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 dePhysiqueThéorique (IRSAMC),CNRS and UPS,Université deToulouse, F-31062 Toulouse, France 1 2 Complex Systems and Theoretical PhysicsUnit, Health and Technology Department,Istituto Superiore diSanità, 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 applicationofthePoincarétheorem 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,Lévywalksanddynamicalcorrelationsinphase- 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 thermodynamics. 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 30). 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)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 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

Inhomogeneous Tsallis distributions in the HMF model

Similar Inhomogeneous Tsallis distributions in the HMF model