Inhomogeneous Tsallis distributions in the HMF model

0.82 MB·English

by  P. H Chavanis

EPJ manuscript No.
(will be inserted by the editor)
Inhomogeneous Tsallis distributions in the HMF model
P.H Chavanis and A. Campa
0 1 Laboratoire dePhysiqueTh´eorique (IRSAMC),CNRS and UPS,Universit´e deToulouse, F-31062 Toulouse, France
1 2 Complex Systems and Theoretical PhysicsUnit, Health and Technology Department,Istituto Superiore diSanit`a,
0 and INFN Roma 1, Gruppo Collegato Sanita, 00161 Roma, Italy
2

n To be included later
a
J
Abstract. We study the maximization of the Tsallis functional at fixed mass and energy in the HMF

3 model.Wegiveathermodynamicalandadynamicalinterpretationofthisvariationalprinciple.Thisleads
1
toq-distributionsknownasstellarpolytropesinastrophysics.Westudyphasetransitionsbetweenspatially

homogeneousandspatiallyinhomogeneousequilibriumstates.Weshowthatthereexistsaparticularindex
]
h qc =3playingtheroleofacanonicaltricriticalpointseparatingfirstandsecondorderphasetransitionsin
c thecanonicalensembleandmarkingtheoccurenceofanegativespecificheatregion inthemicrocanonical
e ensemble.WeapplyourresultstothesituationconsideredbyAntoni&Ruffo[Phys.Rev.E52,2361(1995)]
m and show that theanomaly displayed on their caloric curvecan beexplained naturally by assuming that,
- in this region, the QSSsare polytropes with critical index qc =3. We qualitatively justify the occurrence
t ofpolytropic(Tsallis) distributionswithcompactsupportintermsofincompleterelaxationandinefficient
a
t mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF
s model andthefirstplausible explanation of thesurprising result observednumerically byAntoni& Ruffo
.
t (1995).Inthecourseofouranalysis,wealsoreportaninterestingsituationwherethecaloriccurvepresents
a
both microcanonical first and second order phase transitions.
m
- PACS. 0 5.20.-yClassicalstatisticalmechanics-05.45.-aNonlineardynamicsandchaos-05.20.DdKinetic
d
theory - 64.60.De Statistical mechanics of model systems
n
o
c
[ 1 Introduction of bars in disk galaxies, giving it a physical application.

This model is now known as the Hamiltonian Mean Field
1
v Systems with long-range interactions are numerous in (HMF) model.This abbreviationwascoinedbyAntoni&
9 nature. Some examples include self-gravitating systems Ruffo [8] andit stood at that time either for Hamiltonian
0 (galaxies), two-dimensional turbulence (vortices), chemo- or Heisenberg Mean Field model since the HMF model
1 taxis of bacterial populations (clusters) and some mod- can be viewed as a mean field XY model with long-range
2 els in plasma physics [1]. These systems are fascinating interactions (i.e. not restricted to the nearest neighbors).
1. because they present striking features that are absent In fact, this model was first introduced much earlier (this
0 in systems with short-range interactions such as nega- is notwell-known)by Messer& Spohn[10] andcalledthe
0 tive specific heats in the microcanonical ensemble, nu- cosine model1. They rigorously studied phase transitions
1 merous types of phase transitions, ensembles inequiv- in the canonical ensemble by evaluating the free energy
v: alence, unusual thermodynamic limit, violent collision- and exhibited a second order phase transition between a
i less relaxation, long-lived quasi stationary states (QSS), homogeneousphaseandaclusteredphasebelowacritical
X
non-Boltzmanniandistributions,out-of-equilibriumphase temperature Tc =1/2.
r transitions, re-entrant phases, non-ergodic behavior, slow In their seminalpaper,Antoni& Ruffo [8] studiedthe
a
collisional relaxation, dynamical phase transitions, alge- statistical mechanics of this model in the canonical en-
braicdecayofthe correlationfunctions...We referto[2,3] semble(CE)directly fromthe partitionfunctionandper-
for some recent reviews on the subject. formedN-body simulations in the microcanonicalensem-
In order to understand these strange properties in a ble (MCE). They started from a waterbag2 initial condi-
simple setting, a toy model of systems with long-range
interactions has been actively studied. It consists of N 1 These authors mention that this “cosine model” was sug-
particlesmovingonaringandinteractingviaacosinepo- gested byG. Battle [11].
tential. This model has been introduced by many authors 2 Awaterbagdistributioncorrespondstoauniformdistribu-
[4-8] at about the same period with different motivations tionfunctionf(θ,v)=f0 inthedomain[−θm,θm]×[−vm,vm]
(seeashorthistoryin[9]).Letusmentionforexamplethat surroundedby“vaccum”f(θ,v)=0.Itcanhavedifferentmag-
Pichon[7]introducedthis modeltoexplainthe formation netization 0≤M0 ≤1.
2 P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model
tion with magnetization M M(0) = 1 and plotted in ficities of the Vlasov equation (Casimir constraints). Ap-
0
≡
their Fig. 4 the caloric curve giving the averaged kinetic plying the Lynden-Bell theory to the HMF model (for
temperature T =2E /N as a function of the energy U. waterbag initial conditions), an out-of-equilibrium phase
kin
TheycomparedtheirnumericalcurvewiththeBoltzmann transition was discovered in [18,19]. For a given value
prediction of statistical equilibrium in the canonical en- of the energy U, there exists a critical value of the
semble.Theyfoundagoodagreementathighandlowen- initial magnetization (M ) (U) such that: for M <
0 crit 0
ergies.However,closetothecriticalenergyU =3/4(cor- (M ) (U)thestableLynden-Belldistribution(i.e.most
c 0 crit
respondingtoT =1/2),theresultsdifferfromthecanon- probable state) is homogeneous (non-magnetized) and
c
ical prediction. In particular, the system bifurcates from for M > (M ) (U) the stable Lynden-Bell distribu-
0 0 crit
thehomogeneousbranchatasmallerenergyU 0.6 0.7 tion is inhomogeneous (magnetized). For U = 0.69, the
∼ −
and the caloric curve T(U) displays a region of negative critical magnetization is (M ) = 0.897 [18,19]. More
0 crit
specific heats. Antoni & Ruffo [8] interpreted this result generally, there exists a critical line (M ) (U) in the
0 crit
eitheras(i)anonequilibriumeffect,or(ii)asamanifesta- phase diagram separating homogeneous and inhomoge-
tion of ensembles inequivalence due to the non-additivity neousLynden-Belldistributions.Itwasfoundlater[21,22]
of the energy.However,it is clear from the previous work thatthesystemdisplaysfirstandsecondorderphasetran-
ofInagaki[6]andthe applicationofthePoincar´etheorem sitions separated by a tricritical point. There is also an
that the ensembles are equivalent for the HMF model (as interestingphenomenonofphasereentranceinthe(f ,U)
0
confirmed later by various methods [12,9,2]). Therefore, plane predicted in [18] and numerically confirmed in [23].
