Dynamical and Thermodynamical Stability of Isothermal Distributions in the HMF Model

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(minima or saddle points must be discarded). This is the is able to trigger the instability of the homogeneous phase problem of thermodynamical stability [11,14,15,16,17,30]. below Ec or Tc. These problems of dynamical and thermodynamical sta- Let us now review the results concerning the dynam- bility have been investigated in the past using different ical stability of steady states of the Vlasov equation in methods that we shall briefly review in this introduction. the context of the HMF model. We first consider the lin- Let us first discuss the problem of thermodynamical ear dynamical stability problem. Inagaki & Konishi [13] stability. Messer & Spohn [11] considered a potential en- and Pichon [15] studied the linear dynamical stability of −1 the isothermal (Maxwell) distribution with respect to the ergy of the form U = N iplasma physics and stellar dynamics. They showed that of free energy F [f] at fixed mass M. Considering specif- the system becomes dynamically unstable below the crit- ically the cosine model, they showed that it displays a ical temperature Tc or the critical energy Ec (the same as second order phase transition from a homogeneous phase the ones arising in the thermodynamical approach) leading to an instability similar to the Jeans instability in self- to a clustered phase below a critical temperature Tc. In- agaki [14] studied the thermodynamical stability of the gravitating systems. They showed that only the modes n = 1 grow, leading to the formation of a single clus- modified Konishi-Kaneko [12] system in the microcanoni- ± cal ensemble by considering the maximization of entropy ter. Inagaki & Konishi [13] also compared their theoretical S[f] at fixed mass M and energy E. By using the theory of results with direct numerical simulations, finding a good Poincaréon linear series of equilibria (see, e.g., [10,31] for agreement. Antoni & Ruffo [16] studied the linear dynamical stability of the spatially homogeneous waterbag distri- details), or by studying the sign of the second order vari- ′ ations of entropy (for the homogeneous phase), he showed bution and determined the critical temperature Tc and the ′ that the system displays a microcanonical second order critical energy Ec marking the separation between stable and unstable states. More recently, Choi & Choi [26], Cha- phase transition at a critical energy Ec, corresponding to 1 vanis et al. [17] and Jain et al. [27] completed these studies the critical temperature Tc . Similar results were obtained by Pichon [15] who showed in addition that this type of and obtained explicit expressions for the growth rate and phase transitions could explain the formation of bars in pulsation period of isothermal, polytropic and waterbag disk galaxies. Antoni & Ruffo [16] studied the statistical distributions. On the other hand, Chavanis & Delfini [28] equilibrium state of the HMF model in the canonical en- performed an exhaustive study of the linear dynamical semble directly from the partition function. They simpli- stability of the HMF model by using the Nyquist method. fied it by using the Hubbard-Stratonovich transformation They considered various types of distributions (single and and the saddle point approximation valid for N + . double humped) and derived general stability criteria and → ∞ stability diagrams for both attractive (ferromagnetic) and They evidenced a second order phase transition at T = Tc in the canonical ensemble and performed numerical simu- repulsive (antiferromagnetic) interactions. lations in the microcanonical ensemble. These simulations Let us finally review the results concerning the for- show a discrepancy with the theoretical results close to mal nonlinear dynamical stability of a steady state of the the transition energy Ec, but this discrepancy is not due Vlasov equation. A distribution function is said to be for- to ensembles inequivalence but to nonequilibrium effects mally stable [33] if it is a local minimum or a local maxi- [16,24,32]. More recently, Barré et al. [30] studied the sta- mum of an energy-Casimir functional (i.e. the second vari- tistical mechanics of the HMF model by applying large ations of the energy-Casimir functional are positive defi- deviation technics. They confirmed the existence of mi- nite or negative definite). Formal stability implies linear crocanonical and canonical second order phase transitions stability, but the converse is wrong in general. Yamaguchi and the equivalence of statistical ensembles. Finally, Cha- et al. [25] derived a necessary and sufficient condition of vanis et al. [17] pursued the thermodynamical approach formal nonlinear dynamical stability for spatially homo- of Inagaki [14] based on variational principles. In partic- geneous distribution functions of the HMF model. They ular, they reduced the stability problem to the study of observed that linear stability and formal stability criteria an eigenvalue equation and solved this eigenvalue equa- coincide in that case. Chavanis et al. [17] reconsidered the tion numerically for any energy and analytically close to formal stability problem on a new angle (see also [28,34] the critical point (E ,T ). They proved by this method for more details) which can be extended to more general c c 2 that the statistical ensembles are equivalent and that the situations . They showed that the variational problem for homogeneous states are stable for E > Ec (or T >Tc) the distribution function f(θ, v) is equivalent to a simpler and unstable for E < Ec (or T 0). The thermodynamic limit corresponds to N + in Vlasov dynamical stability and the thermodynamical sta- → ∞2 bility of the Lynden-Bell distributions have been studied such a way that the rescaled energy ǫ = 8πE/kM remains of order unity. We can take k 1/N which is the by Antoniazzi et al. [20] and Staniscia et al. [22] by solving ∼ the variational problem numerically. These studies exhibit Kac prescription. In that case, the energy is extensive, E/N 1, but non-additive. For N + , the mean a rich phase diagram with several types of phase transi- ∼ → ∞ tions showing the complexity of the stability problem in field approximation is exact and the N-body distribu- the general case. tion function is a product of N one body distributions: In this paper, we shall present a new method to de- PN (θ1, v1, ..., θN , vN ,t)= P1(θ1, v1,t)...P1(θN , vN ,t). termine the dynamical and thermodynamical stability of Let us introduce the distribution function f(θ,v,t) = NP (θ,v,t). For a fixed interval of time and N + , the isothermal distributions of the HMF model. The idea is 1 → ∞ to start from the fundamental variational problems (12) evolution of the distribution function f(θ,v,t) is governed and (13) and transform them into equivalent but simpler by the Vlasov equation variational problems until a point at which they can be ∂f ∂f ∂Φ ∂f explicitly solved. This completes the thermodynamical ap- + v =0, (2) proach of Inagaki [14] by proving analytically the stability ∂t ∂θ − ∂θ ∂v of the inhomogeneous phase which was not done in Ina- where gaki’s paper. An interest of our approach is its simplicity and generality. Indeed, it can be extended to solve Vlasov 2π k ′ ′ ′ and Lynden-Bell stability problems. For example, it has Φ(θ,t)= cos(θ θ )ρ(θ ,t) dθ , (3) −2π − been used recently to settle the Vlasov dynamical stabil- Z0 ity of polytropic distributions (see Sec. 8 of [24]). How- is the self-consistent potential generated by the density of ever, in that case, the calculations are less explicit than particles ρ(θ,t)= f(θ,v,t) dv. The mean force acting on for isothermal distributions. It is therefore interesting to a particle located in θ is F = ∂Φ/∂θ(θ,t). Expanding develop the calculations in detail in the case of isothermal the cosine functionR in equationh i (3),− we obtain distributions (where they are fully analytical) in order to clearly illustrate the method. Although we rederive known Φ(θ,t)= Bx cos θ + By sin θ, (4) results in a different manner, we think that the present approach is interesting and potentially useful to tackle more where general problems. In addition, our approach not only determines the strict equilibrium state (global maximum of k B = ρ(θ,t)cos θ dθ, (5) x −2π 3 These criteria have been used in [17,25] to determine the Z formal stability of isothermal and polytropic distributions and in [18] to determine the formal stability of the Lynden-Bell (or k B = ρ(θ,t) sin θ dθ, (6) Fermi-Dirac) distribution. y −2π Z 4 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model are proportional to the magnetization (with the opposite where α is a Lagrange multiplier associated with the con- sign). The magnetization can be viewed as the order pa- servation of mass. Since T is fixed in the canonical ensem- rameter of the HMF model. ble, it is clear that equation (15) is equivalent to equation Let us introduce the mass (14). Therefore, the optimization problems (12) and (13) have the same critical points. Performing the variations in M[ρ]= ρ dθ, (7) equations (14) and (15), we find that the critical points are Z given by the mean field Maxwell-Boltzmann distribution 2 and the mean field energy v f(θ, v)= A′ e−β 2 +Φ(θ) , (16) 1 2 1 E[f]= fv dθdv + ρΦdθ = K + W, (8) ′ −1−α 2 2 where A = e is a constant. Integrating over the ve- Z Z locity, we get the mean field Boltzmann distribution where K is the kinetic energy and W the potential energy. Using equations (4)-(6), the potential energy can be ρ(θ)= Ae−βΦ(θ), (17) expressed in terms of the magnetization as where A = (2π/β)1/2A′. Using the expression (4) of the πB2 potential, the distribution function (16) can be rewritten W = . (9) − k 2 v ′ −β 2 +Bx cos θ+By sin θ We also introduce the Boltzmann entropy f(θ, v)= A e . (18) f It is convenient to write Bx = B cos φ and By = B sin φ S[f]= f ln dθdv, (10) 2 2 1/2 − N with B = (Bx + By ) . In that case, the foregoing ex- Z pression takes the form and the Boltzmann free energy 2 v f(θ, v)= A′ e−β 2 +B cos(θ−φ) . (19) F [f]= E[f] TS[f], (11) − The corresponding density profile is where T =1/β is the temperature. In the microcanonical ensemble, the statistical equilibrium state is determined ρ(θ)= Ae−βB cos(θ−φ). (20) by the maximization problem The amplitude A and the magnetization B are determined max S[f] E[f]= E,M[f]= M . (12) f { | } by substituting equation (20) in equations (5), (6) and (7). This yields In the canonical ensemble, the statistical equilibrium state M A = , (21) is determined by the minimization problem 2πI0(βB) min F [f] M[f]= M . (13) and 2πB I (βB) f { | } = 1 , (22) kM I0(βB) The Boltzmann entropy functional (10) and the maximum entropy principle (12) can be justified from a standard where In(x) is the modified Bessel function of order n. combinatorial analysis (see, e.g., [10,17]). The distribution Equation (22) determines the magnetization B as a func- function f(θ, v) that is solution of (12) is the most probable tion of the temperature T . Then, A is given by equation macroscopic state, i.e. the macrostate that is the most (21). Note that the critical points are degenerate. There represented at the microscopic level, assuming that the exists an infinity of critical points which differ only by accessible microstates (with the proper values of mass and their phase φ, i.e. by the position of the maximum of the energy) are equiprobable (see Appendix C). density profile. They have the same value of entropy or We shall first determine the critical points of these vari- free energy (see below). In the following, we shall take ational problems. This will allow us to set the notations φ = 0 without loss of generality. In that case, Bx = B and that will be needed in the following. The critical points By = 0. Then, the distribution function and the density of the maximization problem (12) are determined by the can be written variational principle 1/2 2 β v f(θ, v)= ρ(θ) e−β 2 , (23) δS βδE αδM =0, (14) 2π − − where β =1/T and α are Lagrange multipliers associated M with the conservation of energy and mass. The critical ρ(θ)= e−βB cos θ, (24) points of the minimization problem (13) are determined 2πI0(βB) by the variational principle where B is determined in terms of T by equation (22). δF + αTδM =0, (15) The study of the self-consistency relation (22) is classical: P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 5

