Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Dynamical stability criterion for inhomogeneousquasi-stationary states in long-range systems
Alessandro Campa and Pierre-Henri Chavanis Complex Systems and Theoretical Physics Unit, Health and TechnologyDepartment, Istituto Superiore di Sanit`a, and INFN Roma1, Gruppo CollegatoSanit`a, 00161 Roma, Italy Laboratoire de Physique Th´eorique (IRSAMC), CNRS and UPS, Universit´e deToulouse, F-31062 Toulouse, FranceE-mail: [email protected], [email protected]
Abstract.
We derive a necessary and sufficient condition of linear dynamical stabilityfor inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF)model. The condition is expressed by an explicit disequality that has to be satisfiedby the stationary state, and it generalizes the known disequality for homogeneousstationary states. In addition, we derive analogous disequalities that express necessaryand sufficient conditions of formal stability for the stationary states. Their usefulness,from the point of view of linear dynamical stability, is that they are simpler, althoughthey provide only sufficient criteria of linear stability. We show that for homogeneousstationary states the relations become equal, and therefore linear dynamical stabilityand formal stability become equivalent.PACS numbers: 05.20.-y, 05.20Dd, 64.60.De
Submitted to:
Journal of Statistical Mechanics: Theory and Experiment tability of inhomogeneous quasi-stationary states
1. Introduction
There are numerous distinctive features that characterize the behaviour of many-bodysystems with long-range interactions, features that are not present in systems withshort-range interactions. These peculiarities concern both the equilibrium properties,such as the inequivalence of ensembles and negative specific heats in the microcanonicalensemble, and the out-of-equilibrium dynamical behaviour, such as the existence of long-lived quasi-stationary states and of out-of-equilibrium phase transitions. The study ofthese properties is interesting in its own, but it is also justified by the many differentphysical systems in which long-range interactions play the prominent role, e.g., self-gravitating systems [1, 2, 3], unscreened Coulomb systems [4], some models in plasmaphysics [5] and in hydrodynamics [6], and trapped charged particles [7]. Recent reviewsgive the state of the art of the subject [8, 9].In this paper, we treat the subject of the long-lived quasi-stationary states(QSS). They are out-of-equilibrium states in which the distribution functions are non-Boltzmannian, and their lifetime increases with the size of the system as given by thenumber N of degrees of freedom; this increase generally scales as a power law in N butit can also be exponential. It has to be emphasized that the QSS’s are not related tothe usual metastable states that are found also in short-range systems. The latter arerealized by local extrema of thermodynamical potential (e.g., they are local maxima ofthe entropy or local minima of the free energy, if these quantities are computed as afunction of an order parameter of the system), in which the system is trapped until it isdriven away by some perturbation, and then heads towards the global extremum, i.e., theequilibrium state. Global and local extrema, i.e., equilibrium and metastable states, areobtained on the basis of the usual Boltzmann-Gibbs statistics. The evaluation of thesestates can be done following different routes; e.g., one can compute the partition functionof the N -body system or work at the level of the one-particle distribution function, butonly with a direct relation to the Boltzmann-Gibbs statistics, that governs equilibrium.On the other hand, QSS’s in principle have nothing to do with some sort ofequilibrium state, global or local, of the system; nevertheless the system can be trappedfor macroscopic times in these states. We want to underline that this fact does notimply at all any failure of the Boltzmann-Gibbs description of the equilibrium states. Itsimply means that the approach to equilibrium, that both in long-range and short-rangesystems should be described, at least approximately, by a kinetic equation, happens oftenin a manner, when long-range interactions are present, in which the system resides formacroscopic times in dynamical states that are very far from the equilibrium states.From this it should be clear that the ultimate reason for the existence of the QSS’sshould be looked into dynamics, i.e., into the representation of the dynamics via akinetic equation. It turns out that the dynamics of many-body long-range systems canbe described with a very good approximation, in a certain time range, by the Vlasovequation for the one-particle distribution function. This equation is also sometimescalled the collisionless Boltzmann equation, since it represents the interactions between tability of inhomogeneous quasi-stationary states tability of inhomogeneous quasi-stationary states
2. The Vlasov equation and the quasi-stationary states
This paper will use the HMF model as a benchmark for our analysis, and therefore itis convenient to introduce the Vlasov equation from the beginning in this framework.Consider N particles of unit mass moving on a circle, with the Hamiltonian of the systemgiven by: H = 12 N X i =1 p i N X i = j V ( θ i − θ j ) , (1)where θ i ∈ [0 , π ] is the angle giving the position of a particle on the circle and −∞ < p i < ∞ is its linear momentum (equal to the velocity since the mass isunitary). The N normalization of the interaction potential V ( θ i − θ j ) is the usualone introduced in order to have an extensive energy; it is equivalent to a system-size-dependent rescaling of time, and it does not affect the study of the properties of thesystem. The Vlasov equation associated to this system, governing the evolution of theone-particle distribution function f ( θ, p, t ) is: ∂f ( θ, p, t ) ∂t + p ∂f ( θ, p, t ) ∂θ − ∂ Φ( θ, t ; f ) ∂θ ∂f ( θ, p, t ) ∂p = 0 , (2)where Φ( θ, t ; f ) is the mean field potential:Φ( θ, t ; f ) = Z ∞−∞ d p ′ Z π d θ ′ V ( θ − θ ′ ) f ( θ ′ , p ′ , t ) . (3)The last equation shows that the Vlasov equation (2) is a nonlinear integrodifferentialequation. It is immediate to see that it conserves the normalization of f ( θ, p, t ): Z ∞−∞ d p Z π d θ f ( θ, p, t ) = 1 , (4) tability of inhomogeneous quasi-stationary states E = Z ∞−∞ d p Z π d θ (cid:18) p θ, t ; f ) (cid:19) f ( θ, p, t ) . (5)The HMF model is obtained when V ( θ − θ ′ ) is a cosine potential. In the following, wewill present the criteria that determine the dynamical stability of stationary states ofthe Vlasov equation (2) for a general potential V ( θ − θ ′ ). These criteria can be triviallygeneralized to arbitrary space dimension. Afterwards, we will specialize to the HMFmodel when the relations will be transformed in explicit conditions for the stationarystates.As we have described in the Introduction, there is a time regime in which thedynamics of the N -body system is, to a high degree of accuracy, represented by the timeevolution of the one-particle distribution function as governed by the Vlasov equation.In this framework, it is natural to expect that the QSS’s will be associated to stationarystates (i.e., time independent states) of this equation. Of course, these stationary statesshould also be stable, i.e., a small perturbation should not drive the system away fromthe stationary state. Then, it is natural to be interested in the dynamical stability ofthe stationary states of the Vlasov equation. It is easy to see that any function of theform f ( θ, p ) = f ( p + Φ( θ ; f )) is stationary; therefore, in principle one is interested indetermining the stability of any such function.The function f ( p + Φ( θ ; f )) is by definition