this negative specific heats region is not a manifestation Coming back to the specific value U = 0.69, direct nu-
of ensembles inequivalence but rather a nonequilibrium mericalsimulations of the HMF model for M <(M )
0 0 crit
effect. have shown that the Lynden-Bell prediction works fairly
well [19]. This agreement is remarkable since there is no
Latora et al. [13,14] again performed microcanoni-
fitting parameter in the theory. This led the authors of
cal simulations of the HMF model. They confirmed the
[19]toarguethat“Lynden-Bell’stheoryexplainsquasista-
“anomaly” (negative specific heats region) reported by
tionary states in the HMF model”. A controversystarted
Antoni&Ruffo [8]andobservedmanyotheranomaliesin
when some of these authors [17,24,2] concluded that “the
thatregionsuchasnon-Gaussiandistributions,anomalous
approachof Tsallis is unsuccessful to explain QSSs”.
diffusion,L´evywalksanddynamicalcorrelationsinphase-
However, caution was made by one author [25,18,26]
space.Furthermore,theyshowedthatthethermodynamic
who arguedthat the Lynden-Belltheorydoes notexplain
limit N + and the infinite time limit t + do
→ ∞ → ∞ everything. Indeed, for initial magnetization M = 1, the
notcommute. They evidenced tworegimes in the dynam- 0
systemis in the non-degeneratelimit sothat the Lynden-
ics. On a short timescale, of the order of the dynamical
Bell entropy reduces to the Boltzmann entropy (with a
time t 1, the system reaches a quasistationary state
D
∼ different interpretation). Therefore, in this limit case, the
(QSS). Then, on a much longer timescale t (N), the
relax
Lynden-Bell theory leads exactly to the same prediction
systemrelaxestowardstheBoltzmanndistributionofsta-
as the usual Boltzmann statistical theory but, of course,
tisticalequilibrium.Theyshowedthattherelaxationtime
for a completely different reason. This observation led to
increasesrapidly(algebraically)withthenumberofparti-
are-interpretation[26]ofthe caloriccurveT(U)obtained
clesN sothat,atthethermodynamiclimitN + ,the
→ ∞ by Antoni & Ruffo [8] and Latora al. [14]. In this curve,
system remains permanently in the QSS. It is thus clear
thetheoreticallineshouldnotbeinterpretedastheBoltz-
from this study that the different anomalies mentioned
mannstatisticalequilibrium state but as the Lynden-Bell
above characterize the QSS, not the statistical equilib-
statisticalequilibriumstate3.Theyturnouttocoincidein
rium state that is reached much later. In particular, the
the case M =1 but this is essentially coincidental. With
results of Antoni & Ruffo [8] and Latora et al. [14] show 0
this new interpretation [26], the comparison reveals that
that the QSS is non-Boltzmannian in the region close to
the Lynden-Belltheory workswellfor largeandlow ener-
the critical energy U . Latora et al. [14] thus proposed to
c
giesbutthatitfailsforenergiesclosetothecriticalenergy.
describe the QSSin terms of Tsallis [15]generalizedther-
Therefore,theLynden-Bell theorydoes notexplain theob-
modynamics leading to q-distributions. Note that there is
servations in the range [0.5,U ].Chavanis[26]interpreted
no reason why the QSS should be Boltzmannian (in the c
this disagreementas a result of incomplete relaxation. In-
usual sense) since it is an out-of-equilibrium structure.
deed, it was emphasized by Lynden-Bell [20] himself that
Therefore, the comparison of the numerical caloric curve
his approach implicitly assumes that the system “mixes
withthe BoltzmannequilibriumcaloriccurveT(U)isnot
efficiently” so that the ergodicity hypothesis which sus-
justified a priori.
tains his statistical theory applies. However, it has been
After the conference in Les Houches in 2002, and in- observedin many cases of violent relaxationthat the sys-
spired by the results in astrophysics and 2D turbulence tem does not mix efficiently so that the Lynden-Bell pre-
presented by one of the authors [16], several groups of
researchers [17,9,18,19] started to interpret the QSSs ob- 3 As mentioned above, there is no reason to compare the
served in the HMF model in terms of Lynden-Bell’s [20] QSS with the Boltzmann prediction that corresponds to the
statisticaltheoryofviolentrelaxationbasedontheVlasov collisional regime reached for t→+∞.By contrast, it is fully
equation. This is a fully predictive theory based on stan- relevant to compare the observed QSS with the Lynden-Bell
dard thermodynamics but taking into account the speci- prediction that applies to thecollisionless regime.
P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model 3
diction fails (see variousexamples quotedin [25,26]). The notaq-distributionsinceitdoesnothavepower-lawtails.
qualitative reason is easy to understand [26]. Since vio- However, Chavanis [26] remarked that a semi-ellipse is a
lent relaxation is a purely inertial process (no collision), q-distribution with q = 3! Since q > 1, this distribution
mixing is due to the fluctuations of the mean field poten- has a compact support. Therefore, the numerical results
tialcausedbythefluctuationsofthedistributionfunction of Campa et al. [34] and Yamaguchi et al. [17] show that
itself[20].However,asthesystemapproachesmetaequilib- the system tends to select a particular Tsallis distribu-
rium (QSS), the fluctuations of the distribution function tion as a QSS7. Furthermore, the index q = 3 seems to
are less and less efficient (by definition!) and the system playaparticularrolesinceCampaet al.[34]obtainedthe
can be trapped in a QSS (a steady solution of the Vlasov same index q =3 in different situations (see their Fig. 4).
equationonthe coarse-grainedscale)thatisnotthe most However, until now, the reason for this particular value
mixed state. This is what happens in astrophysics (ellip- remains unknown. The fact that the QSS has a compact
ticalgalaxiesarenot describedby Lynden-Bell’sdistribu- support is relatively natural in the phenomenology of in-
tion that has infinite mass [27]) and in certain situations complete violent relaxation. Indeed, when mixing is not
of2Dturbulence[28-30].TheresultsofAntoni&Ruffo[8] very efficient, we expect that the high energy states are
andLatoraetal.[14]revealthatthesamephenomenonoc- notsampledbythesystem.Thisleadstoaconfinementof
cursfortheHMFmodelclosetothecriticalenergy4.This the distribution which is a virtue of the Tsallis distribu-
is alsovisible on Fig.8 of Bachelardet al. [31]. There is a tionswith q >1.Asimilar confinementwasobservedina
huge region in the top right of the phase diagram where plasma experiment [28] and a good fit was obtained with
the Lynden-Bell theory does not work. This concerns in a q-distribution with index q = 2 (in our notations) [29].
