Physical interpretation of the canonical ensemble for long-range interacting systems in the absence of ensemble equivalence
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Physical interpretation of the canonical ensemble for long-range interacting systems inthe absence of ensemble equivalence
Marco Baldovin
Dipartimento di Fisica, Universit`a di Roma Sapienza, P.le Aldo Moro 2, 00185, Rome, Italy (Dated: August 29, 2018)In systems with long-range interactions, since energy is a non-additive quantity, ensemble in-equivalence can arise: it is possible that different statistical ensembles lead to different equilibriumdescriptions, even in the thermodynamic limit. The microcanonical ensemble should be consideredthe physically correct equilibrium distribution as long as the system is isolated. The canonical en-semble, on the other hand, can always be defined mathematically, but it is quite natural to wonder towhich physical situations it does correspond. We show numerically and, in some cases, analytically,that the equilibrium properties of a generalized Hamiltonian mean-field model in which ensembleinequivalence is present are correctly described by the canonical distribution in (at least) two differ-ent scenarios: a) when the system is coupled via local interactions to a large reservoir (even if thereservoir shows, in turn, ensemble inequivalence) and b) when the mean-field interaction between asmall part of a system and the rest of it is weakened by some kind of screening.
I. INTRODUCTION
Equilibrium statistical mechanics provides a very accurate description of the statistical features of systems withmany particles. Relevant results can be derived when only short-range interactions are involved and the thermody-namic limit is considered; among them, equivalence of statistical ensembles covers a prominent role, since it allowsthe computation of averages for macroscopic observables according to different statistical descriptions [1]. From atechnical point of view it relies on the validity of the law of large numbers and of the central limit theorem, on theresults of large deviations theory, but also on the concavity of thermodynamic potentials [2]. More difficult cases are: • systems at the critical point where also spatially far parts are strongly interacting, so that the central limittheorem cannot be used (see e.g. [3, 4]); • systems with few degrees of freedom [5–7]; • systems with long-range interactions, in which potentials decay not faster than r − d , where r is the distance and d the spatial dimension [8].The latter case includes rather interesting physical problems, e.g. in plasma, hydrodynamics, self gravitating systemsand lasers [8]. In addition, all systems in which the elements interact via a mean field also belong to this category.In systems with long-range interactions the equivalence of statistical ensembles is not guaranteed: in particularthere are rigorous results for Hamiltonian models with mean field interactions, showing that the thermodynamicpotentials can be non convex; this is due to the non-additivity of energy [9].,graphicx,enumerate As a consequence,the canonical and microcanonical ensembles can give different results, i.e. the average of a macroscopic observable A is sensitive to the choice of the probability density function: h A i m = h A i c . In other words, fixing the energy E of a system does not always lead to the same average one gets by fixing itstemperature to the corresponding value T ( E ) ≡ (cid:0) ∂S∂E (cid:1) − , where S is the microcanonical entropy. These results arerather clear from a mathematical point of view, but their physical meaning may appear not completely obvious, due tosome potential sources of confusion in the “operative definition” of the canonical ensemble for long-range interactingsystems.Microcanonical ensemble always possess a transparent physical interpretation, since it describes the statisticalproperties of isolated Hamiltonian systems. The canonical ensemble, on the other hand, should be used for systemsat fixed temperature; it characterises, in particular, systems of Brownian particles, where the stochastic forces andthe dissipation provide a constraint on the temperature: such mechanism usually originates from the interactionsof the particles with another system (of a different nature) which acts as a stochastic thermal bath. The abovediscussion is valid regardless of the range of the potential, and both microcanonical and canonical ensemble have beenextensively studied also for systems showing long-range interactions [10]. Clearly, every Hamiltonian system (whichis described by the microcanonical