linearly stable if it is possible to choosethe norm of the perturbation δf ( θ, p, t ) at time t = 0 such that this norm remains smallerthan any (small) positive number, provided that the dynamics of the perturbation isgoverned by the linearized Vlasov equation, with the linearization made around thestationary state. Introducing the individual energy: ε ( θ, p ) ≡ p θ ; f ) , (6)the linearized Vlasov equation for δf ( θ, p, t ) is easily obtained as: ∂δf ( θ, p, t ) ∂t + p ∂δf ( θ, p, t ) ∂θ − dΦ( θ ; f )d θ ∂δf ( θ, p, t ) ∂p − ∂ Φ( θ, t ; δf ) ∂θ pf ′ ( ε ( θ, p )) = 0 , (7)where the potential Φ( θ ; f ) is constant in time (and thus the partial derivative withrespect to θ has become a total derivative) and where in the last term the functionaldependence of f on ( θ, p ) only through ε ( θ, p ) has been exploited. The problem ofthe linear stability associated to this equation has been treated long ago by Antonov[13] in astrophysics. In order to have a self-contained presentation, in section 4 wewill reproduce, although in a somewhat different formulation and scope, the Antonovresults, that later will be used to derive the stability conditions on the stationary state f ( p + Φ( θ ; f )). These will be necessary and sufficient conditions of linear stability.As we have previously underlined, we are also interested in less refined stabilitycriteria, that provide only sufficient, but simpler, conditions of linear stability. To thatpurpose, we can use the notion of formal stability of a stationary point of a generaldynamical system. The stationary point is said to be formally stable [28] if a conserved tability of inhomogeneous quasi-stationary states
3. The maximization of a class of functionals
Let L be the functional space of real differentiable functions f ( θ, p ) defined for 0 ≤ θ ≤ π and p ∈ R . We also assume that the functions decay sufficiently fast for p → ∞ sothat the integral of p f is finite. The scalar product in this space is naturally definedby: h g, f i = Z ∞−∞ d p Z π d θ g ( θ, p ) f ( θ, p ) . (8)We consider here the problem [30] of finding the constrained maximum of the functional: S [ f ] = − Z ∞−∞ d p Z π d θ C ( f ( θ, p )) , (9)with the function C ( x ) at least twice differentiable and strictly convex, i.e., with thesecond derivative strictly positive. The constraints are given by two functionals. Thefirst is a linear-quadratic expression: E = Z ∞−∞ d p Z π d θ a ( θ, p ) f ( θ, p )+ 12 Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ f ( θ, p ) b ( θ, p, θ ′ , p ′ ) f ( θ ′ , p ′ ) , (10)that we can call the “total energy” (in analogy with Eq. (5)), and that is constrainedto have a given value E . In this expression, a and b are two given functions, with b possessing the symmetry property b ( θ ′ , p ′ , θ, p ) = b ( θ, p, θ ′ , p ′ ). The second constraint isthe normalization: I = Z ∞−∞ d p Z π d θ f ( θ, p ) = 1 . (11) tability of inhomogeneous quasi-stationary states β and µ the extremum is given by equating to zero thefirst order variation: δS − βδE − µδI = 0 . (12)We thus have: − C ′ ( f ( θ, p )) − β (cid:20) a ( θ, p ) + Z ∞−∞ d p ′ Z π d θ ′ b ( θ, p, θ ′ , p ′ ) f ( θ ′ , p ′ ) (cid:21) − µ = 0 . (13)If we denote the “individual energy” (in analogy with Eq. (6)) by: ε ( θ, p ) ≡ a ( θ, p ) + Z ∞−∞ d p ′ Z π d θ ′ b ( θ, p, θ ′ , p ′ ) f ( θ ′ , p ′ ) , (14)then the extremum relation can be written as: − C ′ ( f ( θ, p )) = βε ( θ, p ) + µ . (15)From the convexity property of C ( x ) it follows that this relation can be inverted to give: f ( θ, p ) = F ( βε + µ ) ≡ f ( ε ) , (16)where F is the inverse function of − C ′ . Inserting this function in Eqs. (10) and (11)we obtain the values of the Lagrange multipliers. It is clear that Eq. (14) is alsoa consistency equation. We note that in order to interpret f ( θ, p ) as a distributionfunction, acceptable functions C ( f ) in Eq. (9) are only those that, through Eq. (16),provide a positive definite function. We also note the identity: f ′ ( ε ( θ, p )) = − βC ′′ ( f ( θ, p )) , (17)that is obtained by differentiating Eq. (15). From the convexity property of C ( x )it follows that β f ′ ( ε ( θ, p )) is negative definite. On the other hand, since we have ∂f∂p = pf ′ ( ε ( θ, p )), the integrability in p requires that, if f ′ ( ε ( θ, p )) has a definite sign,this sign must be negative. Therefore, β is restricted to positive values. We thereforeconclude that the extremization of S at fixed E and I determines distribution functionsof the form f = f ( ε ) with f ′ ( ε ) < S [ f ], for all the allowed displacements δf ( θ, p ), is negative definite. The variation ofthe functional S [ f ] is given, up to second order, by: δS = − Z ∞−∞ d p Z π d θ (cid:20) C ′ ( f ( θ, p )) δf ( θ, p ) + 12 C ′′ ( f ( θ, p ))( δf ( θ, p )) (cid:21) , (18)with the derivatives computed at the extremal point. Then, from Eqs. (15), (16) and(17) we obtain: δS = Z ∞−∞ d p Z π d θ (cid:20) ( βε ( θ, p ) + µ ) δf ( θ, p ) + 12 βf ′ ( ε ( θ, p )) ( δf ( θ, p )) (cid:21) . (19) tability of inhomogeneous quasi-stationary states E and I mustidentically vanish for the allowed displacements δf ( θ, p ). These variations are given by: δE = Z ∞−∞ d p Z π d θ ε ( θ, p ) δf ( θ, p )+ 12 Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) b ( θ, p, θ ′ , p ′ ) δf ( θ ′ , p ′ ) ≡ δI = Z ∞−∞ d p Z π d θ δf ( θ, p ) ≡ − βδE − µδI we have: δS = 12 β Z ∞−∞ d p Z π d θ f ′ ( ε ( θ, p )) ( δf ( θ, p )) − β Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) b ( θ, p, θ ′ , p ′ ) δf ( θ ′ , p ′ ) . (22)The right-hand side of this expression must be negative definite for all alloweddisplacements δf ( θ, p ), i.e., for all those that at first order do not change E and I .The problem of maximizing S [ f ] at constant E and I can be shown to be equivalentto that of minimizing the energy E at constant S and I . In fact, using Lagrangemultipliers 1 /β and − µ/β , the equation of the first order variation is now: δE − β δS + µβ δI = 0 , (23)which is the same as Eq. (12); then the solution is again given by Eq. (13). Now, wehave to study the variation of E , that up to second order is given by the left-hand sideof Eq. (20), i.e.: δE = Z ∞−∞ d p Z π d θ ε ( θ, p ) δf ( θ, p )+ 12 Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) b ( θ, p, θ ′ , p ′ ) δf ( θ ′ , p ′ ) . (24)The variations of S and I must identically vanish; the latter is expressed by Eq. (21),while the former is given by: δS = Z ∞−∞ d p Z π d θ (cid:20) ( βε ( θ, p ) + µ ) δf ( θ, p ) + 12 βf ′ ( ε ( θ, p )) ( δf ( θ, p )) (cid:21) ≡ . (25)Adding to Eq. (23) the zero valued expression − β δS + µβ δI we arrive at: δE = − Z ∞−∞ d p Z π d θ f ′ ( ε ( θ, p )) ( δf ( θ, p )) + 12 Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) b ( θ, p, θ ′ , p ′ ) δf ( θ ′ , p ′ ) . (26)This is the same as Eq. (22) divided by − β <
0. Therefore, Eq. (26) is positive definite,i.e., E is minimum at the stationary state, if Eq. (22) is negative definite. To complete tability of inhomogeneous quasi-stationary states S at constant E and I andthe minimization of E at constant S and I we have to see that in both cases the alloweddisplacements δf ( θ, p ) are the same. This can be deduced in the following way. ForEq. (22) the allowed displacements are all those that at first order give δE = δI = 0.By Eq. (12), they also at first order give δS = 0; then, they are also allowed for Eq.(26). In turn, the allowed displacements for Eq. (26) are all those that at first ordergive δS = δI = 0. By Eq. (23), they also at first order give δE = 0; then, they are alsoallowed for Eq. (22). This concludes the proof.After treating the problem of the linear stability of the stationary states of theVlasov equation, we will use the results of this section to study their formal stability.