B = 0 is always solution while solutions with B = 0 only 8 6 exist for T 0 4 (stable) 1 kM/4 β

K = MT. (25) = 2 η Combining this relation with equation (9), we ﬁnd that 2 ε ,η c c B=0 the total energy E = K + W is given by ε min (stable) 2 0 1 πB -2 -1 0 1 2 3 4 5 E = MT . (26) 2 2 − k ε=8πE/kM The series of equilibria giving T as a function of E is Fig. 1. Series of equilibria (caloric curve) giving the tempera- determined by equations (22) and (26) by eliminating B. ture as a function of energy. The relation that gives the magnetization B as a function of the energy E is determined by equations (22) and (26) by eliminating T . Finally, using equations (10) and (23), These relations apply to the inhomogeneous states (b = 0). the entropy is given by For the homogeneous states (b = 0), we have 6 1 S = M ln T ρ ln ρ dθ, (27) 1 1 1 1 2 − ǫ = , s = ln ǫ, f = + ln η. (35) Z η 2 η η 1 1 up to a term 2 M + 2 M ln(2π)+ M ln M. Using equation The magnetization b takes values between 0 and 1, the (24), it can be rewritten inverse temperature η between 0 and + and the energy ǫ between ǫ = 2 and + . The homogeneous∞ states 1 2πB2 min S = M ln T + M ln I (βB) , (28) exist for any η 0− and for any∞ǫ 0. The inhomogeneous 2 0 − kT ≥ ≥ states exist for any η ηc = 1 and any ǫmin ǫ ǫc = 1. 1 3 The bifurcation point≥ is located at ≤ ≤ up to a term 2 M + 2 M ln(2π). The relation between the entropy S and the energy E is determined by equations 8πE β kM (28), (26) and (22). Using equations (26) and (28), the c c ǫc 2 =1, ηc =1. (36) free energy (11) is given by ≡ kM ≡ 4π Close to the bifurcation point (η η = 1+, ǫ ǫ = 1 1 πB2 c c F = MT MT ln T MT ln I (βB)+ , (29) 1−), we get (see Appendix A): → → 2 − 2 − 0 k up to a term 1 MT 3 MT ln(2π). The relation between 2 1 − 2 − 2 b 2(η 1), b (1 ǫ), η 1+ (1 ǫ), the free energy F and the temperature T is determined ≃ − ≃ r5 − ≃ 5 − by equations (29) and (22). p (37) It is convenient to write these equations in parametric form by introducing the parameter x = βB. Then, we 1 5 have s (1 ǫ), f 1 (η 1)2. (38) ≃−2 − ≃ − 2 − 2πB I (x) b = 1 , (30) ≡ kM I0(x) Close to the ground state (η + , ǫ ǫmin = 2), we get (see Appendix A): → ∞ → − βkM x 1 ǫ +2 2 η = , (31) b 1 , b 1 , η , (39) ≡ 4π 2b(x) ≃ − 4η ≃ − 8 ≃ ǫ +2

8πE 1 2 ǫ = 2b(x)2, (32) s ln(ǫ + 2), f 2+ ln η. (40) ≡ kM 2 η(x) − ≃ ≃− η

From these relations, we can obtain the curves T (E), S 1 2 s = ln η(x) + ln I0(x) 2η(x)b(x) , (33) B(T ), B(E), S(E) and F (T ) in parametric form. These ≡ M −2 − curves are plotted in Figures 1-5 for completeness. These results are well-known and they have been derived in 8πF 2 many papers [4,14,16,17,30] using diﬀerent methods. The f = ǫ(x) s(x). (34) ≡ kM 2 − η(x) present approach, that complements the original approach 6 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model

1 2 B=0 1 0.8 B > 0 η c 0 2 0.6 -1 B > 0 B/kM F/kM π π -2 0.4 b=2 f=8 B=0 -3 0.2 -4 η B=0 c 0 -5 0 1 2 3 4 5 0 1 2 3 4 5 η=βkM/4π η=βkM/4π Fig. 2. Magnetization (order parameter) as a function of tem- Fig. 5. Free energy as a function of temperature. perature.

1 Remark: using the Poincar´etheorem (see, e.g. [10,31]), we can directly conclude from the series of equilibria

0.8 B > 0 that the homogeneous states are microcanonically (resp. canonically) stable for E Ec (resp. T Tc) while they are microcanonically (resp.≥ canonically)≥ unstable for 0.6 E Ec (resp. T Tc). On the other hand, the inhomoge-

B/kM ≤ ≤

π neous states are always stable. The caloric curve therefore 0.4 exhibits a microcanonical second order phase transition b=2 ′ ′′ marked by the discontinuity of β (E)= S (E) at E = Ec 0.2 and a canonical second order phase transition marked by ′ ′′ ε the discontinuity of E (β) = (βF ) (β) at β = βc. The en- min ε c B = 0 sembles are equivalent. In order to illustrate our method, 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 which remains valid in more general situations, we shall 2 ε=8πE/kM however treat both canonical and microcanonical ensem- Fig. 3. Magnetization (order parameter) as a function of en- bles. ergy.

3 Canonical ensemble 1

B=0 3.1 The functionals F [f] and F [ρ] 0 ε c The minimization problem (13) has several interpreta- -1 tions: -2 (i) It determines the statistical equilibrium state of