particular the point U =0.69 and M =1, as anticipated This confinement was justified by a lack of ergodicity in
0
in [18,26]. It is precisely this “no-man’s land” region that thesystem[30].OnecanthereforeinterprettheTsallisdis-
Tsallisandcoworkershaveinvestigated[3].Inthisregion, tributions as an attempt to take into account incomplete
standard statistical mechanics (i.e. Lynden-Bell’s theory) relaxation and non-ergodicity in systems with long-range
does not seem to directly apply5. interactions.In this interpretation,the index q couldbe a
In their early work, Latora et al. [14] tried to fit the measure of mixing8. If we assume that the system mixes
QSS by a q-distribution. They considered a distribution efficiently, then q =1 and we get the Lynden-Bell theory.
with q = 5 < 1 (in our notations) leading to power- If the system does not mix well, the Lynden-Bell theory
−
law tails and introduced by hand an additional cut-off fails and q = 1. Since the value of the q parameter de-
6
to make the distribution normalizable. This procedure is pends on the efficiency of mixing (which is not known a
veryad hoc. Furthermore,evenif we accept it, we canar- priori),itappearsdifficulttodetermineitsvaluefromfirst
gue that it does not provide an impressive fit of the QSS. principles.Furthermore,itsvaluecanchangefromcaseto
Recently, Campa et al. [34] have performed new simula- casesincethedegreeofmixingcanvarydependingonthe
tions for initial magnetizations M = 0 and M = 1 and initial condition (some systems can mix well and others
0 0
found that the QSS is very well-fitted by a semi-ellipse6. less).Finally,wecanarguethattheTsallisentropyisjust
Similar results were obtained earlier by Yamaguchi et al. one generalized entropy among many others and that it
[17] for the M = 0 case. They claimed that the QSS is may not be universal [36]. It may just describe a special
0
type of non-ergodic behavior but not all of them. In fact,
4 Latoraetal.[14]findthattheLargestLyapunovexponent non-ergodic effects can be so complicated that it is hard
fortheQSStendstozero.Inthissense,mixingisnegligibleand tobelievethatthey canbe encapsulatedinasimplefunc-
one expects anomalies in the relaxation process. This is fully tional such as the Tsallis functional or any other [30,25].
consistentwiththeideaofincompleterelaxationandinefficient Nevertheless, we must recognize that some QSSs can be
mixing introduced by Lynden-Bell [20] and further discussed well-fittedbyq-distributions.Evenmorestrikingly,follow-
by Chavanis [25,18,26]. ingthesuggestionof[26],Campaet al. [37]demonstrated
5 Quoting Einstein [32] and Cohen [33], Latora et al. [14] numerically that, during the collisional regime, the time
argued that standard statistical mechanics fails when the dy-
dependent distribution f(v,t) is still very well-fitted by
namics plays a nontrivial role (e.g. long-range correlations or
q(t)-distributionswithatime-dependentindex.Whenthe
fractal structures in phase space). In our point of view, this
index reaches a critical value q (U) (predicted by the
isa correct interpretation although standard thermodynamics crit
theory [26,37]), the distribution function becomes Vlasov
should refer here to Lynden-Bell’s theory, which is the proper
BoltzmannapproachappliedtotheVlasovequation.Thissub-
tlety is not addressed in the paper of Latora et al. [14] since 7 Let us be more precise: (i) For M0 = 1, an isotropic wa-
theydidnotknowthetheoryofLynden-Bellatthattime.Yet, terbag distribution violently relaxes towards a q-distribution
theirgeneral comment canbeappliedtoLynden-Bell’stheory with q = 3 [34]; (ii) For M0 = 0, a waterbag distribution is
as well: if the dynamics is nontrivial and the system does not Vlasovstableanddoesnotundergoviolentrelaxation(itisal-
mix well, standard (Lynden-Bell) statistical mechanics fails. readytheLynden-Bellstate).However,inthecollisionalregime
6 For M0 = 1, this differs from the numerical results of La- (duetofiniteN effects),itbecomesaq-distributionwithq=3
toraet al.[14].However,Campaet al.[34]showthattheordi- [17,34];(iii)forintermediatevaluesofM0,thesystemviolently
nary waterbag initial condition leads to the presence of large relaxes towards the Lynden-Belldistribution [19,34].
sample to sample fluctuations so that many averages are nec- 8 This interpretation was proposed by one of the authors in
essary. They proposed to use isotropic waterbag distributions severalpapers[35,18,26]anditmaybemoreaccuratethanthe
to reducethefluctuations. usualinterpretation: “q is a measure of nonextensivity”[3].
4 P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model
unstable and a dynamical phase transition from the ho- lowingq todeviateslightlyfromthecriticalvalueq =3).
c
mogeneousphase(non-magnetized)totheinhomogeneous These behaviors had never been explained since the orig-
phase (magnetized) is triggered. This explains previous inal paper of Antoni & Ruffo [8]. We provide a plausible
observations on the evolution of the magnetization M(t) explanation in terms of Tsallis (polytropic) distributions.
[14,17,34]. Interestingly, a very similar behavior has been We qualitatively justify the occurence of polytropic dis-
found by Taruya & Sakagami[38] for self-gravitatingsys- tributions with a compactsupport interms of incomplete
tems9. relaxation. Furthermore, the index selected by the sys-
In view of these results, it is interesting to study tem appears to be the critical one (or close to it). This
in more detail the structure and the stability of q- is the first time that a sort of prediction of the q index is
distributions. Note that q-distributions correspond to madeinthatcontext.However,ourapproachdoesnotex-
what have been called stellar polytropes in astrophysics plaineverythingandjustopensadirectionofresearch.In-
[27]. They were introduced long ago by Eddington [40] deed,onehasfirsttoconfirmthe resultsby moredetailed
as particular stationary solutions of the Vlasov equation. comparisons with numerical simulations and, in case of
Theywereusedtoconstructsimpleself-consistentmathe- agreement, try to understand why critical polytropes are
maticalmodelsofgalaxies.Atsometime,theywerefound selected by the system and if this is a general feature.
to provide a reasonable fit of some observedstar clusters,
the so-called Plummer [41] model. Improved observation
ofglobularclustersandgalaxiesshowedthatthe fitisnot 2 Interpretations of the Tsallis functionals
perfect and more realistic models have been introduced
since then [27]. However, stellar polytropes are still im- Inordertomotivateourstudyofpolytropes,weshallfirst
portant for historical reasons and for their mathematical recall different interpretations of the Tsallis functionals
simplicity. The stability of polytropic distributions is an that have been given previously by one of the authors
old problem in stellar dynamics [27]. It has been recon- [47,49].