4. The linear stability of Vlasov stationary states
The energy functional in the previous section, Eq. (10), was more general than the oneassociated to the Vlasov equation, Eq. (5). The former reduces to the latter when thefunctions a ( θ, p ) and b ( θ, p, θ ′ , p ′ ) are related to the kinetic energy and to the potentialenergy of the system, respectively; namely, when a = p and b = V ( θ − θ ′ ). Forconvenience, we rewrite here the relevant expressions. We have the individual energy ε ( θ, p ) = p θ ; f ) , (27)with the mean field potentialΦ( θ ; f ) = Z ∞−∞ d p ′ Z π d θ ′ V ( θ − θ ′ ) f ( θ ′ , p ′ ) , (28)and the total energy E [ f ] = Z ∞−∞ d p Z π d θ (cid:18) p θ, t ; f ) (cid:19) f ( θ, p, t ) . (29)We will consider stationary states that are associated to the extremization of functionalsof the form (9). We have seen that in this case f ( θ, p ) = f ( ε ( θ, p )) with f ′ ( ε ( θ, p )) < γ ( θ, p ) ≡ f ′ ( ε ( θ, p )). As remarkedabove, we follow and complete the treatment of Antonov [13].In the following we will need the usual extension of the scalar product defined inEq. (8) to complex valued functions: h g, f i = Z ∞−∞ d p Z π d θ g ∗ ( θ, p ) f ( θ, p ) , (30)where the asterisk denotes complex conjugation.It is not difficult to see that the linearized Vlasov equation (7), determining thedynamics of δf ( θ, p ), can be cast in the form: ∂δf∂t ( θ, p, t ) = − γ ( θ, p )( DKδf )( θ, p, t ) , (31)where D is the antisymmetric linear differential operator (advective operator):( Dg )( θ, p ) = p ∂∂θ g ( θ, p ) − dΦd θ ∂∂p g ( θ, p ) , (32) tability of inhomogeneous quasi-stationary states K is the linear integral operator:( Kg )( θ, p ) = 1 γ ( θ, p ) g ( θ, p ) − Φ( θ ; g ) . (33)This can be obtained by seeing that Dε = 0; then, the action of D on any function of ε gives zero; in particular Dγ = 0. Finally, it can be easily checked that the operator B ≡ DKD , needed shortly, is hermitian.The stationary point of the dynamics δf ( θ, p ) ≡ (i) allnonsecular solutions of the type δf ( θ, p, t ) = δf ( θ, p,
0) exp( λt ) have eigenvalues λ withnon positive real part; (ii) in the presence of secular terms (i.e., if there are eigenvalueswith an algebraic multiplicity larger than the geometric multiplicity), the associatedeigenvalue must have a negative real part.Before proceeding further, we need to put in evidence the properties of the operators D and K when acting on functions that are either symmetric or antisymmetric in p .From the definition of D in Eq. (32) it is clear that D transforms symmetric functions inantisymmetric functions, and viceversa. Concerning K , we see from its definition in Eq.(33) that it maintains the symmetry of the functions; however, since for antisymmetricfunctions g a ( θ, p ) we have Φ( θ ; g a ) ≡
0, the action of K in this case simplifies in( Kg a )( θ, p ) = 1 γ ( θ, p ) g a ( θ, p ) . (34)We now suppose that δf ( θ, p ; λ ) is the eigenfunction associated to the eigenvalue λ . We then have, from Eq. (31): λδf ( θ, p ; λ ) = − γ ( DKδf )( θ, p ; λ ) , (35)where for simplicity we have dropped the dependence of γ on the coordinates. We nowseparate δf in the symmetric and antisymmetric parts: δf = δf s + δf a . Taking intoaccount the mentioned properties of the operators D and K we obtain λδf s ( θ, p ; λ ) = − ( Dδf a )( θ, p ; λ ) (36)and λδf a ( θ, p ; λ ) = − γ ( DKδf s )( θ, p ; λ ) . (37)If we multiply the second of these equations by λ and substitute λδf s from the first wehave: λ δf a ( θ, p ; λ ) = γ ( DKDδf a )( θ, p ; λ ) = γ ( Bδf a )( θ, p ; λ ) . (38)If λ = 0, Eqs. (36) and (37) prove two properties. The first is that aneigenfunction cannot be either symmetric or antisymmetric, but both components mustbe nonvanishing. The second is that, if δf s + δf a is associated to the eigenvalue λ , then δf s − δf a is associated to the eigenvalue − λ .From Eq. (38) we have: λ δf a ( θ, p ; λ ) γ ( θ, p ) = ( Bδf a )( θ, p ; λ ) . (39) tability of inhomogeneous quasi-stationary states δf a gives: λ Z ∞−∞ d p Z π d θ | δf a ( θ, p ; λ ) | γ ( θ, p ) = h δf a , Bδf a i , (40)while the scalar product of its complex conjugate with δf ∗ a gives, exploiting thehermiticity of B :( λ ∗ ) Z ∞−∞ d p Z π d θ | δf a ( θ, p ; λ ) | γ ( θ, p ) = h δf a , Bδf a i . (41)Since γ is negative definite, Eqs. (40) and (41) imply that, if λ = 0, λ is necessarily real,and therefore λ is either real or pure imaginary. In much the same way, it can be shownthat, if δf ( θ, p ; λ ) and δf ( θ, p ; λ ) correspond to two different non zero eigenvalues,then: h δf a ( λ ) , Bδf a ( λ ) i = 0 . (42)The case λ = 0 will be considered later.Let us now assume that an eigenvalue λ has an algebraic multiplicity larger than itsgeometric multiplicity. Then, if δf ( θ, p ; λ ) is an eigenvector associated to this eigenvalue,there will also exist a solution of Eq. (31) given by [ tδf ( θ, p ; λ ) + δf (1) ( θ, p ; λ )] exp( λt ).Substituting in Eq. (31) and using Eq. (35) we obtain: δf ( θ, p ; λ ) + λδf (1) ( θ, p ; λ ) = − γ ( DKδf (1) )( θ, p ; λ ) . (43)Separating both δf and δf (1) in the symmetric and antisymmetric parts we obtain: δf s ( θ, p ; λ ) + λδf (1) s ( θ, p ; λ ) = − ( Dδf (1) a )( θ, p ; λ ) (44)and δf a ( θ, p ; λ ) + λδf (1) a ( θ, p ; λ ) = − γ ( DKδf (1) s )( θ, p ; λ ) . (45)If we multiply the second of these equations by λ and we substitute λδf (1) s from the firstwe have: λδf a ( θ, p ; λ ) + λ δf (1) a ( θ, p ; λ ) = γ ( Bδf (1) a )( θ, p ; λ ) + γ ( DKδf s )( θ, p ; λ ) . (46)Using Eq. (37) we arrive at:2 λδf a ( θ, p ; λ ) + λ δf (1) a ( θ, p ; λ ) γ ( θ, p ) = ( Bδf (1) a )( θ, p ; λ ) . (47)The scalar product of this expression with δf a gives: Z ∞−∞ d p Z π d θ λ | δf a ( θ, p ; λ ) | + λ δf ∗ a ( θ, p ; λ ) δf (1) a ( θ, p ; λ ) γ ( θ, p ) = h δf a , Bδf (1) a i . (48)Substracting from this equation the one that is obtained by forming the scalar productof the complex conjugate of Eq. (39) with δf (1) ∗ a we have: λ Z ∞−∞ d p Z π d θ | δf a ( θ, p ; λ ) | γ ( θ, p ) = 0 . (49) tability of inhomogeneous quasi-stationary states λ = 0 we deduce that δf a ( θ, p ; λ ) ≡
0, since γ is negative definite; then also δf s ( θ, p ; λ ) ≡ δf ( θ, p ; λ ) ≡
0. This shows that no eigenvalue λ differentfrom 0 has an algebraic multiplicity larger than its geometric multiplicity.We now consider the case λ = 0. We note that the presence of a zero eigenvaluewith an algebraic multiplicity larger than the geometrical multiplicity would implythe presence of a solution of the equation of motion (31) of the form [ tδf ( θ, p ; 0) + δf (1) ( θ, p ; 0)], and therefore the stationary point δf ( θ, p ) ≡ t exist.For λ = 0, Eqs. (36) and (37) become:( Dδf a )( θ, p ; 0) = 0 (50)and ( DKδf s )( θ, p ; 0) = 0 . (51)We note that Eq. (50) implies ( Bδf a )( θ, p ; 0) = 0.In the case the eigenvalue λ = 0 has an algebraic multiplicity larger than thegeometric multiplicity, we obtain, from Eqs. (44) and (45): δf s ( θ, p ; 0) = − ( Dδf (1) a )( θ, p ; 0) (52)and δf a ( θ, p ; 0) = − γ ( DKδf (1) s )( θ, p ; 0) . (53)Substitution of Eq. (52) in Eq. (51) gives ( Bδf (1) a )( θ, p ; 0) = 0. Dividing both sides ofEq. (53) by γ and forming the scalar product with δf a we get: Z ∞−∞ d p Z π d θ | δf a ( θ, p ; 0) | γ ( θ, p ) = −h δf a , DKδf (1) s i = h Dδf a , Kδf (1) s i = 0 , (54)where in the last step we have used Eq. (50). It follows that δf a ( θ, p ; 0) ≡
0, and then δf s ( θ, p ; 0) = 0 in order to have a non trivial solution. Then, from Eq. (52) we obtainthat δf (1) a ( θ, p ; 0) = 0 and ( Dδf (1) a )( θ, p ; 0) = − δf s ( θ, p ; 0) = 0.If there exists a solution of the equation of motion (31) of the form [ t δf ( θ, p ; 0) + tδf (1) ( θ, p ; 0) + δf (2) ( θ, p ; 0)], then the evaluation of (31) at t = 0 gives: δf (1) ( θ, p ; 0) = − γ ( DKδf (2) )( θ, p ; 0) . (55)The usual separation in the symmetric and antisymmetric parts gives: δf (1) s ( θ, p ; 0) = − ( Dδf (2) a )( θ, p ; 0) (56)and δf (1) a ( θ, p ; 0) = − γ ( DKδf (2) s )( θ, p ; 0) . (57)The substitution of Eq. (56) in Eq. (52), taking into account that δf a ( θ, p ; 0) = 0,gives ( Bδf (2) a )( θ, p ; 0) = 0. Finally, it can be shown that a solution of the form[ t δf ( θ, p ; 0) + t δf (1) ( θ, p ; 0) + tδf (2) ( θ, p ; 0) + δf (3) ( θ, p ; 0)] cannot exist. In fact, the tability of inhomogeneous quasi-stationary states t = 0 gives, after separation in the symmetric and antisymmetricparts: δf (2) s ( θ, p ; 0) = − ( Dδf (3) a )( θ, p ; 0) . (58)Substitution in Eq. (57) gives: δf (1) a ( θ, p ; 0) = γ ( Bδf (3) a )( θ, p ; 0) . (59)Dividing both sides by γ and forming the scalar product with δf (1) a we obtain: Z ∞−∞ d p Z π d θ | δf (1) a ( θ, p ; 0) | γ ( θ, p ) = h δf (1) a , Bδf (3) a i = h Bδf (1) a , δf (3) s i = 0 , (60)since ( Bδf (1) a )( θ, p ; 0) = 0. This implies that δf (1) a ( θ, p ; 0) = 0, that in turns gives, fromEq. (52), δf s ( θ, p ; 0) = 0. This is not acceptable, since we already have δf a ( θ, p ; 0) = 0.Summarizing all the results, we have that the initial value of δf ( θ, p ) can bedecomposed in general as: δf ( θ, p ) = X λ =0 c ( λ ) δf ( θ, p ; λ )+ r X j =1 h c j δf j ( θ, p ; 0) + c (1) j δf (1) j ( θ, p ; 0) + c (2) j δf (2) j ( θ, p ; 0) i , (61)where the sum over the nonzero eigenvalues stands also for an integral if the eigenvaluesare continuously distributed. The sum over r different contributions coming from thezero eigenvalue takes into account possible separation into disjoint eigenspaces; some ofthe corresponding functions might have only δf j ( θ, p ; 0) different from zero, and someonly δf j ( θ, p ; 0) and δf (1) j ( θ, p ; 0). The fact that ( Bδf j,a )( θ, p ; 0) = ( Bδf (1) j,a )( θ, p ; 0) =( Bδf (2) j,a )( θ, p ; 0) = 0 implies that if we take the antisymmetric part of δf ( θ, p ) in Eq.