s=S/M B=0 the HMF model in the canonical ensemble. In that ther- -3 B > 0 modynamical interpretation S is the Boltzmann entropy, F is the Boltzmann free energy and T is the thermody- -4 namical temperature. The minimization problem (13) can therefore be interpreted as a criterion of canonical ther- -5 modynamical stability. For isolated systems that evolve at -2 -1 0 1 2 3 2 ε=8πE/kM fixed energy, like the HMF model, the canonical ensemble is not physically justified. For such systems, the proper Fig. 4. Entropy as a function of energy. ensemble to consider is the microcanonical ensemble and the statistical equilibrium state is given by (12). However, the canonical ensemble always provides a sufficient condition of microcanonical stability [6] and it can be useful of Inagaki [14], is the most direct and the most complete. in that respect4. Indeed, if a system is canonically stable Indeed, these relations characterize all the critical points of (12) and (13). Now, we must select among these critical 4 For systems with short-range interactions, the ensembles points those that are (local) maxima of S at fixed E and are equivalent and we can choose the one that is the most M (microcanonical ensemble) and those that are (local) convenient to make the calculations. For systems with long- minima of F at fixed M (canonical ensemble). This is the range interactions, the ensembles may not be equivalent: grand object of the next sections. canonical stability implies canonical stability which itself im- P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 7 at the temperature T , then it is automatically granted to for all perturbations δf that do not change the mass: be microcanonically stable at the corresponding energy δM = 0. On the other hand, the critical points of (42) are E = E(T ). Therefore, we can start by this ensemble and given by equations (24) and (22) where β is prescribed. A consider the microcanonical ensemble only if the canoni- critical point of F [ρ] at fixed mass is a (local) minimum cal ensemble does not cover the whole range of energies. iff On the other hand, for systems in contact with a thermal 1 (δρ)2 bath fixing the temperature, like the Brownian Mean Field δ2F = δρδΦdθ + T dθ > 0, (46) (BMF) model [17], the canonical ensemble is the proper 2 2ρ ensemble to consider and the statistical equilibrium state Z Z is given by (13). for all perturbations δρ that conserve mass: δM = 0. This (ii) It determines a particular steady state of the stability criterion is equivalent to the stability criterion Vlasov equation that is formally nonlinearly dynamically (45) but it is simpler because it is expressed in terms of stable5. The minimization problem (13) can therefore be the density instead of the distribution function [37]. interpreted as a sufficient condition of dynamical stabil- Remark 1: the thermodynamical approach of Messer & ity. In that dynamical interpretation, S is a particular Spohn [11] in the canonical ensemble directly leads to the Casimir (pseudo entropy), F is an energy-Casimir func- minimization problem (42) for the density, and justifies it tional (pseudo free energy) and T is a positive constant. rigorously. Our approach recovers this result by another It can be convenient to develop a thermodynamical anal- method. It also shows that this minimization problem pro- ogy [34] to study this dynamical stability problem and use vides a sufficient condition of dynamical stability with re- a common vocabulary. In this way, the methods devel- spect to the Vlasov equation. oped in thermodynamics can be applied to the dynamical Remark 2: considering the BMF model [17], the mini- stability context. mization problem (13) determines stable steady states of It is shown in Appendix A.2. of [37] that the solution the mean field Kramers equation and the minimization of (13) is given by problem (42) determines stable steady states of the mean field Smoluchowski equation. According to the equivalence 1/2 2 β v (44), we conclude that a distribution function f(θ, v) is a f(θ, v)= ρ(θ) e−β 2 , (41) 2π stable steady state of the mean field Kramers equation iff the corresponding density field ρ(θ) is a stable steady where ρ(θ) is the solution of state of the mean field Smoluchowski equation (see [38] for a more general statement). min F [ρ] M[ρ]= M , (42) ρ { | } Remark 3: the equivalence between (13) and (42) can be extended to a larger class of functionals of the form where S[f]= C(f) dθdv where C is convex. Such functionals 1 (Casimirs)− arise in the Vlasov dynamical stability problem F [ρ]= ρΦdθ + T ρ ln ρ dθ. (43) 2 [28,29,33].R We refer to [17,34,39] for a detailed discussion Z Z of this equivalence. Therefore, the minimization problems (13) and (42) are equivalent: (13) (42). (44) 3.2 The function F (B) ⇔ This equivalence holds for global and local minimization 3.2.1 Global minimization [37]: (i) f(θ, v) is the global minimum of (13) iff ρ(θ) is the global minimum of (42) and (ii) f(θ, v) is a local minimum The equivalence (44) is valid for an arbitrary potential of of (13) iff ρ(θ) is a local minimum of (42). We are therefore interaction u(r, r′). Now, for the HMF model, the problem led to considering the minimization problem (42) which is can be further simplified. Indeed, the potential energy is simpler to study since it involves the density ρ(θ) instead given by equation (9) so that the free energy (43) can be of the distribution function f(v,θ). rewritten Before that, let us compare the conditions of stability issued from (13) and (42). The critical points of (13) πB2 are given by equations (23), (24) and (22) where β is pre- F [ρ]= + T ρ ln ρ dθ. (47) − k scribed. A critical point of F [f] at fixed mass is a (local) Z minimum iff We shall first determine the global minimum of free en- 1 (δf)2 ergy at fixed mass. To that purpose, we shall reduce the δ2F = δρδΦdθ + T dθdv > 0, (45) 2 2f minimization problem (47) to an equivalent but simpler Z Z minimization problem. plies microcanonical stability, but the converse is wrong in gen- To solve the minimization problem (47), we proceed in 6 eral. two steps : we first minimize F [ρ] at fixed M and Bx and 5 This is a refined condition of formal stability with respect to the usual criterion [33] since the mass is treated here as a 6 We have used this method in different situations (see, e.g. constraint (see [28,29] for a more detailed discussion). [34,36,37]). 8 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model

1 By. Writing the variational principle as

T = Tc 0.8 T > Tc δ ρ ln ρ dθ + αδM + µxδBx + µyδBy =0, (48) Z T < Tc we obtain 0.6 B/kM π −λx cos θ−λy sin θ ρ1(θ)= Ae , (49) 0.4 b=2 −1−α k k where A = e , λx = 2π µx and λy = 2π µy. The Lagrange multipliers are determined− by the constraints− M, 0.2 Bx and By. If we write Bx = B cos φ and By = B sin φ 0 then we find that λx = λ cos φ and λy = λ sin φ with 0 2 4 6 8 10 2 2 1/2 λ λ = (λx + λy) . Equation (49) can be rewritten Fig. 6. Graphical construction determining the critical points −λ cos(θ−φ) ρ1(θ)= Ae . (50) of F (B) and their stability. The critical points are determined by the intersection(s) between the curve b = b(λ) defined by Finally, A and λ are determined in terms of M and B equation (52) and the straight line b = λ/(2η). The critical through the equations point is a minimum (resp. maximum) of F (B) if the slope of M the curve b(λ) at that point is smaller (resp. larger) than the A = , (51) straight line b = λ/(2η). 2πI0(λ) and 2πB I (λ) b = 1 . (52) The critical points of F (B), satisfying F ′(B) = 0, corre- ≡ kM I0(λ) spond therefore to Equation (50) is the (unique) global minimum of F [ρ] with 2 2 1 (δρ) λ = x βB. (57) the previous constraints since δ F = 2 T ρ dθ > 0 ≡ (the constraints are linear in ρ so that their second varia- R Substituting this result in equation (52), we obtain the tions vanish). Then, we can express the free energy F [ρ] as self-consistency relation a function of B by writing F (B) F [ρ ]. After straight- ≡ 1 forward calculations, we obtain 2πB I1(βB) = , (58) kM I (βB) πB2 2πB 0 F (B)= + T λ MT ln I0(λ), (53) − k k − which determines the magnetization B as a function of the temperature T . This returns the results of Sec. 2. where λ(B) is given by equation (52). Finally, the mini- Now, a critical point of F (B) is a minimum if F ′′(B) > mization problem (42) is equivalent to the minimization 0 and a maximum if F ′′(B) < 0. Differentiating equation problem (56) with respect to B, we find that min F (B) , (54) B { } 2π dλ F ′′(B)= T 1 . (59) k dB − in the sense that the solution of (42) is given by equations (50), (51) and (52) where B is the solution of (54). Note that the mass constraint is taken into account implicitly Therefore, a critical point is a minimum if in the variational problem (54). Therefore, (42) and (54) dB are equivalent for global minimization. However, (54) is < T, (60) much simpler because, for given T and M, we just have dλ to determine the minimum of a function F (B) instead of and a maximum when the inequality is reversed. the minimum of a functional F [ρ] at fixed mass. We can determine the minimum of the function F (B) Let us therefore study the function F (B) defined by by a simple graphical construction. To that purpose, equations (53) and (52). Its first derivative is we plot b 2πB/kM as a function of λ according to equation (52).≡ This is represented in Figure 6. We find 2πB 2π 2πB I′ (λ) dλ F ′(B)= + T λ + MT 0 . that 2πB/kM 1 for λ + . On the other hand, − k k kM − I (λ) dB → → ∞ 0 2πB/kM λ/2 for λ 0. Therefore, the magnetization ∼ → kM (55) B takes values between 0 and Bmax = 2π . According ′ to equation (57), the critical points of F (B) are deter- Using the identity I0(λ)= I1(λ) and the relation (52), we mined by the intersection of this curve with the straight see that the term in parenthesis vanishes. Then, we get kM line 2πB/kM = (2πT/kM)λ. For T >Tc 4π , there is a unique solution B = 0 corresponding to the≡ homogeneous ′ 2π F (B)= (T λ B). (56) phase. For T

0.4 For η > ηc = 1, we explicitly check that the minimum ′ ′′ T > T satisfying f (b) = 0 and f (b) > 0 is given by the first c T = T c relation in equation (37). 0.2 Remark 1: In Appendix E, we plot the second variations of free energy F ′′(B(T )) (related to the variance of the magnetization) as a function of the temperature and 0 T < T F(B) c recover the previous conditions of stability. Remark 2: since the phase φ does not appear in the -0.2 function (53), this means that the inhomogeneous minima of F (B) for T Tc, this curve has a (unique) global minimum at B = 0. For T Ec and the inhomogeneous curve has a local maximum at B = 0 and a global minimum solution for E < Ec. Since we cover all the accessible range at B(T ) > 0. of energies, we conclude that the ensembles are equivalent. We shall, however, treat the microcanonical ensemble specifically in Sec. 4 since the method can be useful in other contexts where the ensembles are not equivalent solution B = 0 and an inhomogeneous solution B(T ) = 0. (see, e.g. [24]). According to inequality (60), a solution is a minimum6 of F (B) if d(2πB/kM)/dλ < 2πT/kM and a maximum if 3.2.2 Local minimization d(2πB/kM)/dλ > 2πT/kM. Therefore, a critical point of free energy F (B) is a minimum (resp. maximum) if the We shall now show that the minimization problems (42) slope of the main curve is lower (resp. higher) than the and (54) are also equivalent for local minimization. To slope of the straight line at the point of intersection. From that purpose, we shall relate the second order variations this criterion, we easily conclude that: for T >T , the ho- c of F [ρ] to the second variations of F (B) by using a suitable mogeneous solution B = 0 is the global minimum of F (B); decomposition7. for T Tc and the where α is a Lagrange multiplier accounting for the con- inhomogeneous solution for T 0, Eliminating λ between the expressions (61) and (52), we − k 2 ρ Z obtain the free energy f(b) as a function of the magne- (65) tization b for a fixed value of the temperature T (more precisely, for given η, these equations determine f(b) in a for all perturbations δρ that conserve mass: δρdθ = 0. parametric form). For T >Tc and T

k For λ = βB, according to the self-consistency relation δBy = δρ sin θ dθ. (67) −2π (22), we have B(βB) = B. Diﬀerentiating equation (79) Z with respect to λ, we get We can always write the perturbations in the form ′ ′ 2π ′ I1(λ)I0(λ) I0(λ)I1(λ) δρ = δρk + δρ⊥, (68) B (λ)= − 2 . (80) kM I0(λ) where δρk = (µ + νx cos θ + νy sin θ)ρ and δρ⊥ δρ δρk. ′ ′ 1 ≡ − Using the identities I0(λ) = I1(λ), I1(λ) = 2 (I0(λ) + The second condition ensures that all the perturbations 2 I2(λ)) and I0(x) I2(x)= I1(x), and recalling equation are considered. We shall now choose the constants µ, νx x (79), the foregoing− relation can be rewritten and νy such that 2 2 2π ′ 1 I1(λ) 4π B(λ) δρk dθ =0, (69) B (λ)=1 2 2 . (81) kM − λ I0(λ) − k M Z Taking λ = βB and introducing the function λ(B), which k is the inverse of B(λ), we obtain the identity δB = δρ cos θ dθ, (70) x −2π k 2 2 Z 2π 1 1 I1(βB) 4π B ′ =1 2 2 . (82) kM λ (B) − βB I0(βB) − k M k δB = δρ sin θ dθ. (71) y −2π k Comparing this relation with equation (78), we obtain Z This implies that 2π 4π2B2 I = + . (83) kλ′(B) k2M δρ⊥ dθ =0, (72) Z Solving equations (74), (75) and (76) for µ, νx and νy, and using the result (83), we ﬁnd that