sidered recently, for box-confined systems, by Taruya &
Sakagami [42-44] in the framework of Tsallis generalized
thermodynamicsandbyChavanisetal.[45,35,46,47,39]in 2.1 Thermodynamical interpretation
relationtotheirnonlineardynamicalstabilitywithrespect
totheEulerandVlasovequations.Itisthereforeinterest- There is no reason to describe the QSSs that emerge in
ingtoextendthesestudiestothe caseoftheHMFmodel. Hamiltoniansystemswithlong-rangeinteractionsinterms
For the moment, only spatially homogeneous polytropic of Boltzmann statistical mechanics since they correspond
distributions have been considered [9,48]. In the present to out-of-equilibrium structures (in the usual sense). The
paper, we extend the analysis to spatially inhomogeneous QSSs are formed during the collisionless regime while
polytropes.Specifically,westudythemaximizationofthe Boltzmann statistical equilibrium is reached on a much
Tsallis functional at fixed mass and energy and plot the longer timescale at the end of the collisional regime.
correspondingcaloriccurves.We give a thermodynamical Fundamentally, the QSSs are stable steady states of the
and a dynamical interpretation of this variational princi- Vlasov equation (on a coarse-grained scale) and they
ple. We find the existence of a critical polytropic index should be described by Lynden-Bell’s statistical mechan-
q =3 (where results are analytical) which plays the role ics. However,Lynden-Bell’s theory, as any statistical the-
c
of a canonical tricritical point separating first and sec- ory,assumesefficientmixing andergodicity.If the system
ondorderphasetransitionsinthecanonicalensembleand does not mix well, the QSS will differ from the Lynden-
marking the onset of negative specific heats in the mi- Bellprediction(bydefinition).Ifwewanttoapply Tsallis
crocanonicalensemble (there also exists a microcanonical generalized thermodynamics to that context, in order to
tricritical point at q 16.9 and a microcanonical critical takeinto accountnonergodiceffects andincomplete mix-
point at q 6.55). In≃terestingly, this critical value q =3 ing,wemustmodifytheLynden-Bellentropywhichisthe
c
turnsoutt≃obetheoneobservedbyCampaal.[34]intheir Boltzmann entropy associated to the Vlasov equation10.
numerical simulations. Then, we apply our theory to the Therefore, the proper q-entropy to consider is
situation considered by Antoni & Ruffo [8]. We find that
1
the structure of their numerical caloric curve T(U) close S [ρ]= (ρq ρ)dηdrdv. (1)
q
−q 1 −
to the critical energy can be explained naturally if we as- − Z
sumethattheQSSsinthis regionarepolytropeswiththe
Thisentropyappliestothedistributionρ(r,v,η)ofphase
criticalindexq =3(thisresultisspecificallydiscussedin
c levelsη (see[49]fordetails)soasto takeinto accountthe
Sec. 7.2). This yields a transition energy U′ = 5/8 lower
c
than Uc = 3/4 and a region of negative specific heats in 10 Latora et al. [14] did not know the Lynden-Bell theory
thecurveT(U)thatareinqualitativeagreementwiththe and proposed to replace the usual Boltzmann entropy S[f] =
numericalresults(theagreementcouldbeimprovedbyal- − flnfdrdvbytheq-entropySq[f]=−q−11 (fq−f)drdv.
However,asarguedlongagoin[30],thisapproachisingeneral
9 Thereis,however,acrucialdifferenceintheinterpretation. incRorrect since it does not take into account Rthe constraints
Taruya & Sakagami [38] interpret the instability in terms of of the Vlasov equation. Furthermore, since Sq[f] is rigorously
Tsallis generalized thermodynamics while Campa et al. [37] conserved by the Vlasov equation (it is a particular Casimir),
interpret it in terms of Vlasov dynamical instability [39]. thereis no thermodynamical reason to maximize it.
P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model 5
constraints of the Vlasov equation (Casimirs). For q = 1 2.2 Dynamical interpretation
(efficientmixing),werecovertheLynden-Bellentropy.For
q =1,thisfunctionalcoulddescribeincompleteviolentre- We would like to give another interpretation of Tsallis
6
laxationandnon-ergodiceffectsintheframeworkofTsal- functionalthatisnotrelatedtothermodynamicsandthat
lis thermodynamics. Now, for two-levels initial conditions doesnotsufferthelimitationsexposedabove.Itwillshow
and in the non degenerate limit (this corresponds to the that the Tsallis formalism can be useful in a dynamical
waterbag model with initial magnetization M0 =1 in the context, using a thermodynamical analogy [47,39,49].
HMF model), the Lynden-Bell entropy takes a form sim-
Ithasbeenknownforalongtimethatanydistribution
ilar to the Boltzmann entropy and the constraints reduce function of the form f = f(ǫ), where ǫ = v2/2+Φ is the
to the mass and the energy. In that case, the generalized
individualenergy,is asteadystateoftheVlasovequation
Lynden-Bell entropy (1) reduces to
[52]. In particular, q-distributions have been introduced
long ago by Eddington [40] in astrophysics where they
q
1 f f arecalled stellar polytropes12. It is alsoknown that these
S [f]= drdv. (2)
q −q−1Z "(cid:18)η0(cid:19) − η0# distributions are critical points of a certain functional of
the form
ThisissimilartotheTsallisentropyconsideredbyLatora S[f]= C(f)drdv, (3)
et al. [14] but it arises here for a different reason: note −
Z
in particular the bar on f (coarse-grained distribution)
where C is a convex function, at fixed mass and energy.
and the presence of the initial phase level η0. Note that Furthermore, if they are maxima of this functional, they
S [f] is expected to increase while S [f] is conserved. In
q q are nonlinearly dynamically stable with respect to the
this thermodynamical approach, the QSS is obtained by
Vlasov equation. As far as we know, this dynamical sta-
maximizing the Tsallis entropy at fixed mass and energy bility criterionwas first stated by Ipser & Horwitz [54] in
(microcanonical ensemble). This is expected to select the
astrophysics.Furthermore,Ipser[55]introducedthefunc-
mostprobablestateundersomedynamicalconstraints(re- tional S = f1+1/(n−3/2)drdv (in his notations) asso-
sponsibleforincomplete mixing)thatareimplicitly taken −
ciatedtostellarpolytropes.ThisisnothingbuttheTsallis
intoaccountinthe formoftheentropy.Thecanonicalen- functional13. R
semble has no justification in the present context but, as
Let us be more precise and more general. The maxi-
usual, it can be useful to provide a sufficient condition of
mization problem
microcanonical stability [50].