ensemble as long as it is isolated) can be related to a Brownian system (which isinstead correctly described by the canonical ensemble): notable examples are the relation between stellar systems andself-gravitating Brownian particles [11] and that between the Hamiltonian Mean Field model [12] and the BrownianMean Field model [13].As far as Hamiltonian systems with only short-range interactions are considered, the canonical ensemble can bedefined in a different way: it is generally possible to observe the statistical behaviour of a small number of degreesof freedom and regard the rest of the system as a thermal bath constraining the temperature of such small portion.The procedure can be found on textbooks [1] and requires that the Hamiltonian term which represents the reciprocalinteraction is negligible in the thermodynamic limit. In this case the temperature is fixed in a natural way, even inabsence of an “external” reservoir. As soon as long-range interactions are involved, the above procedure cannot beapplied: “surface contributions” to the energy of the small part, due to the interactions with the rest of the system,are no more negligible (i.e. energy is a non-additive quantity) and canonical ensemble cannot be defined in this way.In past years some authors claimed that systems with long-range interactions should be only described by themicrocanonical ensemble [14, 15]. It has also been pointed out that for self-gravitating systems canonical ensemblecould be only defined at a formal level [16]. In the light of the above, other people stressed instead the role of canonicalensemble in describing systems of Brownian particles coupled to external baths [17]. Operative protocols have alsobeen studied in order to model a “physical” thermal reservoir in numerical simulations, and their effects on the systemhave been compared to those of Nos´e-Hoover thermostats and Monte-Carlo integration schemes in non-equilibriumconditions [18–20].In this paper we address the problem of the physical meaning of canonical ensemble when mean-field interactingsystems with non-equivalence of ensembles are involved; in particular, we show by numerical simulations that thecanonical ensemble is the only one that provides the correct equilibrium behavior • when the system is coupled via small local interactions to a large thermal bath; • when the (mean-field) interaction between a small part of the system and the rest of it is very weak.In the following we will study the Generalized Hamiltonian Mean Field (GHMF) model introduced in Ref. [21]. Thissystem is a generalization of the well-known Hamiltonian Mean Field model [12]; it is composed of N rotators whoseHamiltonian (with an additive constant) is: H N = N X i =1 p i N (cid:20) J − m ) + K − m ) (cid:21) (1)where J and K are constant parameters, m is the intensity of a magnetization defined as m = q m x + m y m x = 1 N N X i =1 sin θ i m y = 1 N N X i =1 cos θ i (2)and { θ i , p i } i = 1 , ..., N are canonical variables. The statistical properties of GHMF model can be analyticallystudied using large-deviations techniques [9]. This approach shows that an isolated system can be characterized bynegative specific heat ∂ε/∂T < ε is the specific energy and T the system’s temperature) in a certain energyrange for suitable choices of J and K . Therefore, microcanonical and canonical ensembles are not equivalent, so thatthe graph of T ( ε ) in the latter description is not the inverse of ε ( T ) in the former (it is necessary to introduce aMaxwell construction, since a first order phase transition occurs in the canonical ensemble).The paper is organized as follows. Section II is devoted to the investigation of different protocols to build a“physical” thermal reservoir for the GHMF model. We show by numerical simulations that when the system is coupledto the thermal bath by local interactions, its thermodynamic behavior is described by the canonical ensemble, andensemble inequivalence is clearly evident; this is also true in the not completely trivial case in which the reservoiris a GHMF system as well (therefore exhibiting negative specific heat). In Section III the related problem of theequilibrium properties of a weakly interacting portion of a GHMF system is investigated. We introduce a parameter λ which tunes the mean-field interaction between two portions of the system: λ determines how much each of thetwo subsystems “feels” the mean-field effect of the other, varying between 0 (two isolated GHMF systems) and 1 (aunique GHMF system resulting from the complete mean-field interaction of the two parts). The equilibrium behaviorof a small portion of the system as a function of λ is analyzed using large deviation theory and molecular dynamicssimulations: in the λ ≪ II. LOCALLY COUPLED “THERMAL BATHS” FOR SYSTEMS WITH NON-EQUIVALENCE OFENSEMBLES