(61) and form the scalar product h δf a , Bδf a i , only the eigenfunctions correspondingto the eigenvalues λ different from zero contribute. Precisely, using the orthogonalityproperty (42) we have: h δf a , Bδf a i = X λ =0 | c ( λ ) | h δf a ( λ ) , Bδf a ( λ ) i . (62)With the use of Eq. (41) we get: h δf a , Bδf a i = X λ =0 | c ( λ ) | λ Z ∞−∞ d p Z π d θ | δf a ( θ, p ; λ ) | γ ( θ, p ) . (63)This is the first important expression of this section. From it we deduce that, if thestationary point δf ( θ, p ) = 0 is linearly stable, then necessarily the last expression ispositive for all cases in which not all the coefficients c ( λ ) are zero. In fact, the linearstability requires that all eigenvalues different from zero are pure imaginary (we recallthat if a real negative eigenvalue exists, also its opposite exists and leads to instability;we also recall that the antisymmetric part of any eigenfunction corresponding to aneigenvalue different from zero is nonvanishing). From this and from the fact that γ isnegative definite, our statement follows. tability of inhomogeneous quasi-stationary states δf if one or some of thecoefficients c j , c (1) j , c (2) j are non zero. We know that, if one of the eigenspaces, e.g. theone corresponding to j = j in Eq. (61), has an algebraic multiplicity larger than itsgeometric multiplicity, the stationary point δf ( θ, p ) = 0 is linearly unstable. In thiscase, taking δf ( θ, p ) = δf (1) j ( θ, p ) we have h δf a , Bδf a i = 0 with, according to what wasproved just after Eq. (54), ( Dδf a )( θ, p ) = 0. On the other hand, if all the eigenspacescorresponding to λ = 0 have equal algebraic and geometric multiplicities, then only theterms with c j appear in Eq. (61) as the contribution from the zero eigenvalue. But inthis case, it follows from Eq. (50) that the scalar product (63) can be zero only for afunction δf ( θ, p ) such that ( Dδf a )( θ, p ) = 0.In conclusion, the necessary and sufficient condition for the linear stability is thatthe scalar product (63) is nonnegative, and it is zero only for functions δf ( θ, p ) suchthat ( Dδf a )( θ, p ) = 0.We now rewrite the scalar product h δf a , Bδf a i in another way. Precisely: h δf a , Bδf a i = h δf a , DKDδf a i = −h Dδf a , KDδf a i = − h Dδf a , γ Dδf a i + h Dδf a , Φ( Dδf a ) i , (64)where use has been made of the definition of the operator K in Eq. (33). Using thedefinition of Φ in Eq. (3) we finally obtain: h δf a , Bδf a i = − Z ∞−∞ d p Z π d θ γ ( θ, p ) | ( Dδf a )( θ, p ) | (65)+ Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ ( Dδf a ) ∗ ( θ, p ) V ( θ − θ ′ )( Dδf a )( θ ′ , p ′ ) . We should note two things. Firstly, since our equation of motion is real, it is alwayspossible to choose δf ( θ, p ) real. Secondly, it is not difficult to check, form the lastexpression, that if we consider the whole function δf ( θ, p ) instead of its antisymmetricpart δf a ( θ, p ), we will always have h δf, Bδf i ≥ h δf a , Bδf a i . We then arrive at thefollowing necessary and sufficient condition of linear stability: − Z ∞−∞ d p Z π d θ γ ( θ, p ) (( Dδf )( θ, p )) (66)+ Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ ( Dδf )( θ, p ) V ( θ − θ ′ )( Dδf )( θ ′ , p ′ ) ≥ δf ( θ, p ), with the equality holding only when ( Dδf )( θ, p ) = 0. This is the mainexpression of this section. In treating the particular case of the HMF model, it will bethe basis to obtain a condition on the stationary state f ( ε ( θ, p )).
5. The energy principle and the most refined formal stability criterion
The stability of stationary states of the Vlasov equation has been studied by Kandrup[16, 17], introducing a Hamiltonian formulation for this equation, obtaining a sufficientcondition of linear stability. We give here few details, mainly to show that, for stationary tability of inhomogeneous quasi-stationary states f ( ε ( θ, p )), with f ′ ( ε ( θ, p )) <
0, this condition becomes identical tothe one given in Eq. (66), thus becoming also necessary.This approach is based on a Hamiltonian formulation of the Vlasov equation, andon the observation that the Vlasov dynamics admits an infinite number of conservedquantities, called Casimir invariants, given by: C A [ f ] = Z ∞−∞ d p Z π d θ A ( f ( θ, p )) , (67)for any function A ( x ). Note that functionals of the form (9) are particular Casimirs.The Hamiltonian formulation of the Vlasov equation (2) for f ( θ, p ) is realized by castingit in the form ∂f∂t + { f, ε } = 0 , (68)where ε is the individual energy (27) and the curly brackets denote the Poisson bracket: { a, b } = ∂a∂θ ∂b∂p − ∂a∂p ∂b∂θ . (69)It can be shown that, if f ( θ, p ) is a stationary state of the Vlasov equation of a generalform, then a sufficient condition for its linear stability is the following [17]: the differencebetween the total energy (29) computed at the perturbed state f ( θ, p ) + δf ( θ, p ) andthe total energy computed at the stationary state f ( θ, p ) is positive for all δf ( θ, p )that conserve all the Casimirs. In other words, f ( θ, p ) is linearly stable if it is a localminimum of energy with respect to perturbations that conserve all the Casimirs. Thisforms the most refined formal stability criterion. These so-called “phase-preserving” orsymplectic perturbations can be expressed in the form: f ( θ, p ) + δf ( θ, p ) = e { a, ·} f ( θ, p ) , (70)for some “small” generating function a ( θ, p ). They amount to a re-arrangement of phaselevels by a mere advection in phase space. Expanding to second order in a we have f ( θ, p ) + δf ( θ, p ) = f ( θ, p ) + { a, f } + 12 { a, { a, f }} . (71)The corresponding expansion of the total energy (29) is easily obtained as: E [ f + δf ] − E [ f ] = Z ∞−∞ d p Z π d θ ε ( θ, p ) { a, f } ( θ, p )+ 12 Z ∞−∞ d p Z π d θ ε ( θ, p ) { a, { a, f }} ( θ, p )+ 12 Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ { a, f } ( θ, p ) V ( θ − θ ′ ) { a, f } ( θ ′ , p ′ ) , (72)where the individual energy (27) at the stationary distribution has been used (and forclarity the explicit dependence of the Poisson brackets has been written). Now, it ispossible to exploit the identity: Z ∞−∞ d p Z π d θ c { c , c } = Z ∞−∞ d p Z π d θ c { c , c } (73) tability of inhomogeneous quasi-stationary states c , c , c . Using in addition that for a stationary state { f, ε } = 0, we have that the first line of the right-hand side of Eq. (72), i.e., thefirst order variation of the total energy, vanishes. This shows that the total energyat a stationary distribution is an extremum ( δE = 0) with respect to symplecticperturbations. However, this does not guarantee that it is an extremum with respect toall perturbations. Using again the identity (73), the second order variations of energydeduced from Eq. (72) are: δ E = 12 Z ∞−∞ d p Z π d θ { ε, a } ( θ, p ) { a, f } ( θ, p )+ 12 Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ { a, f } ( θ, p ) V ( θ − θ ′ ) { a, f } ( θ ′ , p ′ ) . (74)The positive definiteness of this expression is a sufficient condition of linear stability.We now suppose that the stationary distribution function is a function of ε ( θ, p ).In this case { a, f } = f ′ ( ε ) { a, ε } . Furthermore, we also have { a, ε } = Da , where D isthe linear differential operator defined in Eq. (32). Using that Dε = 0, and therefore Df ′ ( ε ) = 0, Eq. (74) becomes in this case: δ E = − Z ∞−∞ d p Z π d θ f ′ ( ε ( θ, p )) (( D ˜ a )( θ, p )) (75)+ 12 Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ ( D ˜ a )( θ, p ) V ( θ − θ ′ )( D ˜ a )( θ ′ , p ′ ) , where ˜ a ≡ f ′ ( ε ) a . The positive definiteness of this expression is exactly the necessaryand sufficient condition of linear stability (66), recalling the definition of γ ( θ, p ).It is convenient to introduce the notation δf ( θ, p ) ≡ D ˜ a ( θ, p ). Then, the necessaryand sufficient condition of linear stability can be written − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf ( θ, p )) (76)+ Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) V ( θ − θ ′ ) δf ( θ ′ , p ′ ) ≥ , for any perturbation of the form δf ( θ, p ) ≡ D ˜ a ( θ, p ) where ˜ a ( θ, p ) is any function. Theseperturbations correspond to a mere displacement (by the advective operator D ) of thephase levels, i.e. to dynamically accessible perturbations. It is straightforward to checkby a direct calculation that these perturbations conserve energy and all the Casimirsat first order (this is of course obvious for symplectic perturbations). Indeed, using δf ( θ, p ) = { a, f } and identity (73), we get δE = Z ∞−∞ d p Z π d θ δf ( θ, p ) ε = Z ∞−∞ d p Z π d θ { a, f } ε = Z ∞−∞ d p Z π d θ { f, ε } a = 0 , (77)and δC A = Z ∞−∞ d p Z π d θ δf ( θ, p ) A ′ ( f ) = Z ∞−∞ d p Z π d θ { a, f } A ′ ( f ) tability of inhomogeneous quasi-stationary states Z ∞−∞ d p Z π d θ { f, A ′ ( f ) } a = 0 . (78)