Since the Tsallis entropy (1) includes the Lynden-Bell
max S[f] E[f]=E, M[f]=M , (4)
entropy as a special case (for q = 1), it can give at least f { | }
as good,or better, results. However,we wouldlike to em-
phasize several limitations of Tsallis generalized thermo- determines a steady state of the Vlasov equation of the
dynamics:(i) atpresent,there is no theory predicting the form f = f(ǫ) with f′(ǫ) < 0 that is nonlinearly dynam-
value of q. This prediction is of course very difficult since ically stable. This has been stated by Ellis et al. [56] in
q is related to non-ergodic effects. For the moment, q ap- 2Dturbulence(forthe2DEulerequation)andbyIpser&
pears as a fitting parametermeasuring the degree ofmix- Horwitz[54],Tremaineetal.[57]andChavanis[39]instel-
ingofasystem;(ii)itisnotclearthatallnon-ergodicpro- lar dynamics.The maximizationproblem (4) is similar to
cesses can be described by the Tsallis entropy. Therefore, a condition of microcanonical stability in thermodynam-
thisfunctionalisprobablynotuniversal.Itmay,however, ics.Therefore,wecandevelopathermodynamical analogy
describe a certainclass of non-ergodic behaviors11; (iii) it [49]to investigatethe nonlineardynamicalstability prob-
is not clear that non-ergodic effects can be described by lem. In this analogy, S is called a pseudo entropy. Thus,
a simple entropic functionalsuchas the Tsallis functional theTsallisfunctionalisaparticularpseudoentropywhose
or any other. Maybe,a better approachis to consider the maximization at fixed mass and energy determines dis-
dynamics of mixing and develop kinetic theories and re- tributions (polytropes) that are nonlinearly dynamically
laxation equations of the process of violent relaxation as stable with respect to the Vlasov equation [47,39].
suggested in [51,25,26,49]. This kinetic approach may be
closertotheoriginalideasofEinstein[32]andCohen[33]. 12 The connection between q-distributions and stellar poly-
tropeswasfirstmentionedbyPlastino&Plastino[53].Amore
11 It has been shown that the Tsallis entropy satisfies a lot detaileddiscussionhasbeengivenrecentlybyChavanis&Sire
ofaxiomssatisfiedbytheBoltzmannentropy.Thismakesthis [47].
entropy very “natural” to generalize the Boltzmann entropy. 13 Indeed, the second term in the Tsallis functional (11) is a
However, the axiomatic approach may not be the best justifi- constant (proportional to mass) and the first term coincides
cation of an entropy. The original combinatorial approach of with the Ipser functional with the notation (18). As noted in
Boltzmann seems to be more relevant. The Boltzmann and [47], theTsallis functional (11) is more convenientto takethe
theLynden-Bellentropiescanbederivedfromacombinatorial limit q → 1 (i.e. n → +∞) since it reduces to the Boltz-
analysis by assuming that all the microstates are equiproba- mannfunctionalwhiletheIpserfunctionaltakesatrivialform.
ble. It would be interesting to derive the Tsallis entropy from Therefore, the Tsallis functional allows to make a continuous
a combinatorial analysis by putting some constraints on the linkbetweenisothermal(n=∞)andpolytropic(nfinite)dis-
availability of themicrostates. tributionsusing L’Hoˆpital’s rule.
6 P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model
The minimization problem Remark: the “microcanonical” criterion (4) provides
itself just a sufficient condition of nonlinear dynamical
min F[f]=E[f] TS[f] M[f]=M . (5) stability. There exists an even more refined criterion of
f { − | }
nonlineardynamicalstabilitytakingintoaccountthecon-
alsodeterminesasteadystateoftheVlasovequationthat servation of all the Casimirs (see discussion in [48]).
isnonlinearlydynamicallystable.Infact,(4)and(5)have
the same critical points. However, the criterion (5) is less
refined than (4). Indeed, the minimization problem (5) is 2.3 Generalized H-functions and selective decay
similar to a condition of canonical stability in thermody-
namics. In this analogy, F is called a pseudo free energy.
A generalized H-function is a functional of the coarse-
Now,itisageneralresult[50]thatcanonicalstabilityim- grained distribution function f(r,v,t) of the form
plies microcanonical stability, but the reciprocalis wrong
in case of ensembles inequivalence. Schematically: (5)
(4). In the present dynamical context, this means th⇒at H[f]= C(f)drdv, (10)
−
steady states of the Vlasov equation that are stable ac- Z
cording to the criterion (5) are necessarily stable accord- where C is any convexfunction. Tremaine et al. [57] have
ing to the more constrained criterion (4). However, there shown that the generalized H-functions increase during
may exist steady states of the Vlasov equation that are violent relaxation in the sense that H(t) H(0) for any
stable according to (4) while they fail to satisfy (5). As time t 0 where it is assumed that th≥e initial distri-
shown in [39], this is the case for stellar polytropes with bution ≥function is not mixed so that f(r,v,t = 0) =
indices 3<n<5 in astrophysics. f(r,v,t = 0)14. By contrast, the energy E[f] and M[f]
It can also be shown [49] that (5) is equivalent to
and the mass calculated with the coarse-graineddistribu-
tionfunctionareapproximatelyconserved.Thissuggestsa
min F[ρ] M[ρ]=M , (6)
ρ { | } phenomenologicalgeneralizedselectivedecayprinciple(for
H):“dueto mixing,thesystemmaytendto aQSSthat
where −
maximizes a certain H-function (non universal) at fixed
1 ρ p(ρ′) massandenergy”[49]15.Inthatcontext,the Tsallisfunc-
F = 2 ρΦdr+ ρ ρ′2 dρ′dr. (7) tionalsH[f]= 1 (fq f)drdv canbe interpretedas
Z Z Z −q−1 −
particular generalized H-functions [49].
Moreprecisely,adistributionfunctionf(r,v)issolutionof R
(5) iff the corresponding density profile ρ(r) is solution of
(6),wherep(ρ)istheequationofstatedeterminedbyC(f)
3 Polytropic distributions: general theory
(see[49]formoredetails).Finally,(6)isclearlyequivalent
to
For any of the reasons exposed previously, we think that
min [ρ,u] M[ρ]=M , (8) itis useful to study polytropic (Tsallis)distributionfunc-
ρ,u {W | }
tionsandinvestigatetheirstabilitythroughthevariational
where problems(4) and(5). We firstdevelopageneraltheoryof
polytropes followingthe lines of [46,47]. It will be applied
= ρ ρ p(ρ′)dρ′dr+ 1 ρΦdr+ ρu2 dr. (9) specifically to the HMF model in Sec. 4.
W ρ′2 2 2
Z Z Z Z
Now, it can be shownthat this minimization problemde-
3.1 Polytropic distributions in phase space
termines a steady solution of the barotropic Euler equa-
tion that is formally nonlinearly dynamically stable [27].