′ δρ⊥ cos θ dθ =0, δρ⊥ sin θ dθ =0. (73) ′ 2πBλ (B) νx = λ (B)δBx, µ = δBx, (84) Z Z − − kM The conditions (69), (70) and (71) lead to the relations

2π δBy 2π νy = . (85) µM νxB =0, (74) − k M I − k − Therefore, the perturbation δρk takes the form

kνx δBx = µB I, (75) 2πB − 2π δρ = λ′(B) + cos θ ρ(θ)δB k − kM x 2π δB kν sin θρ(θ) y . (86) δB = y (M I), (76) − k M I y − 2π − − By construction, δρ and δρ are orthogonal for the scalar where we have deﬁned k ⊥ product weighted by 1/ρ in the sense that

2 I ρ cos θ dθ. (77) δρ δρ⊥ ≡ k dθ =0. (87) Z ρ This forms a system of three algebraic equations that de- Z termines the three constants µ, νx and νy. Using the equi- Indeed, we have librium distribution (64), we ﬁnd after simple algebra that δρ δρ⊥ k dθ = (µ + ν cos θ + ν sin θ)δρ dθ =0, 1 I (βB) ρ x y ⊥ I = M 1 1 , (78) Z Z − βB I (βB) (88) 0 2 where we have used the identity I0(x) I2(x) = x I1(x). where we have used equations (72) and (73) to get the last Let us deﬁne the function B(λ) by the− relation equality. As a result, we obtain

2 2 2 2πB(λ) I1(λ) (δρ) (δρ ) (δρ ) = . (79) dθ = k dθ + ⊥ dθ. (89) kM I0(λ) ρ ρ ρ Z Z Z P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 11

Using equations (86) and (83), we obtain after simpliﬁca- 4 Microcanonical ensemble tion 4.1 The functionals S[f] and S[ρ] (δρ )2 2π 4π2 1 k dθ = λ′(B)(δB )2 + (δB )2. ρ k x k2 M I y The maximization problem (12) has several interpreta- Z − (90) tions: (i) It determines the statistical equilibrium state of the Therefore, the second order variations of free energy given HMF model in the microcanonical ensemble. In that ther- by equation (65) can be written modynamical interpretation S is the Boltzmann entropy. The maximization problem (12) can therefore be inter- 2 π ′ 2 δ F = (T λ (B) 1)(δBx) preted as a criterion of microcanonical thermodynamical k − stability. 2 π 2π T 1 (δρ⊥) (ii) It determines a particular steady state of the + 1 (δB )2 + T dθ. (91) k k M I − y 2 ρ Vlasov equation that is formally nonlinearly dynamically − Z stable8. The maximization problem (12) can therefore be Using identity (59), we obtain interpreted as a suﬃcient condition of dynamical stability. In that dynamical interpretation, S is a particular Casimir 2 1 ′′ 2 δ F = F (B)(δBx) (pseudo entropy). As explained previously, it is convenient 2 to develop a thermodynamical analogy [34] to study this 2 π 2π T 1 (δρ⊥) dynamical stability problem and use a common vocabu- + 1 (δB )2 + T dθ. (92) k k M I − y 2 ρ lary. − Z It is shown in Appendix A.1. of [37] that the solution For the inhomogeneous phase B = 0, using equation (78) of (12) is given by and the self-consistency relation (22),6 we get 1/2 2 β v 2π f(θ, v)= ρ(θ) e−β 2 , (98) I = M T. (93) 2π − k In that case, equation (92) reduces to where β =1/T is determined by the energy constraint

2 1 2 1 ′′ 2 1 (δρ⊥) E = MT + W, (99) δ F = F (B)(δBx) + T dθ. (94) 2 2 2 ρ Z and ρ(θ) is the solution of On the other hand, for the homogeneous phase B = 0, equation (78) leads to max S[ρ] M[ρ]= M , (100) ρ { | } M I = . (95) 2 where 1 Using equation (206), equation (92) reduces to S[ρ]= M ln T ρ ln ρ dθ. (101) 2 − Z 1 1 (δρ )2 δ2F = F ′′(0) (δB )2 + (δB )2 + T ⊥ dθ. Eliminating the temperature thanks to the constraint 2 x y 2 ρ Z (99), we can write the entropy in terms of ρ alone as (96) 1 S[ρ]= ρ ln ρ dθ + M ln(E W [ρ]). (102) These relations show that ρ is a local minimum of F [ρ] at − 2 − fixed mass iff B is a local minimum of F (B). Indeed, if Z F ′′(B) > 0 then δ2F > 0 since the last term is positive. Therefore, the maximization problems (12) and (100) are On the other hand, if F ′′(B) < 0 it suffices to consider equivalent: a perturbation of the form (68) with δρ⊥ = 0 and δρk 2 (12) (100). (103) given by equation (86) to conclude that δ F < 0 for this ⇔ perturbation. This implies that ρ is not a local minimum This equivalence holds for global and local maximization of F [ρ] since there exists a particular perturbation that [37]: (i) f(θ, v) is the global maximum of (12) iff ρ(θ) is decreases the free energy. This is the case for the homoge- the global maximum of (100) and (ii) f(θ, v) is a local neous solutions when T

(100) which is simpler to study since it involves the density S[f]= 1 (f q f) dθdv associated to polytropic dis- − q−1 − ρ(θ) instead of the distribution function f(v,θ). tributions. Such functionals (Casimirs) arise in the Vlasov R Before that, let us compare the conditions of stability dynamical stability problem. We refer to [24,43] for a de- issued from (12) and (100). The critical points of (12) tailed discussion of this equivalence. are given by equations (23), (24), (22) and (26) where E is prescribed. Furthermore, a critical point of S at ﬁxed mass and energy is a (local) maximum iﬀ (see Appendix 4.2 The function S(B) B):

(δf)2 1 4.2.1 Global maximization δ2S = dθdv β δρδΦdθ < 0, (104) − 2f − 2 Z Z The equivalence (103) is valid for an arbitrary potential of for all perturbations δf that do not change the mass and interaction u(r, r′). Now, for the HMF model, the problem the energy at first order: δM = δE = 0. On the other can be simplified further. Indeed, the potential energy is hand, the critical points of (100) are given by equations given by equation (9) so that the energy (99) and the (24), (22) and (26) where E is prescribed (see Appendix entropy (102) can be rewritten A.1 of [37]). Furthermore, a critical point of S at fixed mass is a (local) maximum iff (see Appendix A.1 of [37]): 1 πB2 E = MT , (106) (δρ)2 1 2 − k δ2S = dθ δρδΦdθ − 2ρ − 2T Z Z 2 1 2 1 πB Φδρ dθ < 0, (105) S[ρ]= ρ ln ρ dθ + M ln E + . (107) −MT 2 − 2 k Z Z for all perturbations δρ that conserve mass: δM = 0. This We shall first determine the global maximum of entropy stability criterion is equivalent to the stability criterion at fixed mass. To that purpose, we shall reduce the maxi- (104) but it is simpler because it is expressed in terms of mization problem (100) to an equivalent but simpler max- the density instead of the distribution function. imization problem. Remark 1: the thermodynamical approach of Kiessling To solve the maximization problem (100), we proceed [42] in the microcanonical ensemble rigorously justifies the in two steps: we first maximize S[ρ] at fixed M and Bx maximization problem (12). and By. Writing the variational problem as Remark 2: comparing the stability criteria (45) and (104), we see that canonical stability implies microcanon- δ ρ ln ρ dθ αδM µ δB µ δB =0, ical stability in the sense that a (local) minimum of F at − − − x x − y y fixed mass is necessarily a (local) maximum of S at fixed Z (108) mass and energy. Indeed, if inequality (45) is satisfied for all perturbations that conserve mass, then inequality (104) and proceeding as in Sec. 3.2.1, we obtain is satisfied a fortiori for all perturbations that conserve mass and energy. However, the reciprocal is wrong in case −λ cos(θ−φ) of ensembles inequivalence. Therefore, we just have the ρ1(θ)= Ae , (109) implication (13) (12). This result can also be obtained where A and λ are determined by the constraints M and by comparing the⇒ stability criteria (46) and (105). Indeed, B through since the last term in equation (105) is negative, it is clear M that if inequality (46) is satisfied, then inequality (105) A = , (110) is automatically satisfied. In general this is not reciprocal 2πI0(λ) and we may have ensembles inequivalence. However, if we and consider a spatially homogeneous system for which Φ is 2πB I1(λ) uniform, the last term in equation (105) vanishes (since b = . (111) ≡ kM I (λ) the mass is conserved) and the stability criteria (46) and 0 (105) coincide. Therefore, for spatially homogeneous sys- Equation (109) is the (unique) global maximum of S[ρ] 2 tems, we have ensembles equivalence. with the previous constraints since δ2S = 1 (δρ) dθ < Remark 3: according to the two interpretations of (12) − 2 ρ 0 (the constraints are linear in ρ so that their second order recalled at the beginning of this section, we note that R thermodynamical stability implies dynamical stability (for variations vanish). Then, we can express the entropy S as a function of B by writing S(B) S[ρ ]. After straight- isothermal distributions). However, the converse is wrong ≡ 1 since (12) provides just a sufficient condition of dynami- forward calculations, we obtain cal stability. More refined stability criteria are discussed 2πB M πB2 in [28,29]. S(B)= M ln I (λ) λ + ln E + , 0 − k 2 k Remark 4: the equivalence between (12) and (100) can be extended to a larger class of functionals of the form (112) P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 13 where λ(B) is given by equation (111). Finally, the maxi- E > E mization problem (100) is equivalent to the maximization c problem 0.2 max S(B) , (113) E = E B { } c 0 in the sense that the solution of (100) is given by equa- S(B) tions (109)-(111) where B is the solution of (113). Note that the energy and mass constraints are taken into ac- -0.2 B(E) E < E count implicitly in the variational problem (113). There- c fore, (100) and (113) are equivalent for global maximiza- -0.4 tion. However, (113) is much simpler because, for given E 0 0.2 0.4 0.6 0.8 1 and M, we just have to determine the maximum of a func- 2πB/kM tion S(B) instead of the maximum of a functional S[ρ] at Fig. 8. Entropy S(B) as a function of magnetization for a fixed mass and energy. given value of energy. For E>Ec, this curve has a (unique) Let us therefore study the function S(B) defined by global maximum at B = 0. For E 0.