6. Less refined formal stability criteria: sufficient conditions of stability
As we have seen above, the minimization of E with respect to symplectic perturbations,i.e. dynamically accessible perturbations that conserve all the Casimirs, is a necessaryand sufficient condition of linear stability. It is also the most refined criterion offormal stability since all the constraints of the Vlasov equation are taken into accountindividually. Less refined formal stability criteria, that provide only sufficient (albeitsimpler) conditions of linear stability can be obtained by relaxing some constraints. As we have seen in section 3, a stationary state of the form f ( θ, p ) = f ( ε ( θ, p )) with f ′ ( ε ) < S of the form (9), with the function C ( x ) given by Eq. (16), under the constraints given by the normalization I , Eq. (11),and the total energy E , Eq. (29); but also by extremizing the total energy E at constant S and I . Furthermore, we have proven that a maximum of S at fixed E and I is aminimum of E at fixed S and I and viceversa. Since we have seen that for distributionsof the form f ( ε ( θ, p )) the minimization of E with respect to symplectic perturbations(i.e. perturbations that conserve all the Casimirs) is a necessary and sufficient conditionof linear stability, it is clear that the minimization of E with respect to all perturbationsthat conserve S and I (i.e., two particular Casimirs) gives a sufficient condition of linearstability. In turn, this means that the maximization of S at constant E and I gives thesame sufficient condition.Specializing Eq. (26) to our case, we immediately obtain our sufficient condition: − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf ( θ, p )) (79)+ Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) V ( θ − θ ′ ) δf ( θ ′ , p ′ ) ≥ δf ( θ, p ) that at first order give δS = δI = 0, or equivalently δE = δI = 0.Another way to see that Eq. (79) gives a sufficient condition, if Eq. (76) gives anecessary and sufficient condition, is the following. Considering that Dε = 0, we havethat all functions of the type δf = D ˜ a give δE = δI = 0 at first order. Therefore, wearrive at the conclusion that the condition for the maximization of S at constant E and I ,or for the minimization of E at constant S and I , is stronger than the condition for lineardynamical stability. Said differently, if inequality (79) is satisfied for all perturbations δf ( θ, p ) that conserve E and I at first order, it is a fortiori satisfied for all perturbationsthat conserve E , I and all the Casimirs at first order. However, the reciprocal is wrong.Therefore, Eq. (79) gives only a sufficient condition of linear stability.At this stage, it is interesting to note some analogies with thermodynamics. Inparticular, the formal stability obtained by maximizing S at constant E and I can tability of inhomogeneous quasi-stationary states S as a“pseudo entropy”. Less refined stability properties can be found by relaxing one orboth constraints (see below).On the other hand, taking S as being the Boltzmann entropy, we note thatthermodynamical stability (in the usual sense) implies Vlasov linear dynamical stability.However, the converse may not be true in the general case, i.e. the Maxwell-Boltzmanndistribution could be linearly stable according to (76) without being a maximum ofBoltzmann entropy at fixed energy and normalization (i.e. a thermodynamical state). Following the usual procedure of thermodynamics, we pass from the “microcanonical”problem of maximizing S , Eq. (9), at constant E and I , to the “canonical” problemof maximizing the “pseudo free energy” S − βE (equivalent to minimizing E − β S )at constant I . Introducing the Lagrange multiplier µ , we obtain again the first ordervariational problem: δS − βδE − µδI = 0 , (80)equivalent to Eq. (12), and therefore the same extremizing stationary state. Withoutrepeating again the computations made in section 3, it is now clear that the conditionof maximum (i.e., of formal stability) is given by the relation − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf ( θ, p )) (81)+ Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) V ( θ − θ ′ ) δf ( θ ′ , p ′ ) ≥ δf ( θ, p ) that at first order give δI = 0. Relaxing also the constraint of normalization is associated to the passage to the “grand-canonical” problem. Namely, we look for the maximum of the “pseudo grand-potential” S − βE − µI (or the minimum of E − β S + µβ I ) without any constraint. The first ordervariational problem will be again given by Eq. (80), thus obtaining the same stationarystate, while the condition of formal stability is given by the relation − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf ( θ, p )) (82)+ Z ∞−∞ d p Z π d θ Z ∞−∞ d p ′ Z π d θ ′ δf ( θ, p ) V ( θ − θ ′ ) δf ( θ ′ , p ′ ) ≥ δf ( θ, p ).This unconstrained problem corresponds to the usual energy-Casimir method [28]. tability of inhomogeneous quasi-stationary states We have found that the necessary and sufficient condition of linear dynamical stabilityfor a stationary state f ( ε ( θ, p )) of the Vlasov equation is given by Eq. (66). We haveproven that this is equivalent to the fact that the stationary distribution function f satisfies (locally) the problem:min f { E [ f ] | all Casimirs } . (83)This is the most refined criterion of formal stability as it takes into account an infinityof constraints. By relaxing some constraints, we have then found that progressivelyless refined, sufficient conditions of linear stability are given by the following problems.First, the “microcanonical” stability problem:max f { S | E, I } , (84)equivalent to: min f { E | S, I } . (85)Then, the “canonical” stability problem:max f { S − βE | I } . (86)Finally, the “grand-canonical” stability problem:max f { S − βE − µI } . (87)The solution of an optimization problem is always solution of a more constraineddual problem [29]. Therefore, a distribution function that satisfies the “grand-canonical” stability problem (no constraint) will satisfy the “canonical” stabilityproblem, a distribution that satisfies the “canonical” stability problem (one constraint)will satisfy the “microcanonical” stability problem, and a distribution that satisfiesthe “microcanonical” stability problem (two constraints) will satisfy the infinitelyconstrained stability problem (83). This is the analogous of what happens in the studyof the stability of macrostates in thermodynamics. Of course, the converse of thesestatements is wrong and this is similar to the notion of ensembles inequivalence inthermodynamics. We have the chain of implications(87) ⇒ (86) ⇒ (85) ⇔ (84) ⇒ (83) . (88)The usefulness of these less refined conditions of linear stability will be clear after thestability conditions, that now appear as conditions to be satisfied by the perturbation tothe stationary distribution function, will be transformed, in the application to the HMFmodel, in explicit conditions on the stationary distribution function itself. It will beshown that the less refined conditions of stability are associated to simpler expressions,and therefore, in a concrete calculation, one might use the simpler expressions if themore refined ones appear to be practically unfeasible. The procedure is to start by thesimplest problem and progressively consider more and more refined stability problems tability of inhomogeneous quasi-stationary states Remark:
The connection between the optimization problems (83)-(87) was firstdiscussed in relation to the Vlasov equation in [31], in Sec. 8.4 of [26], and in Sec. 3.1 of[32]. Similar results are obtained in 2D fluid mechanics for the Euler-Poisson system [33].Criterion (83) is equivalent to the so-called Kelvin-Arnol’d energy principle, criterion(87) is equivalent to the standard Casimir-energy method introduced by Arnol’d [34]and criterion (84) is equivalent to the refined stability criterion given by Ellis et al. [35].