Let us consider the Tsallis functional
From the implication (8) (6) (5) (4), we con-
⇔ ⇔ ⇒
clude that a distribution f(r,v) is stable with respect
1
to the Vlasov equation [according to (4)] if the corre- S = (fq fq−1f)drdv. (11)
−q 1 − 0
sponding density profile ρ(r) is stable with respect to the − Z
Euler equation [according to (8)]. As shown in [39], this
We have introduced a constant f in order to make the
provides a nonlinear generalization of the Antonov first 0
expression homogeneous. This constant will play no role
law in astrophysics: “a spherical galaxy f = f(ǫ) with
in the following since the last term is proportional to the
f′(ǫ)<0isnonlinearlydynamicallystablewithrespectto
theVlasov-Poissonsystemifthecorrespondingbarotropic 14 NotethatthetimeevolutionofthegeneralizedH-functions
star is nonlinearly dynamically stable with respect to the isnotnecessarilymonotonic(nothingisimpliedconcerningthe
Euler-Poisson system”. Interestingly, this also provides a relative values of H(t) and H(t′) for t,t′ >0).
new interpretation [39] of this law in terms of ensembles 15 Thecloserelationshipbetweenthisphenomenologicalprin-
inequivalence.Indeed, the Antonovfirstlaw has the same ciple and thecriterion of nonlinear dynamical stability (4), as
status as the fact that “canonical stability implies micro- well as the limitations of this phenomenological principle, are
canonical stability” in thermodynamics. discussed in detail in [25,49].
P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model 7
1
mass that is conserved. Therefore, we could equally work where we have set A = [β(q 1)/q]q−1 and ǫm =
−
with the Ipser functional qµ/[β(q 1)]. Such distributions have a compact sup-
−
port in phase space since they vanish at ǫ = ǫ . At
1 m
S = fqdrdv, (12) a given position, the distribution function vanishes for
−q 1
− Z v vm(r) = 2(ǫm Φ(r)). (ii) For 0 < q < 1, the
≥ −
but the first expression allows us to make the connection distribution function can be written
p
with isothermal distributions when q 1. Indeed, for 1
q 1, we recover the Boltzmann functi→onal f =A(ǫ0+ǫ)q−1, (20)
→
1
f wherewehavesetA=[β(1 q)/q]q−1 andǫ0 =qµ/[β(1
S = fln drdv. (13) − −
q)]. Such distributions are defined for all velocities. At a
− f
Z (cid:18) 0(cid:19) givenposition,thedistributionfunctionbehaves,forlarge
We shall consider the maximization of the Tsallis func- velocities, as f v−2/(1−q) v−(d−2n). We shall only
tional at fixed energy and mass consider distribu∼tion functions∼for which the density and
the pressure
v2 1
E = f drdv+ ρΦdr, (14)
2 2 1
Z Z ρ= fdv, p= fv2dv, (21)
d
Z Z
M = ρdr. (15)
are finite. This implies d/(d+2)<q <1 (n< 1).
Z −
Someinterpretationsofthisvariationalproblemhavebeen
given in Sec. 2. Here, S is either a generalized entropy
3.2 Polytropic equation of state
(thermodynamical interpretation) or a pseudo entropy
(dynamical interpretation).By anabuse of language,and
For any distribution function of the form f = f(ǫ), the
to simplify the terminology, we shall call it simply an en-
density and the pressure are functions of Φ(r) such that
tropy. We will work in a space of dimension d since our
ρ = ρ[Φ(r)] and p = p[Φ(r)]. Eliminating Φ(r) between
formalism can have application for different systems.
theseexpressions,weobtainabarotropicequationofstate
Thecriticalpointsofentropyatfixedenergyandmass
p(r) = p[ρ(r)]. Furthermore, it is easy to see that f =
are determined by the condition
f(ǫ) implies the condition of hydrostatic equilibrium (see
δS βδE αδM =0, (16) Appendix A):
− −
where β = 1/T and α are Lagrange multipliers (T is the p+ρ Φ=0. (22)
∇ ∇
inverse temperature and α the chemical potential). This
Let us determine the equation of state corresponding to
yields the q-distributions (or polytropic distributions):
the polytropic distribution (17). For n d/2, the density
≥
(q 1)β v2 1/(q−1) and the pressure can be expressed as (see Appendix B):
f(r,v)= µ − +Φ(r) , (17)
− q 2 Γ(d/2)Γ(1 d/2+n)
(cid:26) (cid:20) (cid:21)(cid:27)+ ρ=AS (ǫ Φ)n2d/2−1 − , (23)
d m
− Γ(n+1)
whereµ=[fq−1 (q 1)α]/q.Thenotation[x] =xifx
0 − − + ≥
0 and [x] = 0 if x 0. As is customary in astrophysics,
+
we define the polytr≤opic index n by the relation16 p= ASd (ǫ Φ)n+12d/2−1Γ(d/2)Γ(1−d/2+n),
m
n+1 − Γ(n+1)
d 1
n= + . (18) (24)
2 q 1
−
The distribution function f(r,v) depends only on the in- where Γ(x) denotes the Gamma function and Sd the sur-
dividualenergyǫ=v2/2+Φ(r),i.e.f =f(ǫ).Therefore,it faceofaunitsphereinddimensions.Forn< 1,theden-
−
isasteadystateoftheVlasovequation.Weshallconsider sity and the pressure can be expressed as (see Appendix
q >0 so that C(f) is convex and β >0 so that f′(ǫ)<0, B):
which corresponds to the physical situation. For n = d/2
Γ( n)Γ(d/2)
(q + ), we recover the Fermi distribution [18]. For ρ=AS (ǫ +Φ)n2d/2−1 − , (25)
→ ∞ d 0 Γ(d/2 n)
n + (q 1), we recoverthe isothermal distribution.
−
→ ∞ →
Weneedtodistinguishtwocasesdependingonthesign
ofq 1.(i) For q >1(n d/2),the distribution function AS Γ( n)Γ(d/2)
can−be written ≥ p= d (ǫ0+Φ)n+12d/2−1 − . (26)
−n+1 Γ(d/2 n)
1 −
f =A(ǫm−ǫ)+q−1, (19) Eliminating the field Φ(r) between the expressions (23)-
(24) and (25)-(26), one finds that
16 This relation is sometimes presented as a “fundamental”
relationrelatingtheq-parametertothepolytropicindexn[3]. 1
In our sense, this is just a definition of notations, nomore. p=Kργ, γ =1+ . (27)
n
8 P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model
This is the classical polytropic equation of state. This is 3.4 Other expressions of the distribution function
thereasonwhythedistributions(17)arecalledpolytropic
distributions. Furthermore, γ is the ordinary polytropic We can write the polytropic distribution function (17)
index and n is a polytropic index commonly used in as- in different forms that all have their own interest. This
trophysics [58]. The polytropic constant K is given for willshowthatdifferentnotionsof“temperature”existfor
n d/2 by polytropic distributions.