′ I1(λ) 2πB dλ 2π πB M S (B)= M λ + πB2 . I0(λ) − kM dB − k k E + k and a minimum if the inequality is reversed. Since the last (114) term in equation (120) is negative, we recover the fact ′ that canonical stability implies microcanonical stability. Using the identity I0(x) = I1(x) and the relation (111), we see that the term in parenthesis vanishes. Then, we get Indeed, if the critical point is a minimum of free energy (F ′′(B) > 0), then it is a fortiori a maximum of entropy 2π B (S′′(B) < 0). Furthermore, we know that the series of S′(B)= λ , (115) k T − equilibria E(T ) is monotonic (see Sec. 2). Therefore, the homogeneous solution is a maximum of S(B) for E > Ec where T is determined by equation (106). The critical since it is a minimum of F (B) for T >Tc. On the other points of S(B), satisfying S′(B) = 0, correspond to hand, the inhomogeneous solution is a maximum of S(B) for E < Ec since it is a minimum of F (B) for T Ec and the inhomogeneous state for E < Ec. On the other hand, since the 1 πB2 ensembles are equivalent for homogeneous solutions (see E = MT , (118) 2 − k Remark 2 in Sec. 4.1) and since we have established that the homogeneous solution is a saddle point of free energy which determine the magnetization as a function of the for T 0. Differentiating equation To complete our analysis, it can be useful to plot the (115) with respect to B, and recalling that the tempera- function S(B) for prescribed mass and energy. Using equa- ture T is a function of B given by equation (106), we find tions (112) and (111), the normalized entropy s S/M that can be expressed in terms of λ according to ≡ 2π 1 dλ 4πB2 S′′(B)= . (119) 2 k T − dB − kMT 2 I1(λ) 1 2I1(λ) s(λ) = ln I0(λ) λ + ln ǫ + 2 . (122) − I0(λ) 2 I0(λ) Using equation (59), we note that Eliminating λ between expressions (122) and (111), we F ′′(B) 8π2B2 S′′(B)= . (120) obtain the entropy s(b) as a function of the magnetization − T − k2MT 2 b for a fixed value of the energy E (more precisely, for Therefore, a critical point of S(B) is a maximum if given ǫ, these equations determine s(b) in a parametric form). For E > Ec and E < Ec, this function displays the dλ 1 4πB2 two behaviors described above, as illustrated in Figure 8. > , (121) For E E so that λ, b 0, we find that the entropy dB T − kMT 2 → c → 14 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model takes the approximate form and for B = 0: 2 1 F ′′(0) 1 1 2 ǫ +4 4 2 2 2 s(b) ln ǫ + 1 b b , (ǫ 1). δ S = (δBx) + (δBy) ≃ 2 ǫ − − 4ǫ2 → −2 T 1 (δρ )2 (123) ⊥ dθ. (129) −2 ρ For ǫ< 1, we explicitly check that the maximum satisfying Z s′(b) = 0 and s′′(b) < 0 is given by the second relation in Finally, using identity (120), we arrive at equation (37). 1 1 (δρ )2 Remark: In Appendix E, we plot the second variations δ2S = S′′(B)(δB )2 ⊥ dθ, (130) of entropy S′′(B(E)) (related to the variance of the mag- 2 x − 2 ρ netization) as a function of energy and recover the previ- Z for B = 0 and ous conditions of stability. 6 1 1 (δρ )2 δ2S = S′′(B) (δB )2 + (δB )2 ⊥ dθ, 4.2.2 Local maximization 2 x y − 2 ρ Z (131) We shall now show that the maximization problems (100) and (113) are also equivalent for local maximization. To for B = 0. These relations imply that ρ is a local maximum that purpose, we shall relate the second order variations of S[ρ] at fixed mass iff B is a local maximum of S(B). of S[ρ] and S(B) by using a suitable decomposition. Indeed, if S′′(B) < 0, then δ2S < 0 since the last term is A critical point of (100) is determined by the varia- negative. On the other hand, if S′′(B) > 0, it suffices to tional principle consider a perturbation of the form (68) with δρ⊥ = 0 and 2 δρk given by equation (86) to conclude that δ S > 0 for δS αδM =0, (124) − this perturbation. This implies that ρ is not a local maximum of S[ρ] since there exists a particular perturbation where α is a Lagrange multiplier accounting for the con- that increases the entropy. This is the case for homoge- servation of mass. This leads to the distribution (see Ap- neous solutions when E < E . Therefore, (100) and (113) pendix A.1. of [37]): c are equivalent for local maximization. Combining all our results, we conclude that the variational problems (12), M −βB cos θ ρ = e , (125) (100) and (113) are equivalent for local and global maxi- 2πI (βB) 0 mization where the temperature is determined by the energy ac- (12) (100) (113). (132) cording to equation (106). The magnetization B is ob- ⇔ ⇔ tained by substituting equation (125) in equations (5)-(6) leading to the self-consistency relation (22). Using equations (105) and (3), a critical point of S[ρ] at fixed mass 5 Conclusion is a (local) maximum iff In this paper, we have presented a new method to settle (δρ)2 π the stability of homogeneous and inhomogeneous isother- δ2S = dθ + ((δB )2 + (δB )2) − 2ρ kT x y mal distributions in the HMF model. This method starts Z 2 2 from general variational principles and transforms them 4π B 2 2 2 (δBx) < 0, (126) into equivalent but simpler variational principles until a −k MT point at which the problem can be easily solved. For for all perturbations δρ that conserve mass: δρdθ = 0. isothermal distributions, this method returns, as expected, We note that the second order variations of entropy (126) the same results as those obtained in the past by different are related to the second order variations ofR free energy procedures [4,29,30], but we would like to emphasize why (65) by our approach is interesting and complementary to other methods. 1 4π2B2 First of all, it is based on general optimization prob- δ2S = δ2F (δB )2. (127) −T − k2MT 2 x lems: the maximization of S[f] at fixed mass M and energy E or the minimization of F [f] at fixed mass M. Writing the perturbation δρ in the form (68) with equation These optimization problems provide either conditions of (86) and using expressions (94) and (96), we obtain for thermodynamical stability (in microcanonical and canoni- B = 0: 6 cal ensembles respectively) or sufficient conditions of dynamical stability (more or less refined) with respect to 1 F ′′(B) 8π2B2 δ2S = + (δB )2 the Vlasov equation. Therefore, our approach allows us to −2 T k2MT 2 x treat thermodynamical and dynamical stability problems 1 (δρ )2 with the same formalism. This is not possible if we follow ⊥ dθ. (128) −2 ρ an approach starting directly from the density of states Z P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 15 g(E) or the partition function Z(β) which only applies method and to [45] for refined stability criteria (in the to the thermodynamical problem [11,16,30]. As a conse- context of 2D turbulence). quence, our procedure still works if the Boltzmann func- Thirdly, our method determines not only the strict tional is replaced by more general functionals of the form caloric curve β(E) (corresponding to global maxima of S S[f]= C(f) drdv, where C(f) is convex. In that case, at fixed E and M, or global minima of F at fixed M) but the maximization− of S at fixed M and E or the minimiza- it also provides the whole series of equilibria containing all tion of FR at fixed M provide conditions of formal dy- the critical points of S at fixed E and M, or the critical namical stability for arbitrary steady states of the Vlasov points of F at fixed M. In particular, it allows us to deter- equation of the form f = f(ǫ) with f ′(ǫ) < 0 [29]. This mine metastable states that correspond to local maxima dynamical stability problem has been studied recently for of S at fixed E and M, or local minima of F at fixed M. polytropic distributions in [24]. Our procedure could also There are no such metastable states in the HMF model be employed for the Lynden-Bell entropy [18], although for isothermal distributions, but they can exist in other the calculations would be more complicated. We have cho- situations [22,24]. Our approach can be used to determine sen to treat here the isothermal case in detail since the cal- the sign of the second order variations of the thermody- culations are fully analytical. It offers therefore a simple namical potential in order to settle whether the critical illustration of the method. point is a global maximum, a local maximum or a saddle Secondly, the optimization problems (12) and (13) on point. In particular, we have been able to relate the sec- which our approach is based are both intuitive and rigor- ond variations of the different functionals in order to show ous. The thermodynamical approach is intuitive because that the equivalence between the optimization problems the Boltzmann entropy S[f] can be obtained from a simple is not only global but also local. combinatorial analysis. In that case, the entropy is pro- For these reasons, the approach developed in the portional to the logarithm of the disorder where the dis- present paper is an interesting complement to other meth- order measures the number of microstates associated with ods [4,16,29,30] and it could find application and useful- a given macrostate. Therefore, maximizing entropy S[f] ness in more general situations. at fixed mass and energy amounts to selecting the most probable macroscopic state consistent with the dynamical constraints9. In the context of the HMF model, this A Asymptotic expansions thermodynamical approach has been initiated in [14,17] but was not performed to completion since the stability of In this section, we give the asymptotic expansions of b(x), the inhomogeneous phase was not proven (at least analyt- η(x), ǫ(x), s(x) and f(x) for x 0 (corresponding to ically) by this method. This has been done in the present the bifurcation point) and x + →(corresponding to the paper. The dynamical approach is also intuitive because ground state). → ∞ we qualitatively understand that the stability of a dy- For x 0, we have namical system is linked to the fact that the system is → in the minimum of a certain “potential”. For infinite di- 3 5 x x x 6 mensional systems, the nonlinear dynamical stability of a b(x)= + + o(x ), (133) 2 − 16 96 steady state of the Vlasov equation relies on the energy- Casimir method and its generalizations [29]. In the context of the HMF model, this approach has been followed x2 x4 η(x)=1+ + o(x6), (134) in [17,25,29]. The optimization problems (12) and (13) 8 − 192 are also rigorous because they have been given a precise justification by mathematicians. In statistical mechanics, the canonical criterion (13) has been justified rigorously 5x2 7x4 ǫ(x)=1 + + o(x6), (135) in [11] and the microcanonical criterion (12) in [42]. En- − 8 48 sembles inequivalence has been formalized in [6]. On the other hand, the formal and nonlinear dynamical stabil- 5x2 41x4 ity of a steady state of the Vlasov equation have been s(x)= + + o(x6), (136) discussed extensively in the mathematical literature. We − 16 768 refer to [33] for a survey on the standard energy-Casimir