7. The linear dynamical stability of Vlasov stationary states of the HMFmodel
For the HMF model we have V ( θ − θ ′ ) = − cos( θ − θ ′ ). The extremization of a functionalof the type (9) leads to a function of the type (see Eq. (16)): f ( θ, p ) = F (cid:20) β (cid:18) p − M x ( f ) cos θ − M y ( f ) sin θ (cid:19) + µ (cid:21) , (89)with β >
0, and with M x ( f ) and M y ( f ) given by self-consistency equations M x ( f ) = Z ∞−∞ d p ′ Z π d θ ′ cos θ ′ f ( θ ′ , p ′ ) (90)and M y ( f ) = Z ∞−∞ d p ′ Z π d θ ′ sin θ ′ f ( θ ′ , p ′ ) . (91)These are the two components of the magnetization. In this case the mean field potentialΦ( θ ; f ) is Φ( θ ; f ) = − M x ( f ) cos θ − M y ( f ) sin θ , (92)and therefore the individual energy is given by: ε ( θ, p ) ≡ p θ ; f ) = p − M x ( f ) cos θ − M y ( f ) sin θ . (93)Substituting Eqs. (92), (90) and (91) in Eq. (29), we obtain the total energy: E = Z ∞−∞ d p Z π d θ p f ( θ, p ) −
12 ( M x ( f ) + M y ( f )) . (94)The Vlasov equation for the HMF model reads: ∂f ( θ, p ) ∂t + p ∂f ( θ, p ) ∂θ − ( M x ( f ) sin θ − M y ( f ) cos θ ) ∂f ( θ, p ) ∂p = 0 . (95)For the HMF model, we are interested in studying the stability of stationary solutionsof Eq. (95) given by functions of the form f ( θ, p ) = f ( p − M x ( f ) cos θ − M y ( f ) sin θ ).Without loss of generality we can suppose that M y ( f ) = 0 and therefore that f ( θ, p ) = F (cid:20) β (cid:18) p − M cos θ (cid:19) + µ (cid:21) , (96) tability of inhomogeneous quasi-stationary states M ≡ M x ( f ) (dropping the explicit dependence on f ). In this case we have:Φ( θ ; f ) = − M cos θ . (97)The individual energy is: ε ( θ, p ) ≡ p θ ; f ) = p − M cos θ , (98)while the total energy is: E = Z ∞−∞ d p Z π d θ p f ( θ, p ) − M . (99)The linearized Vlasov equation, governing the linear dynamics of δf ( θ, p, t ) aroundthe stationary distribution f ( θ, p ) of the form (96), is obtained by linearizing Eq. (95)(in our case with M y ( f ) = 0 and M x ( f ) = M ), and it is given by: ∂∂t δf = − p ∂∂θ δf + M sin θ ∂∂p δf + pf ′ ( ε ( θ, p )) ∂∂θ Φ( θ ; δf ) , (100)where Φ( θ ; δf ) must contain also the contribution of δf to a magnetization in the y direction: Φ( θ ; δf ) = − Z ∞−∞ d p ′ Z π d θ ′ cos( θ − θ ′ ) δf ( θ ′ , p ′ ) . (101)Eq. (100) is in the form of Eq. (31), with the linear operator D now taking the form:( Dg )( θ, p ) = p ∂∂θ g ( θ, p ) − M sin θ ∂∂p g ( θ, p ) . (102)This operator has the property that Dε = 0 and Dγ = 0, with ε given in Eq. (98) andwhere we again use for simplicity the notation γ ( θ, p ) for the negative definite function f ′ ( ε ). The linearized Vlasov equation can then be written, similarly to Eq. (31), as: ∂δf∂t ( θ, p, t ) = − γ ( θ, p )( DKδf )( θ, p, t ) , (103)where for the HMF model the operator K is defined by:( Kg )( θ, p ) = 1 γ ( θ, p ) g ( θ, p ) − Z ∞−∞ d p ′ Z π d θ ′ cos( θ − θ ′ ) g ( θ ′ , p ′ ) . (104)We are now in the position to follow the general results presented in section 4.Exploiting the particularly simple expression of the interaction potential in the HMFmodel we have, from Eq. (66), that the necessary and sufficient condition for the linearstability of f ( θ, p ) is: − Z ∞−∞ d p Z π d θ γ ( θ, p ) (( Dδf )( θ, p )) (105) − (cid:18)Z ∞−∞ d p Z π d θ cos θ ( Dδf )( θ, p ) (cid:19) − (cid:18)Z ∞−∞ d p Z π d θ sin θ ( Dδf )( θ, p ) (cid:19) ≥ . We note that in Eqs. (105) the first term in the left-hand side is positive definite, whilethe second and third terms are negative definite. tability of inhomogeneous quasi-stationary states D , that implies that the functional subspaceorthogonal to the kernel of the operator is transformed in itself. In fact, if g belongs tothe kernel of D , and g is orthogonal to g , then: h g , Dg i = −h Dg , g i = 0 . (106)The kernel is made of the functions which depend on ( θ, p ) through (cid:16) p − M cos θ (cid:17) . Wemay therefore transform the problem (105) in the problem of satisfying the relation − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf ( θ, p )) (107) − (cid:18)Z ∞−∞ d p Z π d θ cos θδf ( θ, p ) (cid:19) − (cid:18)Z ∞−∞ d p Z π d θ sin θδf ( θ, p ) (cid:19) ≥ Z ∞−∞ d p Z π d θ (cid:18) p − M cos θ (cid:19) s δf ( θ, p ) = 0 s = 0 , , . . . . (108)However, we should take into account the case in which the stationary distributionfunction f has a power law decay for large p . We therefore substitute the previousconditions with: Z ∞−∞ d p Z π d θ h (cid:20) p − M cos θ (cid:21) (cid:18) p − M cos θ (cid:19) s δf ( θ, p ) = 0 s = 0 , , . . . , (109)where h is a function that assures integrability; it may be chosen, e.g., equal toexp h − (cid:16) p − M cos θ (cid:17)i .Since the function γ ( θ, p ) is even in θ , it is useful to separate δf in its even andodd parts in θ , i.e. δf ( θ, p ) = δf e ( θ, p ) + δf o ( θ, p ). In this way, our problem to satisfythe relation in Eq. (107) subject to the conditions given in Eq. (109), is separated in apair of separate problems. Precisely, for the even part we have to satisfy: − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf e ( θ, p )) − (cid:18)Z ∞−∞ d p Z π d θ cos θδf e ( θ, p ) (cid:19) ≥ δf e ( θ, p ) such that the relations Z ∞−∞ d p Z π d θ h (cid:20) p − M cos θ (cid:21) (cid:18) p − M cos θ (cid:19) s δf e ( θ, p ) = 0 s = 0 , , . . . (111)are verified. For the odd part we have to satisfy − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf o ( θ, p )) − (cid:18)Z ∞−∞ d p Z π d θ sin θδf o ( θ, p ) (cid:19) ≥ δf e such that cos θδf e has avanishing integral will trivially give a positive value for the left-hand side of Eq. (110),and similarly any δf o such that sin θδf o has a vanishing integral will trivially give apositive value for the left-hand side of Eq. (112). Therefore the only problems can come tability of inhomogeneous quasi-stationary states δf e and δf o that do not have these mentioned properties. Secondly, sinceboth (110) and (112) are quadratic functions of δf e and δf o , respectively, the sign ofthe expression is not changed by the multiplication of δf e or δf o by any number. Wecan therefore study the sign of the left-hand sides of (110) and (112) also by imposinga linear condition on δf e and δf o .Therefore we proceed in the following way. We look for the extremum of the left-hand side of Eq. (110), constrained by the conditions (111), with the further convenientconstraint: Z ∞−∞ d p Z π d θ cos θδf e ( θ, p ) = 1 . (113)Similarly, we look for the extremum of the left-hand side of Eq. (112) under theconstraint: Z ∞−∞ d p Z π d θ sin θδf o ( θ, p ) = 1 . (114)It is useful at this point to introduce the following definitions: α ( h ) s ≡ Z ∞−∞ d p Z π d θ h (cid:20) p − M cos θ (cid:21) γ ( θ, p ) (cid:18) p − M cos θ (cid:19) s = Z ∞−∞ d p Z π d θ h (cid:20) p − M cos θ (cid:21) γ ( θ, p ) ( ε ( θ, p ; f )) s (115)and η ( h ) s ≡ Z ∞−∞ d p Z π d θ h (cid:20) p − M cos θ (cid:21) γ ( θ, p ) cos θ (cid:18) p − M cos θ (cid:19) s = Z ∞−∞ d p Z π d θ h (cid:20) p − M cos θ (cid:21) γ ( θ, p ) cos θ ( ε ( θ, p ; f )) s , (116)where the dependence on the function h is explicitly indicated.We begin with the problem related to δf e . Introducing the Lagrange multipliers2 µ ( h ) s for the constraints (111) and 2 ν for the constraint (113), respectively, theconditioned extremum of the left-hand side of Eq. (110) is given by the equation: − γ ( θ, p ) δf e ( θ, p ) − (1 + ν ) cos θ − ∞ X s =0 µ ( h ) s h (cid:20) p − M cos θ (cid:21) (cid:18) p − M cos θ (cid:19) s = 0 . (117)It is clear that this extremum is a minimum, since the the second variation is simply − γ >