≥
(i) Thermodynamical temperature T = 1/β: the poly-
−1
1 Γ(d/2)Γ(1 d/2+n) n tropicdistribution(17)directlycomesfromthevariational
K = AS 2d/2−1 − (28)
n+1 d Γ(n+1) principle (16). Therefore, β = 1/T is the Lagrange mul-
(cid:26) (cid:27)
tiplier associated with the conservation of energy. This is
and for n< 1 by
the properthermodynamicaltemperatureto consider,i.e.
−
−1 β =(∂S/∂E)M istheconjugateoftheenergywithrespect
1 Γ( n)Γ(d/2) n
K = AS 2d/2−1 − . (29) totheentropy.Note,however,thatT =1/β doesnothave
d
−n+1 Γ(d/2 n) thedimensionofanordinarytemperature(squaredveloc-
(cid:26) − (cid:27)
ity).
The polytropic constant K plays the role of the temper-
(ii) Dimensional temperature θ =1/b: we can define a
ature T in isothermal systems (q = 1, n = , γ = 1)
iso ∞ quantitythathasthedimensionofatemperature(squared
anditissometimescalleda“polytropictemperature”.For
velocity) by setting b = β/qµ. If we define furthermore
polytropic distributions, the relation between the poly-
f = µ1/(q−1), the polytropic distribution (17) can be
tropic temperature K and the thermodynamical temper- ∗
rewritten
ature T =1/β is
1
T =CnK2n2−nd, (30) f =f 1 b(q 1)ǫ q−1. (34)
∗
− −
where C is given for n d/2 by (cid:20) (cid:21)+
n
≥
2n 2 Comparing Eq. (34) with Eqs. (19) and (20) we find that
C = 2(n+1)2n−d S 2d/2−1Γ(d/2)Γ(1−d/2+n) 2n−d ǫ = 1/[b(q 1)] and A = f /ǫ1/(q−1) for n d/2 and
n d m ∗ m
2n d+2 Γ(n+1) − ≥
− (cid:20) (cid:21) ǫ = 1/[b(1 q)] and A = f /ǫ1/(q−1) for n < 1. Sub-
(31) 0 − ∗ 0 −
stituting these expressions in Eqs. (23) and (25), we find
that the relation between the density and the potential
and for n< 1 by
− can be written
2n 2
C = 2[−(n+1)]2n−d S 2d/2−1Γ(d/2)Γ(−n) 2n−d . n
n 2n+d 2 d Γ(d/2 n) ρ=ρ∗ 1 b(q 1)Φ , (35)
− − (cid:20) − (cid:21) − −
(32) (cid:20) (cid:21)+
where ρ is given for n d/2 by
We note that K is a monotonically increasing function of ∗ ≥
T. This remark will have some importance in the follow-
d/2
ing. Sd 2n d Γ(d/2)Γ(1 d/2+n)
ρ = f − − ,
∗ ∗
2 b Γ(n+1)
(cid:18) (cid:19)
(36)
3.3 Polytropic distributions in physical space
and for n< 1 by
The density is obtained by integrating Eq. (17) on the −
velocityleadingtoEqs.(23)and(25).UsingEqs.(28)and
d/2
S d 2n Γ(d/2)Γ( n)
(29), we find that the density is related to the potential ρ = df − − . (37)
∗ ∗
Φ(r) by 2 b Γ(d/2 n)
(cid:18) (cid:19) −
1
γ 1 γ−1
ρ(r)= λ − Φ(r) , (33) (iii) Polytropic temperature K: eliminating the poten-
(cid:20) − Kγ (cid:21)+ tial between Eqs. (19) and (23), and between Eqs. (20)
whereλ=ǫ /(K(n+1))forn d/2andλ=ǫ /( K(n+ and (25), and using Eqs. (28) and (29), we can express
m 0
1)) for n < 1. As noted in [4≥7], the polytropic d−istribu- the distribution function in terms of the density accord-
tion in phas−e space f = f(ǫ) given by Eq. (17) has the ing to
same mathematical form as the polytropic distribution in
physical space ρ=ρ(Φ) given by Eq. (33) with γ playing 1 v2/2 n−d/2
f = ρ(r)1/n , (38)
the role of q and K playing the role of T = 1/β. In this Z − (n+1)K
(cid:20) (cid:21)+
correspondence, γ is related to q by Eqs. (27) and (18)
leadingtoγ =((d+2)q d)/(dq d+2)andK is related where Z is given for n d/2 by
to T by Eq. (30). Polyt−ropic dist−ributions (including the ≥
isothermal one) are apparently the only distributions for Γ(d/2)Γ(1 d/2+n)
Z =S 2d/2−1 − [K(n+1)]d/2,(39)
which f(ǫ) and ρ(Φ) have the same mathematical form. d Γ(n+1)
P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model 9
and for n< 1 by the potential17. The coefficient of proportionality is re-
−
lated to the polytropic index by (γ 1)/γ =1/(n+1)=
−
Γ( n)Γ(d/2) 2(q 1)/[(d+2)q d]. The relation (48) can also be ob-
Z =Sd2d/2−1 Γ−(d/2 n) [−K(n+1)]d/2. (40) tain−ed directly from−Eq. (19) [or Eq. (20)] noting that
−
This is the polytropic counterpart of the isothermal dis- f =A(ǫ Φ)n−d/2 1 v2/2 n−d/2, (49)
tribution.TheconstantK playsarolesimilartothe tem- m− − ǫ Φ
(cid:20) m− (cid:21)+
perature T in an isothermal distribution. In particular,
iso
and comparing with Eq. (44).
itis uniforminapolytropicdistributionasis the temper-
(v) Energy excitation temperature T(ǫ): for any dis-
atureinanisothermalsystem.ThisiswhyK issometimes
tribution function of the form f = f(ǫ), one may define
called a polytropic temperature.
a local energy dependent excitation temperature by the
(iv) Kinetic temperature T (r): the kinetic tempera-
kin
relation
ture is defined by
1 dlnf
= . (50)
1
T (r)= v2 (r)=p(r)/ρ(r). (41) T(ǫ) − dǫ
kin
dh i
For the isothermal distribution, T(ǫ) coincides with the
For a polytropic distribution, using the equation of state usual temperature T . For the polytropic distribution
iso
(27), we get (17), T(ǫ) = qµ/β (q 1)ǫ. This excitation tempera-
− −
ture has a constant gradient dT/dǫ=1 q related to the
T (r)=Kρ(r)γ−1. (42) polytropic index. The other parameter µ−is related to the
kin
value of the energy where the temperature reaches zero.