4 9 5x This result can also be derived from kinetic theory. For f(x)=1 + o(x6). (137) isolated systems, the evolution of the distribution function is − 128 governed by a kinetic equation that monotonically increases For x + , we have entropy S[f] while conserving mass and energy until the maxi- → ∞ mum entropy state is reached. Similarly, for systems in contact 1 1 −3 with a heat bath, the evolution of the distribution function is b(x)=1 2 + o(x ), (138) governed by a kinetic equation that monotonically decreases − 2x − 8x free energy F [f] while conserving mass until the minimum free energy state is reached [44]. These H-theorems are another way x 1 3 3 η(x)= + + + + o(x−3), (139) to justify the optimization problems (12) and (13). 2 4 16x 16x2 16 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model

4 1 ǫ(x)= 2+ + o(x−3), (140) C Connection between statistical mechanics − x − x2 and variational principles

1 1 s(x)= ln x + (1 ln π)+ + o(x−3), (141) In this section, we discuss the connection between the sta- − 2 − 16x2 tistical mechanics of systems with long-range interactions and variational principles. In the microcanonical ensemble, the accessible conﬁg- 2 + 2 ln π + 4 ln x f(x)= 2+ urations (those having the proper value of energy) are − x equiprobable. Therefore, the density probability of the ln π + 2 ln x + o(x−3). (142) conﬁguration (θ1, v1, ..., θN , vN ) is PN (θ1, v1, ..., θN , vN )= − x2 1 δ(E H) where g(E) is the density of states g(E) −

g(E)= δ(E H) dθ dv ...dθ dv . (149) B Derivation of inequality (104) − 1 1 N N Z We consider a perturbation δf around a distribution f. In other words, g(E)dE gives the number of microstates The exact variations of mass (7) and energy (8) for any with energy between E and E+dE. The entropy is defined perturbation are by S(E) = ln g(E). Let us introduce the (coarse-grained) one-body distribution function f(θ, v). A microstate is de- ∆M = δf dθdv, (143) termined by the specification of the exact positions and velocities θi, vi of the N particles. A macrostate is de- Z termined by{ the} specification of the density f(θ, v) of particles in each cell [θ,θ + dθ] [v, v + dv] irrespectively{ } v2 1 of their precise position in the× cell. Let us call Ω[f] the ∆E = δf dθdv + Φδρ dθ + δρδΦ dθdv. 2 2 unconditional number of microstates corresponding to the Z Z Z (144) macrostate f. For N + , equation (149) can be formally rewritten → ∞ Considering small perturbations δf, the variations of entropy (10) up to second order is g(E) Ω[f]δ(E[f] E)δ(M[f] M) f, (150) ≃ − − D Z (δf)2 ∆S = (ln f + 1)δf dθdv dθdv. (145) where E[f] is the mean field energy (8) and M[f] is the − − 2f Z Z mass (7). The entropy of the macrostate f is defined by Let us now assume that f is a critical points of entropy at the Boltzmann formula S[f] = ln Ω[f]. The Boltzmann fixed mass and energy. It is determined by the variational entropy can be obtained by a standard combinatorial analysis leading, for N + , to equation (10). Therefore, principle (14), leading to → ∞ using Ω[f]= eS[f], equation (150) can be rewritten v2 ln f +1= β + Φ α. (146) − 2 − g(E) eS[f]δ(E[f] E)δ(M[f] M) f. (151) ≃ − − D Z Substituting this relation in equation (145), we obtain The unconditional density probability of the distribution 1 S[f] v2 (δf)2 f is P0[f] = A e (where is the hypervolume of the ∆S = β + Φ + α δf dθdv dθdv. system in phase space). TheA microcanonical density prob- 2 − 2f Z Z ability of the distribution f is P [f] = 1 eS[f]δ(E[f] (147) g(E) − E)δ(M[f] M). − Now, using the conservation of mass and energy ∆M = Integrating over the velocities in equation (149), a clas- ∆E = 0, we get sical calculation leads to N/2 2 2π N−2 (δf) 1 g(E)= [2(E U)] 2 dθ ...dθ , (152) ∆S = dθdv β δρδΦ dθdv. (148) N − 1 N − 2f − 2 Γ 2 Z Z Z Requiring that the critical point be a maximum of entropy where U(θ1, ..., θN ) is the potential energy (second term at fixed mass and energy leads to inequality (104). We re- in the r.h.s. of equation (1)). Let us introduce the (coarse- fer to [29,46] for generalizations of this result to a larger grained) one-body density ρ(θ) and denote by Ω[ρ] the class of functionals in the Vlasov dynamical stability con- unconditional number of microstates θi corresponding text. to the macrostate ρ. For N + , equation{ } (152) can be → ∞ P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 17 formally rewritten10 The computation of Ω(B) is classical [4] and is briefly reproduced in Appendix D, with some complements, for N g(E) e 2 ln(E−W [ρ])Ω[ρ]δ(M[ρ] M) ρ, (153) the sake of self-consistency. For N + , we have ≃ − D → ∞ Z 2πB − λ+M ln I0(λ) where W [ρ] is the mean field potential energy (second Ω(B)= e k . (160) term in the r.h.s. of equation (8)). The unconditional number of microstates Ω[ρ] can be obtained by a clas- Therefore, the density of states (159) can finally be written sical combinatorial analysis leading, for N + , to − ρ ln ρdθ → ∞ Ω[ρ] = e . Therefore, equation (153) can be g(E) eS(B) dB, (161) ≃ rewritten R Z g(E) eS[ρ]δ(M[ρ] M) ρ, (154) where S(B) is given by equation (112). The unconditional ≃ − D density probability of the magnetization B is P0(B) = Z 2πB 1 − k λ+M ln I0(λ) where S[ρ] is given by equation (102). The uncondi- A e . The microcanonical density probabil- tional density probability of the distribution ρ is P [ρ]= 1 S(B) 0 ity of the magnetization B is P (B)= g(E) e . 1 − ρ ln ρdθ In the canonical ensemble, the density prob- A e (where A is the volume of the system in physical space). The microcanonical density probability ability of the configuration (θ1, v1, ..., θN , vN ) is R 1 −βH of the distribution ρ is P [ρ]= 1 eS[ρ]δ(M[ρ] M). PN (θ1, v1, ..., θN , vN ) = e where Z(β) is the g(E) − Z(β) For the HMF model, let us introduce the magnetiza- partition function tion vector N N −βH k k Z(β)= e dθ1dv1...dθN dvN . (162) B = cos θ , B = sin θ . (155) x −2π i y −2π i Z i=1 i=1 X X The free energy is defined by F (β) = 1 ln Z(β). Intro- − β The potential energy can be expressed in terms of the ducing the unconditional number of microstates Ω[f] cor- magnetization as responding to the macrostate f, we obtain for N + : → ∞ πB2 kN U = + . (156) − k 4π Z(β) e−βE[f] Ω[f] δ(M[f] M) f ≃ − D Therefore, the density of states (152) can be rewritten Z S[f]−βE[f] N−2 e δ(M[f] M) f 2πN/2 πB2 kN 2 ≃ − D g(E)= 2 E + Z N −βF [f] Γ 2 k − 4π e δ(M[f] M) f, (163) Z ≃ − D 2πBx 2πBy Z δ + cos θ δ + sin θ × k i k i where F [f]= E[f] TS[f] is the Boltzmann free energy i ! i ! − X X (11). The canonical density probability of the distribution 2πBx 2πBy 1 −βF [f] d d dθ ...dθ , f is P [f]= Z(β) e δ(M[f] M). × k k 1 N − Integrating over the velocities in equation (162), we (157) get or, equivalently, 2π N/2 N−2 Z(β)= e−βU dθ ...dθ . (164) 2πN/2 πB2 kN 2 β 1 N g(E)= 2 E + Z Γ N k − 4π 2 Z Introducing the unconditional number of microstates Ω[ρ] 2πB 2πB 2πB 2πB Ω x , y d x d y , corresponding to the macrostate ρ, we obtain for N × k k k k → + : (158) ∞ Z(β) e−βW [ρ]Ω[ρ] δ(M[ρ] M) ρ where Ω(B) denotes the unconditional number of mi- ≃ − D crostates θi corresponding to the macrostate B. For Z N + ,{ we} have e− ρ ln ρdθ−βW [ρ] δ(M[ρ] M) ρ → ∞ 2 ≃ − D N πB Z 2 ln E+ R g(E) e k Ω(B) dB. (159) −βF [ρ] ≃ e δ(M[ρ] M) ρ, (165) Z ≃ − D 10 Z Like in the main part of the paper, for the sake of concise- ness, we do not explicitly write the constant terms that are where F [ρ] is given by equation (43). The canonical independent on ρ in the expression of the entropy (the term in density probability of the distribution ρ is P [ρ] = the exponential). They can be restored easily. 1 e−βF [ρ]δ(M[f] M). Z(β) − 18 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model