In the following we consider three different ways of building a “thermal reservoir” in numerical simulations. Eachreservoir is coupled to a small GHMF system (1) with J = 1, K = 10. It has been shown [22] that this choice ofthe parameters leads to first-order phase transitions in both microcanonical and canonical ensembles; the latter is adirect consequence of the non-equivalence.In this Section we consider “local” couplings: each particle of the system interacts with only one particle of thebath. The coupling potential is given by an Hamiltonian term λV coup ( δ ), where λ is a (small) constant which indicatesthe strength of the interaction and V coup is a function of the angular distance δ between the two particles. We choose: V coup ( δ ) = A − B cos δ − C cos δ (3)with A = J/ K/ B = J/ K/ C = K/
8, which is the interaction term of Hamiltonian (1) when N = 2.There is no particular reason to make this choice for V coup ( δ ), and the results should be quite independent of its form,provided that its contribution to the total Hamiltonian is negligible.Unless otherwise specified, molecular dynamics simulations reported in the present and in the following Sectionare performed using a second-order Velocity Verlet scheme, in which we take time steps short enough to get energyfluctuations of order O (∆ E/E ) ≈ − . Since we are interested in long-range interacting models at equilibrium, wecompute averages, as far as we can, after thermalization, i.e. after the system has departed from possible metastablestates. Such process can take very long times, depending on the total number of particles N (see Ref. [23–25]):for this reason, in our simulations we choose relatively small values of N (but still in the limit N ≫ N ≈ O (10 ) for the system and N res ≈ O (10 ) for the reservoir. Initial values for positions and momenta are chosenaccording to Gaussian distributions, and then rescaled in order to get the needed total energy; however we stress that,since averages are computed after long thermalization times, our equilibrium results should hold independently of theparticular choice of initial conditions. A. Stochastic heat-bath
First we study a bath composed of N res particles held at a fixed T by a stochastic term in its evolution equation: thisterm should model the effect of several “collisions” occurring on the rotators of the reservoir. We choose N res = N ,where N is the number of elements in the analyzed system, so that every particle of the system is coupled to exactlyone particle of such reservoir; consequently, the complete Langevin equation describing the motion of a single rotatorin the bath (identified by an angular position ξ i and a momentum π i ) reads: ( ˙ ξ i = π i ˙ π i = − τ π i + q Tτ η i ( t ) − λ ddξ i V coup ( θ i − ξ i ) (4)where τ is a characteristic time of the system, η i ( t ) is a delta-correlated Gaussian noise with zero mean such that h η i ( t ) η j ( t ′ ) i = δ ij δ ( t − t ′ ) and θ i is the angular position of the coupled particle in the system. Here the Boltzmann’sconstant is 1. On the other hand, the motion equations for a particle of the system are: ( ˙ θ i = p i ˙ p i = − ( J + Km )( m x sin θ i − m y cos θ i ) − λ ddθ i V coup ( θ i − ξ i ) . (5)All simulations follow the protocol below:1. during the time interval 0 < t < t the system is decoupled from the reservoir ( λ = 0) and it evolves determin-istically;2. the temperature T of the system is computed by averaging the observable p i over all particles for 0 < t < t ;3. the temperature of the bath is set equal to T ;4. for t > t the coupling is switched on ( λ >
0) and the total system evolves according to the stochastic positionVerlet algorithm for Langevin equations discussed in Ref. [26]. T ε (a) Stable statesUnstable/Metastable statesMaxwell construction T ε (b) Stable statesUnstable/Metastable statesMaxwell construction