Foraspatiallyinhomogeneouspolytrope,thekinetictem-
perature T (r) is position dependent and differs from
kin
the thermodynamical temperature T =1/β. The velocity 3.5 Entropy and free energy in physical space
of sound c2 =p′(ρ) is given by
s Substituting the polytropic distribution function (38) in
the Tsallis entropy (11), we find after some calculations
c2(r)=Kγρ(r)γ−1 =γT (r). (43)
s kin (see Appendix C) that
It is also position dependent and differs from the velocity d
S = n β pdr fq−1M . (51)
dispersion (they differ by a factor γ). Using Eq. (41), the − − 2 − 0
distribution function (38) can be written (cid:18) (cid:19)(cid:18) Z (cid:19)
On the other hand, the energy (14) can be written
ρ(r) v2/2 n−d/2 d 1
f =Bn[2πTkin(r)]d/2(cid:20)1− (n+1)Tkin(r)(cid:21)+ , (44) E = 2Z pdr+ 2Z ρΦdr. (52)
Therefore, the free energy F =E TS is given by
−
Γ(n+1) 1
B = , (n d/2) (45) F = ρΦdr+n pdr, (53)
n Γ(1 d/2+n)(n+1)d/2 ≥ 2
− Z Z
up to unimportant constant terms. It can also be written
B = Γ(d/2−n) (n< 1). (46) F = 1 ρΦdr+ K ργ ργ−1ρ dr, (54)
n Γ( n)[ (n+1)]d/2 − 2 γ 1 − 0
− − Z − Z (cid:16) (cid:17)
where
Note that for n d/2, the maximum velocity can be ex-
≥ γ 1 T
pressed in terms of the kinetic temperature by ργ−1 = − fq−1. (55)
0 K q 1 0
−
vm(r)= 2(n+1)Tkin(r). (47) ThiscanbeviewedasafreeenergyF =E KSassociated
with a Tsallis entropy in physical space S−= 1 (ργ
Using Γ(z +a)/Γ(z) pza for z + , we recover the γ−1 −
isothermal distribution∼for n +→. O∞n the other hand, ργ0−1ρ)dr where γ plays the role of q and K theRrole of
→ ∞ T.Again,itisinterestingtonotethat,forpolytropes,the
from Eqs. (42) and (33), we immediately get T (r) =
kin free energy in phase space F[f] has a form similar to the
K(λ (γ 1)Φ(r)/Kγ) so that
− − free energy in position space F[ρ] [47].
T = γ−1 Φ. (48) 17 For any distribution of the form f = f(ǫ), we have ρ =
kin
∇ − γ ∇ ρ(Φ) and p = p(Φ) so that the kinetic temperature (velocity
dispersion) Tkin = p/ρ is a function Tkin = Tkin(Φ) of the
This shows that, for a polytropic distribution, the kinetic potential. For a polytropic distribution function, this relation
temperature (velocity dispersion) is a linear function of turnsout to be linear.
10 P.H Chavanis and A.Campa: Inhomogeneous Tsallis distributions for theHMF model
4 Application to the HMF model
n=1
x=2
0.5 c Complete
4.1 Generalities
polytrope
(x=∞)
0.4
The HMF model is a system of N particles of unit mass
mpot=ent1iaml.oTvhinegdyonnama iccisrcolfetahnedseipnaterrtaicclteisngisvgioavearnceodsibnye ρ(θ) 0.3 Cpoolmytprloeptee
Incomplete
the Hamilton equations [6,8,9]: 0.2 (x=3) polytrope
(x=1)
dθ ∂H dv ∂H
i = , i = , 0.1
dt ∂vi dt −∂θi -θ θ
c c
N 0
1 k -3 -2 -1 0 1 2 3
H = v2 cos(θ θ ), (56) θ
2 i − 4π i− j
i=1 i6=j Fig. 1. Density profile for the polytropes n = 1. We have
X X
represented incomplete (x>xc) and complete (x<xc) poly-
where θi [ π,π]and <vi < denote the position tropes.
∈ − −∞ ∞
(angle)andthe velocityofparticlei andk isthe coupling
constant. The thermodynamic limit corresponds to N
+ insuchawaythattherescaledenergyǫ=8πE/kM→2
∞ kinetictemperaturecanbeexpressedintermsofthepres-
remains of order unity (this amounts to taking k 1/N
∼ sure (21) so that
andE/N 1).Inthatlimit,themeanfieldapproximation
∼
becomes exact.The totalmass andtotalenergy aregiven
1 2π πB2
by E = pdθ =E +W. (64)
kin
2 − k
M = ρdθ, (57) Z0
Z
v2 1 4.2 Complete and incomplete polytropes
E = f dθdv+ ρΦdθ =E +W, (58)
kin
2 2
Z Z Let us now study polytropic distributions in the context
where f(θ,v) is the distribution function, ρ(θ) = fdv of the HMF model. According to Eq. (33), their density
the density, Ekin the kinetic energy and W the potential profile is given by
R
energy. The potential is related to the density by
1
γ 1 γ−1
k 2π ρ(θ)=A 1 − Φ(θ) , (65)
Φ(θ)= cos(θ θ′)ρ(θ′)dθ′, (59) − KγAγ−1
−2π − (cid:20) (cid:21)+
Z0
where A = λ1/(γ−1). We can assume without loss of gen-
and the mean force acting on a particle located in θ is
F = Φ′(θ). For a cosine interaction, the potential can erality that the distribution is symmetrical with respect
h i − to the x axis (θ = 0). In that case, the potential can be
be written
written
Φ(θ)= B cosθ B sinθ, (60) Φ= Bcosθ, (66)
− x − y −
where
where k 2π
k 2π B = ρ(θ)cosθdθ, (67)
B = ρ(θ)cosθdθ, (61) 2π
x 2π Z0
Z0
is the magnetization (B = 0 and B = B ). The density
k 2π y x
profile can be rewritten
B = ρ(θ)sinθdθ, (62)
y
2π
Z0 1
arethetwocomponentsofthemagnetizationvectorupto γ 1 γ−1
ρ(θ)=A 1+ − xcosθ , (68)
a multiplicative factor (note the change of sign with re- γ
(cid:20) (cid:21)+
spectto[9]).SubstitutingEq.(60)inthepotentialenergy
(58), we find that the energy can be rewritten where
B
x= . (69)
v2 πB2 KAγ−1
E = f dθdv , (63)
2 − k For γ 1, we recover the isothermal distribution (20)
Z and th→e relation x = βB of [9]. For x > 0, the density
where B2 = B2 +B2. It is a nice property of the HMF profileisconcentratedaroundθ =0andforx<0,weget
x y
modelthatthepotentialenergycanbeexpressedinterms a symmetrical density profile concentrated around θ =π.
of the order parameter B played here by the magneti- Likeforthegravitationalinteraction,wecanhavecom-
zation. This is not the case for more complex potentials plete or incomplete polytropes [46]. By definition, com-
such as the gravitationalpotential. Note, finally, that the plete polytropes have a compact support (the density