For the HMF model for which the potential energy can the saddle point approximation. In the microcanonical en- be expressed in terms of the magnetization, the partition semble, we obtain function (164) can be rewritten g(E)= eS(E) eNs[φ∗], (171) ≃ 2π N/2 Z(β)= e−βU i.e. β Z 1 2πBx 2πBy lim S(E)= s[φ∗], (172) δ + cos θ δ + sin θ N→+∞ N × k i k i i ! i ! X X where φ∗ is the solution of the maximization problem 2πBx 2πBy d d dθ1...dθN , × k k max s[φ] E,M . (173) φ { | } (166) This leads to the variational problems (12), (100) and or, equivalently, (113). In the canonical ensemble, we obtain N 2 2 2π β πB − kN −βF (β) −βNf[φ∗] Z(β)= e k 4π Z(β)= e e , (174) β ≃ Z 2πB 2πB 2πB 2πB i.e. Ω x , y d x d y . (167) × k k k k 1 lim F (β)= f[φ∗], (175) N→+∞ N For N + , we get → ∞ where φ∗ is the solution of the minimization problem 2 β πB Z(β) e k Ω(B) dB ≃ min f[φ] M . (176) φ 2 Z { | } πB 2πB β − λ+M ln I0(λ) e k e k dB ≃ This leads to the variational problems (13), (42) and (54). Z The preceding discussion shows the connection be- e−βF (B)dB, (168) ≃ tween the statistical mechanics of systems with long-range Z interactions (based on the calculation of the density of where F (B) is given by equation (53). The canonical states and of the partition function) and variational prin- density probability of the magnetization B is P (B) = ciples (based on the maximization of entropy or minimiza- 1 −βF (B) tion of free energy). It also shows how the variational Z(β) e . Let us now denote by φ a generic global variable such problems (12), (100), (113) and (13), (42), (54) are re- as f(θ, v), ρ(θ) or B. We also recall that for systems lated to each other. These results can be made rigorous with long-range interactions, for which the mean field ap- by using the theory of large deviations. We refer to Ellis proximation is exact in the proper thermodynamic limit [47] for a mathematical presentation of this theory and N + , we have the extensive scalings S[φ] = Ns[φ], to Barré et al. [30] and Touchette [48] for its application E[φ→] = Ne∞ [φ], F [φ] = Nf[φ]. Accordingly, the preceding to physical problems. In the present paper, we have con- results can be formally written sidered a different approach. We started from the fundamental variational problems (12) and (13) that can be motivated by a simple combinatorial analysis. This is the g(E) eNs[φ] δ(c [φ] c ) φ, (169) ≃ MCE − MCE D historical approach of the problem finding its roots in Z Boltzmann’s work. This is also the traditional approach and used by physicists to determine the statistical equilibrium state of self-gravitating systems (see, e.g., [8,10,49,50]), −βNf[φ] two-dimensional point vortices (see, e.g., [1,51]) and the Z(β) e δ(cCE[φ] cCE) φ, (170) ≃ − D HMF model (see, e.g., [14,17]). Therefore, describing the Z statistical mechanics of systems with long-range interac- where the δ-functions take into account the con- tions from the fundamental variational problems (12) and straints as described above. The microcanonical num- (13) and reducing them to simpler but equivalent forms ber of microstates corresponding to the macrostate φ (100), (113), (42) and (54) as we have done here is an in- is Ω[φ] = eNs[φ] δ(c [φ] c ) and the micro- MCE − MCE teresting presentation that complements the one followed canonical probability of the macrostate φ is P [φ] = in [4]. A bonus of this approach is that it remains valid 1 eNs[φ] δ(c [φ] c ). Similarly, the canonical g(E) MCE − MCE when the variational problems (12) and (13) have a dy- number of microstates corresponding to the macrostate namical interpretation in relation to the (formal) nonlin- −βNf[φ] φ is Ω[φ] = e δ(cCE[φ] cCE) and the ear stability of the system with respect to the Vlasov equa- − canonical probability of the macrostate φ is P [φ] = tion [29]. In that case, S[f] is a Casimir functional of the 1 −βNf[φ] e δ(cCE[φ] cCE ). For N + , we can make form S[f] = C(f) dθdv (sometimes called a pseudo Z(β) − → ∞ − R P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 19 entropy) that is more general than the Boltzmann func- Recalling that the potential energy can be expressed in tional arising in the thermodynamical approach. In par- terms of the magnetization according to equation (156), ticular, in the dynamical stability problem, the variational the density probability of the magnetization in micro- problems (12) and (13) cannot be obtained from a theory canonical and canonical ensembles is given by of large deviations since their physical interpretations are N/2 completely different. 2πBx 2πBy 1 2π P , = Remark 1: the density of states (149) and the partition k k g(E) Γ N 2 function (153) can also be calculated from field theoret- N−2 ical methods (see e.g. Horwitz & Katz [52] and de Vega πB2 kN 2 2πB 2πB 2 E + Ω x , y , & Sanchez [53] for self-gravitating systems and Antoni & × k − 4π k k Ruffo [16] for the HMF model). These approaches are valu- (180) able but they are also considerably more abstract than the one based on variational principles. Remark 2: the mean field Boltzmann distribution (16) 2πB 2πB 1 2π N/2 can also be obtained from the first equation of the Yvon- P x , y = k k Z(β) β Bogoliubov-Green (YBG) hierarchy [54], but the condition 2 of stability (related to the correlation functions appearing β πB − kN 2πBx 2πBy e k 4π Ω , , (181) in the next equations of the YBG hierarchy) is more diffi- × k k cult to obtain than with the approach based on variational principles. where Remark 3: for N + , the density of states (resp. → ∞ partition function) is dominated by the global maximum 2πBx 2πBy 2πBx Ω , = δ + cos θ of entropy at fixed mass and energy (resp. minimum of k k k i i ! free energy at fixed mass) according to (171) (resp. (174)). Z X Nevertheless, local entropy maxima (resp. free energy min- 2πBy δ + sin θ dθ ...dθ , ima), i.e. metastable states, are also fully relevant because × k i 1 N i ! they have very long lifetimes scaling like eN [55,56]. This X is particularly true in the case of classical self-gravitating (182) systems for which there is no global entropy maximum (resp. free energy minimum) [8,10]. is the unconditional number of microstates with magnetization B. The calculation of this integral is classical [4]. Using the Fourier representation of the δ-function