Figure 1: T ( ε ) vs ε for the GHMF model interacting with a stochastic thermal bath, before (a) and after (b) the coupling.Points have been marked in different ways according to the initial specific energy. All simulations have been performed withthe following parameters: N = 100, λ = 0 . τ = 20, t = 8 · . Averages in the second figure have been computed over atime interval ∆ t = 1 . · , starting at t = 1 . · ≫ t . The process is repeated for several starting specific energy ε of the system.The above setting could sound quite unphysical; we remark however that its study is certainly useful in order tocheck wether the system can actually reach the correct equilibrium distribution through the dynamics: such possibilitycould be questioned if the system starts from metastable states, since in this case thermalization times are potentiallyhuge. In addition, this stochastic approach can give useful insight about the typical waiting times to be expected inthe deterministic simulations.The results are shown in Fig. 1, in which each point represents a simulation. As long as the system is isolated,its T ( ε ) dependence is given by the microcanonical caloric curve, which consists not only of stable states, but also ofunstable and metastable ones [9], i.e. states whose ε does not minimize free energy when T is fixed. This is quiteevident in the second graph of Fig. 1: when the system is coupled to the reservoir, after some time it reaches the“true” equilibrium state at the same temperature (which is fixed by the bath) but with a different specific energy.For metastable states this process can take, as it is well known, very long times even for a relatively small number ofparticle, and this explains the residual point in the “forbidden” branch of the curve.We stress that this simple stochastic approach clearly shows that, at least for this particular choice of the physicalparameters, dynamics does select the correct equilibrium distribution (in accessible comuputational times). Thisconsideration is very important, since it suggests the possibility of similar results also in deterministic simulations. B. Hamiltonian reservoir with short-range interactions
The following protocol simulates a thermal bath by using an Hamiltonian system. In a more general fashion it hasbeen already introduced in Ref. [18] in order to study the non equilibrium behavior of the Hamiltonian Mean Fieldmodel (system (1) with K = 0). The reservoir consists of a chain of N res ≫ N first-neighbors rotators; N of them,randomly chosen, are in turn coupled to the system, trough the λV coup ( δ ) pair potential (see Eq. (3)). Let us remarkthat in Ref. [18] each particle of the system was in contact with S particles in the bath; choosing S ∝ N − / , onereproduces the “surface-like” effect in the thermodynamic limit. Here we are considering the case S = 1, with theadditional constraint that each rotator of the reservoir can be coupled to no more than one particle of the system.The total Hamiltonian is: H tot = H N ( { θ i , p i } ) + N res X i =1 π i γ N res +1 X i =1 (1 − cos( ξ i − ξ i − )) + λ N X i =1 V coupl ( ξ r i − θ i ) (6)where { ξ i , π i } are the coordinates of the particles in the reservoir ( ξ ≡ ξ N +1 ≡
0) and { r i } are distinct integersrandomly chosen in the interval [1 , N res ]. Simulating the total Hamiltonian at different energies E tot , we can sketchthe T ( ε ) dependence for the GHMF system. Fig. 2 shows that also in this case, once equilibrium has been reached,canonical ensemble provides the correct statistical description (besides some long-lasting metastable states). As T ε Stable statesUnstable/Metastable statesMaxwell construction
Figure 2: T ( ε ) vs ε for the GHMF model interacting with a Hamiltonian reservoir with short-range interactions. Parameters: N = 80, N res = 800, γ = 10, λ = 0 .
02. Averages have been computed over a time interval ∆ t = 4 · , after t = 8 · timeunits. already noticed in Ref. [18], thermal equilibrium is not “assumed” by the simulation protocol (as it happens whenstochastic terms are involved), instead it is reached by the system in a rather physical way. C. GHMF reservoir
It could be not completely obvious what does it happen when the reservoir is constituted by another, larger, GHMFsystem. In Ref. [27] violations of the zero-th law of Thermodynamics have been found for a long-range interactingmodel in which, as in the GHMF, statistical ensembles are not equivalent; it has been shown that, if two isolatedsystems with equal size share the same temperature T , but their specific heat is negative, they will reach a differenttemperature T when coupled each other.It can be easily seen, trough a microcanonical approach quite similar to the one used in Ref. [27], that this is notthe case when the ratio N /N between the sizes of the two systems is very high: in such situation the temperatureof the larger one does not change significantly, while, as expected, the thermodynamic behavior of the smaller one isdescribed by the canonical ensemble. Indeed, if one defines α ≡ N / ( N + N ) and indicates by ε and ε the specificenergies of the two systems, the most probable value of ε at fixed total energy E can be computed in general bymaximizing