0. Therefore the necessary and sufficient condition is that the disequality (110)is satisfied for the extremal δf e ( θ, p ) determined by Eq. (117). Denoting furthermore ξ = − (1 + ν ), Eq. (117) gives: δf e ( θ, p ) = ξγ ( θ, p ) cos θ − ∞ X s =0 µ ( h ) s γ ( θ, p ) h [ ε ( θ, p ; f )] ( ε ( θ, p ; f )) s . (118)Substituting this expression in Eqs. (111), we obtain the system of equations: ∞ X s ′ =0 µ ( h ) s ′ α ( h ) s + s ′ = ξη ( h ) s s = 0 , , . . . , (119) tability of inhomogeneous quasi-stationary states ξ Z ∞−∞ d p Z π d θ γ ( θ, p ) cos θ − ∞ X s =0 µ ( h ) s η ( h ) s = 1 . (120)From the system (119), we may obtain the multipliers µ ( h ) s as a function of the multiplier ξ . We see in particular that the multipliers µ ( h ) s are proportional to ξ . We thereforeintroduce the “normalized” multipliers ˜ µ ( h ) s , given by the solution of the system ofequations: ∞ X s ′ =0 ˜ µ ( h ) s ′ α ( h ) s + s ′ = η ( h ) s s = 0 , , . . . . (121)We have that µ ( h ) s = ξ ˜ µ ( h ) s ; substituting in Eq. (120) we obtain: ξ = 1 R ∞−∞ d p R π d θ γ ( θ, p ) cos θ − P ∞ s =0 ˜ µ ( h ) s η ( h ) s . (122)The relation (110) for δf e equal to the extremal function given by Eq. (118) cannow be easily obtained, taking into account Eqs. (113), (121) and (122). Introducingthe further short-hand notation ∞ X s =0 ˜ µ ( h ) s η ( h ) s ≡ z ( γ ) (123)(where we put in evidence the dependence on the stationary distribution functionthrough γ ), we have: 1 z ( γ ) − R ∞−∞ d p R π d θ γ ( θ, p ) cos θ ≥ . (124)We have thus obtained a relation involving only the stationary distribution function.This is the main expression of this paper.The relation valid in the case of the linear dynamical stability of homogeneous (i.e.,with M = 0) stationary distribution functions is easily obtained. In fact, in that case η ( h ) s = 0; therefore ˜ µ ( h ) s = 0 and thus z ( γ ) = 0. Then, taking into account that theintegral of cos θ is equal to π , Eq. (124) becomes in this case:1 + π Z ∞−∞ d p γ ( p ) ≥ . (125)This is identical with the expression generally found in the literature for the linearstability of homogeneous distribution functions in the HMF model (see, e.g., Ref.[20, 22, 23, 24, 26]); for this comparison we have to consider that for homogeneousdistribution functions γ ( p ) = f ′ ( p ) /p . We will show that for homogeneous distributionfunctions all extremal problems lead to the same result. This explains why the sameexpression is obtained from the “canonical” formal stability problem [23] (in principle,this approach only gives a sufficient condition of linear stability, but our present resultsshow that it is in fact sufficient and necessary).The result just obtained for the problem associated to the even part δf e ( θ, p ) canbe immediately transformed in that for the problem associated to the odd part δf o ( θ, p ). tability of inhomogeneous quasi-stationary states µ ( h ) s are present. Then,we easily obtain the further condition, analogous to Eq. (124), that has to be satisfiedby γ ; namely: 1 + Z ∞−∞ d p Z π d θ γ ( θ, p ) sin θ ≥ . (126)For homogeneous distribution functions this relation becomes equal to Eq. (125).However, it can be easily shown that for inhomogeneous distribution functions of theform given in Eq. (96), i.e., when M is strictly positive, Eq. (126) is satisfied as anequality, independently of the particular form of the function and of the value of itsparameters. In fact: Z ∞−∞ d p Z π d θ γ ( θ, p ) sin θ ≡ Z ∞−∞ d p Z π d θ sin θ ∂F∂ε = 1 M Z ∞−∞ d p Z π d θ sin θ ∂F∂θ = − M Z ∞−∞ d p Z π d θ cos θF = − . (127)This equality is clearly associated to a δf o ( θ, p ) that simply rotates, at first order, thedistribution function (96). This shows that, for inhomogeneous distribution functions(96), any odd δf o ( θ, p ) will satisfy Eq. (112).Summarizing the results of this section, the distribution function (96) is linearlydynamically stable iff the relation (124) is satisfied. This relation reduces to the simplerform given in Eq. (125) for a homogeneous (i.e., with M = 0) distribution function.As we have shown in the general case, the most refined formal stability criterion(83) leads to the same necessary and sufficient condition.
8. The formal stability of Vlasov stationary states of the HMF model:sufficient conditions of stability
The problem related to the “microcanonical” formal stability is obtained by specializingto the HMF potential the expressions given in section 6.1. We thus obtain: − Z ∞−∞ d p Z π d θ γ ( θ, p ) ( δf ( θ, p )) (128) − (cid:18)Z ∞−∞ d p Z π d θ cos θδf ( θ, p ) (cid:19) − (cid:18)Z ∞−∞ d p Z π d θ sin θδf ( θ, p ) (cid:19) ≥ δf ( θ, p ) such that the constraints of normalization and of total energy are satisfiedat first order. Using the expression of the total energy for the HMF model, we havethat the allowed δf ( θ, p ) have to satisfy: δE = Z ∞−∞ d p Z π d θ (cid:18) p − M cos θ (cid:19) δf ( θ, p ) = 0 , (129)and δI = Z ∞−∞ d p Z π d θ δf ( θ, p ) = 0 . (130) tability of inhomogeneous quasi-stationary states s = 0 and s = 1, and thefunction h is the constant unitary function. In fact, the constraints (130) and (129) arenothing more than the constraints (111) for s = 0 and s = 1, respectively, and in thecase h = 1. Therefore, the only thing we need before writing down the result for thiscase is to adapt the definition of the parameters α s and η s , and of the multipliers ˜ µ s , tothe present situation. We thus define α s ≡ Z ∞−∞ d p Z π d θ γ ( θ, p ) ( ε ( θ, p ; f )) s (131)and η s ≡ Z ∞−∞ d p Z π d θ γ ( θ, p ) cos θ ( ε ( θ, p ; f )) s . (132)We therefore introduce the “normalized” multipliers ˜ µ s , given by the solution of thesystem of equations: X s ′ =0 ˜ µ s ′ α s + s ′ = η s s = 0 , . (133)The condition on γ ( θ, p ) analogous to Eq. (124) can now be immediately writtendown. It is given by: 1 w ( γ ) − R ∞−∞ d p R π d θ γ ( θ, p ) cos θ ≥ , (134)where now the short-hand notation w ( γ ) stands for: X s =0 ˜ µ s η s ≡ w ( γ ) . (135)Summarizing, the distribution function (96) is formally stable with respect to the“microcanonical” criterion iff the relation (134) is satisfied. This relation reduces tothe simpler form given in Eq. (125) for a homogeneous (i.e., with M = 0) distributionfunction, since in that case ˜ µ s = 0. Thus, dynamical linear stability and formal stabilitylead to identical conditions for stationary homogeneous distribution functions.Although it is not evident from the two expressions (134) and (124), we know thatif the former relation is satisfied, so is the latter, since we had found that the necessaryand sufficient condition for the formal stability is also a sufficient condition for the lineardynamical stability. At this point, it is straighforward to derive the relation for the less refined formalstability problems. In particular, the “canonical” formal stability condition is completelyanalogous to the “microcanonical” formal stability condition, with the difference thatthe allowed δf ( θ, p ) in Eq. (128) have to satisfy only the normalization constraint, i.e., tability of inhomogeneous quasi-stationary states s = 0. Inparticular, the system (133) reduces to the single equation:˜ µ α = η . (136)Then, the condition on γ ( θ, p ) now becomes:1 η α − R ∞−∞ d p R π d θ γ ( θ, p ) cos θ ≥ , (137)or, more explicitly, 1 ( R ∞−∞ d p R π d θ γ ( θ,p ) cos θ ) R ∞−∞ d p R π d θ γ ( θ,p ) − R ∞−∞ d p R π d θ γ ( θ, p ) cos θ ≥ . (138)Again, for M = 0 this condition becomes identical to Eq. (125), since in that case η = 0.For consistency, it is interesting to note that the equality in Eq. (138) correspondsto the condition of marginal stability found by another method in the Appendix F of[26]. Finally, the “grand-canonical” formal stability condition has no constraint at all.Therefore, the condition on γ ( θ, p ) is simply: − R ∞−∞ d p R π d θ γ ( θ, p ) cos θ ≥ . (139)Again, for M = 0 this condition becomes identical to Eq. (125).We remark the following point. The problem associated to the “grand-canonical”formal stability, Eq. (87), does not constraint the value of the normalization I , thattherefore can be different from 1. It is clear however, that if we want to study the “grand-canonical” formal stability of a distribution function which is an extremum also for theother stability problems, we have to restrict ourselves only to normalized distributionfunctions.In the Appendix, just as a useful exercise, we show that, as we should expect,the inhomogeneous Maxwell-Boltzmann distribution function is “canonically” formallystable, and therefore also “microcanonically” formally stable and linearly dynamicallystable, but it is not “grand-canonically” stable. This shows the role of the constraintsand the importance of considering sufficiently refined stability criteria.