D Distribution of the magnetization +∞ dq δ(x)= eiqx , (183) 2π In this Appendix, we determine the distribution of the Z−∞ magnetization by a direct calculation and show its connection with the entropy S(B) and free energy F (B). The we obtain density probability of the magnetization vector deﬁned by N−2 Nh(qx,qy ) equation (155) is Ω = (2π) dqxdqye , (184) Z 2πB 2πB 2πB P x , y = δ x + cos θ where k k k i Z i ! 2π X h(q , q )= i (q B + q B ) + ln J (q), (185) 2πB x y kM x x y y 0 δ y + sin θ P (θ , ..., θ ) dθ ...dθ . (177) × k i N 1 N 1 N i ! 2 2 X and q = qx + qy. Recalling that k 1/N, the function ∼ In the microcanonical ensemble, the N-body distribution h does notq depend on N. For N + , we can make the function is P (θ , v , ..., θ , v ) = 1 δ(E H). Inte- → ∞ N 1 1 N N g(E) − saddle point approximation grating over the velocities, a classical calculation gives ∗ ∗ Ω eNh(qx,qy ) (186) N/2 N−2 1 2π 2 ∼ PN (θ1, ..., θN )= N [2(E U)] , (178) ∗ ∗ g(E) Γ 2 − where (qx, qy) corresponds to the maximum of h(qx, qy). The vanishing of ∂h/∂qx and ∂h/∂qy leads to where U is the potential energy (second term in the r.h.s. of equation (1)). In the canonical ensemble, the 2π J1(q) qx N-body distribution function is PN (θ1, v1, ..., θN , vN ) = i Bx =0, (187) 1 −βH kM − J0(q) q Z(β) e . Integrating over the velocities, we obtain

N/2 1 2π −βU 2π J1(q) qy PN (θ1, ..., θN )= e . (179) i B =0, (188) Z(β) β kM y − J (q) q 0 20 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model where we have used J ′ (x) = J (x). Setting q = iλ E Variance of the magnetization 0 − 1 x − x 2 2 and qy = iλy, we ﬁnd that q = iλ where λ = λx + λy. − In this Appendix, we show the connection between the Substituting these expressions in equationsq (187) and variance of the magnetization in canonical and micro- (188) and using J1(iλ) = iI1(λ) and J0(iλ) = I0(λ), we canonical ensembles and the second order derivatives of ﬁnd that λ is determined by entropy and free energy. The unconditional density probability of the magneti- 2πB I (λ) = 1 . (189) zation (192) can be written kM I0(λ) 1 P (B)= eNh(B), (197) Then, we get 0 A

∗ Bx ∗ By where we have introduced the function qx = iλ , qy = iλ . (190) B B 2πB h(B)= λ + ln I0(λ), (198) Substituting these values in equations (185) and (186), we − kM ﬁnally obtain where λ(B) is deﬁned by equation (189). Note that h(B)

2πB can be interpreted as the entropy of the magnetization for N − λ+ln I0(λ) Ω(B) e [ kM ]. (191) a Poissonian distribution of angles. For N + , the ∼ distribution is strongly peaked around its maximum.→ ∞ The Therefore, for N + , the unconditional density prob- most probable value of the magnetization B corresponds ability of the magnetization,→ ∞ i.e. the one corresponding to to the maximum of h(B). Using equation (189), the ﬁrst a Poissonian (uncorrelated) distribution of angles, is derivative of h(B) is

2πB 2πλ 1 N − λ+ln I0(λ) ′ P (B)= e [ kM ]. (192) h (B)= , (199) 0 A − kM This result can also be obtained from the theory of large so that the most probable value of the magnetization is deviations by a direct application of the Cramer theorem λ = B = 0 (we will see that it corresponds indeed to (see equations (6) and (7) in [30]). a maximum). Let us expand h(B) around its maximum We now take into account the correlations by using h(0) = 0. The second derivative of h(B) is the N-body distribution functions (178) and (179). Ac- 2π dλ cording to equations (180) and (191) the distribution of h′′(B)= . (200) the magnetization in the microcanonical ensemble is given, −kM dB for N + , by → ∞ Using identity (82), we obtain 2 M πB 2πB 1 2 ln E+ k − k λ+M ln I0(λ) 2 PMCE (B)= e . ′′ 8π g(E) h (0) = 2 2 < 0, (201) −k M (193) justifying that B = 0 really is the maximum of h(B). It can be written Therefore, for √NB 1, the distribution of the magnetization is the Gaussian:∼ 1 S(B) B 4 2 2 PMCE ( )= e , (194) 4π − π B g(E) P (B)= e k2M . (202) 0 k2M where S(B) is the entropy deﬁned in equation (112). Ac- This result can be directly obtained from the central limit cording to equations (181) and (191), the distribution of theorem. The variance of the unconditional distribution the magnetization in the canonical ensemble is given, for of magnetization is N + , by → ∞ 2 2 2 k M βπB 2πB 1 − λ+M ln I0(λ) B 0 = . (203) P (B)= e k k . (195) h i 4π2 CE Z(β) Let us now consider the distribution of the magneti- It can be written zation in the canonical ensemble given by equation (196). The most probable value of B corresponds to the min- 1 −βF (B) imum of free energy F (B) as studied in section 3.2.1. PCE(B)= e , (196) Z(β) The vanishing of F ′(B) leads to equation (58). The second derivative of F (B) at the extremum point is given by where F (B) is the free energy deﬁned in equation (53). equation (59) where λ′(B) is given by equation (82). P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model 21

In the homogeneous phase B = 0, the variance of the (118). The second derivative of S(B) is related to the sec- magnetization is ond derivative of free energy by equation (120). In the homogeneous phase B = 0, the variance of the 2 B2 = . (204) magnetization is h iCE βF ′′(0) 2 B2 = . (211) Using equation (82), we obtain h iMCE −S′′(0) 4π λ′(0) = . (205) Using equations (120) and (206), we obtain kM 8π2 T According to equation (59), we have S′′(0) = βF ′′(0) = 1 c . (212) − −k2M − T 2 8π Tc βF ′′(0) = 1 . (206) Using equation (118) with B = 0, this can be expressed k2M − T in terms of the energy as Therefore, using equation (204), we obtain 8π2 E S′′(0) = 1 c . (213) k2M 1 −k2M − E B2 = . (207) h iCE 4π2 1 T /T − c Therefore, using equation (211), we obtain

We first note that, according to equation (206), the homo- 2 ′′ 2 k M 1 geneous phase is a minimum of free energy (F (0) > 0) B MCE = . (214) ′′ h i 4π2 1 E /E for T >Tc and a maximum of free energy (F (0) < 0) − c for T Ec and a minimum of entropy (S (0) > 0) recover the result (203) valid for a Poissonian distribution. for E < Ec in agreement with the discussion of section + 4.2.1. On the other hand, the variances of the magnetiza- Finally, we note that the variance diverges for T Tc . This result was previously obtained in [17,54] (expressed→ tion in canonical and microcanonical ensembles coincide: 2 2 in terms of the variance of the force F 2 = B2 /2) from B CE = B MCE. h i h i h Ini theh inhomogeneousi phase B = 0, the variance of the second equation of the YBG hierarchy. 6 In the inhomogeneous phase B = 0, the variance of the magnetization is the magnetization is 6 1 (∆B)2 = . (215) 1 h iMCE −S′′(E) (∆B)2 = . (208) h iCE βF ′′(B) Using equations (120) and (210), we obtain Using equations (82) and (58), we obtain 2 2 ′′ 8π 1 Tc B 2 1 kM 2πB S (B)= 2 T 8π2B2 + 2 . = T . (209) −k M 2 2 2 − T T ! λ′(B) 2π − − kM − Tc − k M (216) According to equation (59), we have Therefore, the variance of the magnetization is given by equations (215), (216) where B(T ) is given by equation 8π2 1 T ′′ c (117). It can be expressed in terms of the energy by using βF (B)= 2 T 8π2B2 . (210) k M 2 2 2 − T ! equation (118). Figure 10 shows that S′′(B) is always neg- − Tc − k M ative so that the inhomogeneous phase is always a maxi- Therefore, the variance of the magnetization is given by mum of entropy in agreement with the results of section equations (208) and (210) where B(T ) is given by equation 4.2.1. The variances of the magnetization in canonical and ′′ (58). Figure 9 shows that F (B(T )) is always positive so microcanonical ensembles do not coincide in the inhomo- 2 2 that the inhomogeneous phase is always a minimum of geneous phase: (∆B) CE = (∆B) MCE. free energy in agreement with the graphical construction In order to representh i these6 h results graphicallyi (see Fig- of section 3.2.1. ures 9 and 10), it is convenient to introduce the dimen- Let us finally consider the distribution of the magneti- sionless variables defined in section 2. The variance of the zation in the microcanonical ensemble given by equation magnetization in the homogeneous phase is (194). The most probable value of B corresponds to the maximum of entropy S(B) as studied in section 4.2.1. The 2 1 1 ′ N b = = , (217) vanishing of S (B) leads to equation (117) with equation h i 1 η 1 1/ǫ − − 22 P.H. Chavanis: Dynamical and thermodynamical stability of isothermal distributions in the HMF model

10 On the other hand, close to the ground state (ǫ, η) ( 2+, + ): → 8 − ∞ 2 2 1 (ǫ + 2) N (∆b) CE , (222) > 2 2 6 h i ∼ 8η ∼ 32 b) ∆ 4 N<( 1 (ǫ + 2)2 CE N (∆b)2 . (223) h iCE ∼ 16η2 ∼ 64 2 MCE We emphasize that althought the ensembles are equiv- 0 0 1 2 3 alent regarding the caloric curve β(E), the variance of η the magnetization in the inhomogeneous phase diﬀers in Fig. 9. Variance of the magnetization as a function of the the two ensembles. Therefore, numerical simulations of the inverse temperature. We have represented the variance in the isolated HMF model (microcanonical ensemble) and of the canonical (full line) and microcanonical (dashed line) ensem- dissipative BMF model (canonical ensemble) should lead to diﬀerent values of (∆B)2 . bles. h i

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