the total entropy s tot ( ε , ε ) = αs ( ε ) + (1 − α ) s ( ε ) (7)with the constraint ε tot ≡ E/ ( N + N ) = αε + (1 − α ) ε , where s ( ε ) is the entropy of the GHMF model. Criticalpoints of entropy (7) are obtained for values of ε such that the temperatures of the two subsystems are equal, i.e.: s ′ ( ε ) − s ′ (cid:18) ε tot − αε − α (cid:19) = 0 (8)Anyway, if different solutions ε ( n )1 , n = 1 , , ... of Eq. (8) do exist (i.e. if s ( ε ) is not a strictly concave function), theone that corresponds to the stable equilibrium, ε ∗ , must fulfill s tot (cid:18) ε ∗ , ε tot − αε ∗ − α (cid:19) ≥ s tot ε ( n )1 , ε tot − αε ( n )1 − α ! ∀ n . (9)The above inequality can be studied in the α ≪ α h s ( ε ∗ ) − s ( ε ( n )1 ) i ≥ s ′ ( ε tot ) α ( ε tot − ε ( n ) ) − s ′ ( ε tot ) α ( ε tot − ε ∗ ) (10)which immediately leads to the integral condition Z ε ( n )1 ε ∗ T − ( ε ′ ) dε ′ ≤ T − ( ε tot )( ε ( n )1 − ε ∗ ) ∀ n (11) T ε (a) Maxwell construction T ε (a) Maxwell constructionUnstable/Metastable states T ε (a) Maxwell constructionUnstable/Metastable statesStable states T ε (a) Maxwell constructionUnstable/Metastable statesStable states T ε (a) Maxwell constructionUnstable/Metastable statesStable states T ε (a) Maxwell constructionUnstable/Metastable statesStable states T ε (a) Maxwell constructionUnstable/Metastable statesStable states 1 1.5 2 1 2 3 4 ε T 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 1.5 2 2.5 3 3.5 4 ε ε (b) Stable statesUnstable/Metastable states
Figure 3: Case of a GHMF system coupled to a GHMF reservoir. (a) Caloric curve for the system, T ( ε ) vs ε , (main plot)where T is estimated by a direct average h p i i on the particles of the system; the caloric curve for the reservoir, T ( ε ) vs ε , isalso shown (inset). (b) Relation between ε and ε . Points are marked in different ways according to the average energy of thereservoir in the considered simulation. Parameters: N = 80, N res = 800, λ = 0 .
02. Averages have been computed over a timeinterval ∆ t = 4 . · . because of the relation T − ( ε ) ≡ s ′ ( ε ). The above condition is nothing but the Maxwell construction; one can thereforeconclude that in the limit α ≪
1, i.e. when it is possible to identify a reservoir composed of N res = N particles anda small system made of N = N rotators coupled to it, with N ≪ N res , the equilibrium behavior of the second isdescribed by the canonical ensemble at temperature T ( ε tot ). Since ε = ε tot − αε − α ≈ ε tot + α ( ε tot − ε ) + O ( α ) (12)it is also proved that T ( ε tot ) ≈ T ( ε ) + O ( α ) (if ε tot is not too close to a microcanonical phase transition), i.e. thetemperature of the small system is determined by the one of the reservoir, as expected, even if the reservoir is in an“unstable” state with negative specific heat.The above considerations can be tested by numerical simulations on a system of the kind: H tot = H N ( { θ i , p i } ) + H N res ( { ξ i , π i } ) + λ N X i =1 V coup ( θ i − ξ i ) (13)where H N and V coup have been defined in Eq. (1) and (3). In Fig. 3, panel (a), we see that the T ( ε ) dependencefor the small system is in a rather good agreement with the theoretical prediction, where T is estimated by theaverage h p i i on the N particles of the small system itself. Not surprisingly, in some cases the system is trapped ina metastable state. As expected, the reservoir (inset) can assume every specific energy at equilibrium, even thoseleading to negative specific heat. Panel (b) of Fig. 3 shows the relation between the specific energies of the bath ( ε )and that of the system ( ε ), compared to the theoretical curve. III. EQUILIBRIUM BEHAVIOR OF A WEAKLY INTERACTING PORTION OF A MEAN-FIELDSYSTEM
Let us consider the (quite reasonable) situation in which a mean-field interacting system is split into two parts S and S , in such a way that the effective mean field acting on each particle of S depends strongly on the degrees offreedom X of S itself and weakly on those of S (i.e. X ), and vice versa. In real physical systems this could beobtained by some kind of screening between the two parts, or by simply distancing them to a range in which meanfield interactions are no more a valid approximation.Consider the case of GHMF model, and call m ( X ) and m ( X ) the magnetization vectors of the two subsystems,whose components are defined according to Eq. (2). It is reasonable to assume that the effective fields acting on S and S can be described by ( m ∗ ( X , X ) = (1 − λ ) m ( X ) + λ m ( X , X ) m ∗ ( X , X ) = (1 − λ ) m ( X ) + λ m ( X , X ) (14)where m ( X , X ) is the mean field of the total system without splitting and λ ∈ [0 ,
