoni-
cal and microcanonicalensembles. This is the case of a system with long-range
interaction that undergoes a first-order phase transition. We refer to [56] for
details. In summary the Boltzmann entropy for a system with these features is
not a concave function, consequently it cannot be the Legendre transformation
of the Helmholtz free-energy density, and β (ǫ) is a not-invertible function. In
B
cases like this, the canonical ensemble has not foundation since it cannot be
derived from the microcanonicalensemble, unlike the case of the extensive sys-
tems [57], where it can. Therefore, the case of the long-range interactions are
outside the class of systems to which our proof applies, although we consider
Boltzmann entropy the correct definition also for this class of systems.
In the following we will consider two explicit systems, one of which has two
conserved quantities, accordingly in this section we give our proof for a system
with this feature. The restriction of our derivation to the case of a system
where energy is the sole conserved quantity is straightforward. Let us consider
an arbitrary classical many-body Hamiltonian system with k =2 first integrals
ofmotion,H andafurtherconservedquantityV whichisininvolutionwithH.
In order to compare the canonical and the microcanonical description for such
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