9. Discussion and conclusions
In this paper we have derived a necessary and sufficient condition of linear stability for astationary state of the Vlasov equation. This condition is expressed by Eq. (124), whichis the core of the paper. Less refined conditions of formal stability, which are sufficient,although not necessary, for linear dynamical stability, have also been obtained. tability of inhomogeneous quasi-stationary states ∗ . Secondly, actual computations for lineardynamical stability will always require a degree of approximation, since the infinite sumimplicit in the system (121) and in the definition of z ( γ ), Eq. (123), will have to bereplaced by some finite representation. This has led us to treat also the less refinedformal stability conditions. At the price to have only sufficiency, more manageableexpressions are to be expected.We would like to conclude with some comments about the relevance of the Vlasovstable stationary states from the point of view of thermodynamics.If the system is initially in a state that is not a stationary state of the Vlasovequation, it can be argued that there will be a rather fast evolution until a stablestationary state is reached. However, some care must be exercised about the sense inwhich this statement has to be taken. Analogously to the Liouville theorem for the N -body distribution function of Hamiltonian systems, the time evolution of the one-body distribution function as governed by the Vlasov equation is such that its phaselevels are conserved (in fact, the Vlasov equation states exactly the equality to zero ofthe convective derivative of f ). In particular, an initial two levels f , i.e. an f whichis constant in a given region of the ( θ, p ) plane and zero outside of this region, will betwo levels for all the following evolution. How can we expect such a function to evolvetowards a smooth stable stationary state characterized by a continuity of phase values?This can be realized only in a coarse-grained sense, when we study a sort of smeared one-body distribution function, in which the value of f at each point is substituted by theaverage of f taken in a small neighbourhood of the given point. If there is an efficientmixing of the dynamics, we may expect that, no matter how small is the averagingneighbourhood (provided it is not vanishing), the averaged f will evolve towards thestable stationary state of the Vlasov equation. This is exactly the framework in whichthe Lynden-Bell theory of violent relaxation has been proposed [14].It is a typical reasoning in thermodynamics or statistical mechanics to argue thatthe distribution functions of a system, in particular the one-body distribution function,will evolve according to the maximization of a functional given some constraints. Forexample, the final Boltzmann-Gibbs state of the one-body distribution function will begiven by the maximization of the Boltzmann entropy S B [ f ] = − Z ∞−∞ d p Z π d θ f ( θ, p ) ln f ( θ, p ) (140)subject to the constraints of normalization, Eq. (4), and given total energy, Eq.(5); the potential Φ( θ ; f ) will have to be determined self-consistently. The use ofthe Boltzmann-Gibbs entropy (140) is fully justified to characterize the state reachedafter the “collisional” regime has taken over; it will be the most mixed state given the ∗ Interestingly, in stellar dynamics, using the Antonov criterion or the energy principle, it can beshown that all spherical galaxies with f = f ( ε ) and f ′ ( ε ) < tability of inhomogeneous quasi-stationary states f ( θ, p ) = f ( ε ( θ, p )) with f ′ ( ε ) <
0, i.e. a particular steady state of the Vlasovequation. As such, it extremizes a functional of the form S [ f ] = − Z ∞−∞ d p Z π d θ C (cid:0) f ( θ, p ) (cid:1) , (141)at fixed normalization and energy. It can be shown that if this distribution functionis a maximum of S at fixed I and E , then it is Lynden-Bell thermodynamically stable(see [33, 39] for the 2D Euler equation and Sec. V of [38] for the Vlasov equation).According to the present study, if it is a maximum of S at fixed I and E , it is alsogranted to be linearly dynamically Vlasov stable. More generally, it can be shown thatthe coarse-grained distribution function associated with a Lynden-Bell thermodynamicalequilibrium is always dynamically stable (because it is a minimum of energy with respectto phase preserving perturbations), even if it is not a maximum of S at fixed E and I (see Sec. 7.8. of [33] for the 2D Euler equation).On the other hand, since it is not guaranteed that the mixing is always completelyefficient (this is referred to as “incomplete relaxation”), one may argue that in caseswhen it is not efficient the dynamics will evolve trying to maximize, always in the coarse-grained sense, other functionals of the form (141) that are not consistent with Lynden-Bell’s theory (see [40] and Sec. XII of [38]). This is an essentially phenomenologicalapproach. For example, the Tsallis functional is one particular case of such functionals(that are called generalized H -functions [40]). The constraints of normalization andtotal energy are always present, and it is immediate to see that, if the functional tomaximize is of the form (141) then the solution will always be a function of the form f ( θ, p ) = f ( p + Φ( θ ; f )). The problem at hand in this case is to show that a function ofthis form obtained by extremizing (141) at fixed I and E is really a maximum, i.e., thevalue of (141) decreases if we perturb f without changing the values of the constraints.This “microcanonical” stability problem has been studied for the Tsallis distributionsin [27].We are thus led to the conclusion that functions of the form f ( θ, p ) = f ( p +Φ( θ ; f ))are relevant also in a thermodynamical sense (with respect to the collisionless dynamics).However, we have shown that, while for homogeneous states formal “microcanonical”stability implies linear dynamical stability and viceversa, for inhomogenous states onlythe first implication is true. Thus, there might be inhomogeneous states which areformally “microcanonically” unstable (therefore not relevant from a thermodynamicalpoint of view), that nevertheless are dynamically linearly stable. tability of inhomogeneous quasi-stationary states Appendix: The stability of the Maxwell-Boltzmann distribution function inthe HMF model
As an exercise we can apply the results of section 8 to the magnetized Maxwell-Boltzmann distribution function, that realizes the Boltzmann-Gibbs global equilibrium.For this case all the quantities can be obtained analytically, basically because in thiscase γ = − βf . We can consider from the beginning the “canonical” formal stabilityproblem, given by Eq. (137). It is immediate to obtain in this case that α = − β and η = − βM . We also have: Z ∞−∞ d p Z π d θ γ ( θ, p ) cos θ = − β (cid:0) M + ∆ M (cid:1) , (142)where we have denoted with ∆ M the variance of the magnetization, i.e., the expectationvalue h (cos θ − M ) i . We then find that Eq. (137) reduces to: β ∆ M ≤ . (143)We first note that, in the homogeneous case, the last expression reduces to the knownrelation β ≤
1, i.e., β ≤
2. However, it is not difficult to show that, when β >
M >
0, Eq. (143) is always satisfied, on the basis of the graphical construction thatgives β as a function of M , based on the relation M = I ( βM ) I ( βM ) , (144)where I and I are the modified Bessel functions of order 1 and 0, respectively. Wethen have: β ∆ M = β ∂∂ ( βM ) I ( βM ) I ( βM ) = ∂∂M I ( βM ) I ( βM ) < , (145)where the last disequality is a consequence of the graphical solution of Eq. (144). Then,we have proven the “canonical” formal stability, hence the “microcanonical” formalstability and the linear stability. On the contrary, it can be seen that the “grand-canonical” formal stability (139) does not hold for magnetized states. In fact, from Eq.(142), we find that this stability would require βM + β ∆ M ≤ , (146)that in turn, using Eqs. (144) and (145), becomes: β "(cid:18) I ( βM ) I ( βM ) (cid:19) + ∂∂ ( βM ) I ( βM ) I ( βM ) ≤ . (147)However, using the results in Appendix B of Ref. [41], it is proven that the left-handside of the last expression, for M > β −
1. Since Boltzmann-Gibbs magnetized states are realized for β >
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