1] is a real parameter whichquantifies the interaction, so that λ = 0 when the subsystems are completely isolated and λ = 1 when there’s noscreening at all. The total Hamiltonian reads H tot ( X , X ) = N + N X i =1 p i N u ( m ∗ ) + N u ( m ∗ ) (15)where N and N are the number of particles in S and S , and u ( x ) ≡ J − x ) + K − x ) . The equilibrium properties for given values of λ and α ≡ N / ( N + N ) of this Hamiltonian system can be derivedexactly, in the thermodynamic limit, by using large deviations techniques (see Appendix).Let us note that, since long-range interactions are involved, one can introduce different definitions for the energyof S , depending on which extent the non-negligible interactions with S are taken into account. In this context,anyway, the following microcanonical average E = h N X i =1 p i N u ( m ∗ ) i (16)seems to be a quite reasonable choice.Let us focus first on the λ ≪ H tot ( X , X ) = H N ( X ) + H N ( X ) + λH int ( X , X ) (17)where H N is the GHMF model Hamiltonian (1) for a system of N particles and the λH int , whose average is negligiblewith respect to those of H N and H N , includes all interactions terms between the two systems. All interactionsin this system are long-range; nonetheless, in this particular limit, we recover the conditions that are needed in thewell-known derivation of canonical ensemble from a microcanonical description. Assuming that ergodicity holds, inthis limit one expects a thermodynamic behavior quite similar to those that have been discussed in Section II.In the opposite limit, namely λ .
1, the energy range in which m ∗ = m ∗ gradually shrinks. Above certain criticalvalue ¯ λ ( α ), condition m ∗ = m ∗ (or, equivalently, m = m ) always holds at equilibrium, and for λ = 1, the T ( ε )curve will coincide with the microcanonical one, as long as the above definition of energy is considered.In Fig. 4 the two situations are shown for a fixed (small) value of α , and numerical simulations are compared toanalytical calculations.As far as mean field interactions are concerned, some general considerations about the thermodynamics of a smallpiece S of the total system S tot can be outlined. If definition (16) is considered, the caloric curve of S is the same ofan isolated system: in particular negative specific heat can be observed, because of the action of the mean field of S tot which keeps the subsystem in unstable energy regions; when the effect of the total mean field is weakened trough somescreening, but not enough to prevent heat exchange between the two subsystems, unstable and metastable states areno more accessible for S and the canonical description is recovered in the limit.On the other hand, if one defines the energy of the subsystem as the sum of all terms of the total Hamiltonian whichdepend on X only, the caloric curve tends to the one of the ideal gas, since average kinetic energy is of order αN and the average potential energy is of order α N , because of Kac’s prescription [9]. ε ε (a) Exact calculation ε ε (b) Exact calculation
Figure 4: Specific energy ε of the small portion vs specific energy ε of the large one, for λ = 0 .
05 (a) and λ = 0 . N = 50, N = 800( α = 1 / λ = 0 .
02. Averages have been computed over time intervals ∆ t = 3 . · (left) and ∆ t = 10 (right). IV. FINAL REMARKS
In this paper we have investigated several different physical situations in which the equilibrium behavior of a long-range interacting system with ensemble inequivalence is described by the canonical distribution. The aim of suchapproach is to clarify the physical interpretation of this statistical ensemble, which can always be defined from amathematical point of view.First we have studied, by numerical simulations, the case in which a small system is in contact with a large reservoir;then we have analyzed the equilibrium behavior of a small portion of a mean-field system, partially isolated from therest of it by some kind of screening. In both cases, the studied degrees of freedom interact weakly with the remainingpart of the system; nonetheless, energy can still be exchanged, so that the larger part of the system determines thetemperature of the smaller one. This is indeed a physically relevant way to construct the canonical ensemble.Our results show that the canonical distribution is physically meaningful also when inequivalence of statisticalensemble is present, as far as the above conditions hold. Since such assumptions are verified in rather interestingcases, the usage of canonical ensemble for long-range interacting systems seems quite natural and fully justified froma physical point of view.
V. APPENDIX
In this Appendix we use large deviations techniques to investigate the equilibrium behavior, in the thermodynamiclimit, of the Hamiltonian system (15). Large deviations are a well-known tool for the study of mean-field systems[22]. This approach can be used if the Hamiltonian depends only on n ≪ N mean quantities µ j ( X ) with the form µ j = P Ni =1 g ( q i , p i ), j = 1 , ..., n , or if the energy contribution of other terms is negligible in the thermodynamic limit.With the above assumptions it is possible to compute the so-called entropy of macrostates¯ s (¯ µ , ..., ¯ µ n ) ≡ N ln Z d X δ ( µ ( X ) − ¯ µ ) δ ( µ ( X ) − ¯ µ ) ...δ ( µ n ( X ) − ¯ µ n ) , (18)which is maximal in the equilibrium macrostates of the microcanonical ensemble. Even if Hamiltonian (15) is not inthe requested form, its ¯ s (¯ µ , ..., ¯ µ n ) can be easily computed. Indeed, Hamiltonian (15) can be written as H tot ( X , X ) = ¯ H ( κ , m x , m y , κ , m x , m y ) = ¯ H ( w , w ) (19)where κ = 1 N N X i =1 p i m x = 1 N N X i =1 cos θ i m y = 1 N N X i =1 sin θ i κ = 1 N N + N X i = N +1 p i m x = 1 N N + N X i = N +1 cos θ i m y = 1 N N + N X i = N +1 sin θ i and w = ( κ , m x , m y ) w = ( κ , m x , m y )once one recognizes that m = α m + (1 − α ) m , where α = N N + N . The microcanonical entropy, depending on totalenergy E , can be written as S tot ( E ) = ln Z d X d X δ ( H ( X , X ) − E )= ln Z d ¯ w d ¯ w d X d X δ ( ¯ H ( ¯ w , ¯ w ) − E ) δ ( ¯ w − w ( X )) δ ( ¯ w − w ( X ))= ln Z d ¯ w d ¯ w δ ( ¯ H ( ¯ w , ¯ w ) − E ) exp [ N ¯ s ( ¯ w , ¯ w )] (20)with ¯ s ( ¯ w , ¯ w ) ≡ N ln Z d X d X δ ( ¯ w − w ( X )) δ ( ¯ w − w ( X )) . (21)Since S tot ( E ) N ≈ sup ( ¯ w , ¯ w ) | ¯ H ( ¯ w , ¯ w )= E ¯ s ( ¯ w , ¯ w )assuming that one can compute the entropy of macrostate (21), the problem of computing the microcanonical entropyis thus reduced to that of finding a constrained supremum. This is indeed the case, since¯ s ( ¯ w , ¯ w ) = αN ln Z d X δ ( ¯ w − w ( X )) + 1 − αN ln Z d X δ ( ¯ w − w ( X )) ≡ α ˜ s (¯ κ , ¯ m x , ¯ m y ) + (1 − α )˜ s (¯ κ , ¯ m x , ¯ m y ) (22)where ˜ s (¯ κ, ¯ m x , ¯ m y ) is the entropy of macrostates for the GHMF model, that can be computed as discussed in Ref. [9].The final result is ¯ s ( κ , κ , m , m ) = 12 (1 + ln π ) + α κ ) + 1 − α κ )+ α [ − m B inv ( m ) + ln[ I ( B inv ( m ))]]+ (1 − α ) [ − m B inv ( m ) + ln[ I ( B inv ( m ))]] (23)where I n ( x ) is the n -th modified Bessel function of the first kind and B inv ( x ) is the inverse of B ( x ) ≡ I ( x ) /I ( x ).Let us notice that, due to the form of the Hamiltonian, in entropy (23) only the moduli m and m of vectors m , m appear: the task of maximizing this quantity with the constraint ¯ H (¯ κ , ..., ¯ m y ) = E can be performed numerically. Acknowledgments
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