hmf model

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Digital Object Identifier (DOI) 10.1007/s00161-003-0170-0

Continuum Mech. Thermodyn. (2004) 16: 245255

Original article

Dynamics and thermodynamics of a model with long-range interactions

A. Pluchino, V. Latora, A. Rapisarda

Dipartimento di Fisica e Astronomia, Universita di Catania, and INFN Sezione di Catania,

Via S. Sofia 64, 95123 Catania, Italy

Received July 8, 2003 / Accepted October 27, 2003

Published online February 11, 2004 Springer-Verlag 2004

Communicated by M. Sugiyama

Abstract.The dynamics and thermodynamics of particles/spins interacting via long-range forces

display several unusual features compared with systems with short-range interactions. The Hamil-

tonian mean field (HMF) model, a Hamiltonian system ofNclassical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium

properties of the model can be derived analytically in the canonical ensemble: in particular, the

model shows a second-order phase transition from a ferromagnetic to a paramagnetic phase.

Strong anomalies are observed in the process of relaxation towards equilibrium for a particular

class of out-of-equilibrium initial conditions. In fact, the numerical simulations show the presence

of quasi-stationary states (QSSs), i.e. metastable states that become stable if the thermodynamic

limit is taken before the infinite time limit. The QSSs differ strongly from BoltzmannGibbs

equilibrium states: they exhibit negative specific heats, vanishing Lyapunov exponents and weak

mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly decaying correla-

tions, and aging. Such a scenario provides strong hints for the possible application of Tsallisgeneralized thermostatistics. The QSSs have recently been interpreted as a spin-glass phase of

the model. This link indicates another promising line of research, which does not preclude to the

previous one.

Key words: phase transitions, Hamiltonian dynamics, long-range interaction, out-of-equilibrium

statistical mechanics

PACS: 05.70.Fh, 89.75.Fb, 64.60.Fr, 75.10.Nr

1 Introduction

Since the original paper by Ising [1], magnetic models on a lattice have been extensively used over the years

to investigate the statistical physics of interacting many-body systems. In particular, many generalizations of

the Ising model have been proposed, among them models with long-range interactions. The thermodynamics

and dynamics of systems of particles interacting with long-range forces are particularly interesting because

they display a series of anomalies compared with systems with short-range interactions [2]. By long-range

interaction, it is usually meant that the modulus of the potential energy decays, at large distances, not faster than

the inverse of the distance to the power of the spatial dimension. The main reason for the observed anomalies

is that systems with long-range forces in general violate extensivity and additivity, two basic properties used

Correspondence to: A. Rapisarda (e-mail: [email protected])

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246 A. Pluchino et al.

to derive the thermodynamics of a system. Extensivity means that the thermodynamic potentials scale with the

system size, i.e. that the specific thermodynamic potentials (the thermodynamic potentials per particle) do not

diverge in the thermodynamic limit. Additivity means that if we divide the system into two macroscopic parts,

the thermodynamic potentials of the whole system are, in the thermodynamic limit, the sum of those of the

two components. Although extensivity can be artificially restored by an ad-hoc introduction of anN-dependentcoupling constant (the so called Kacs prescription [3]) in the interaction, the problem of non-additivity is

still present [2]. Thus, the study of long-range magnetic systems on lattices can give insight into the statistical

propertiesof long-range potentials, and is therefore useful for investigating the statistical mechanicsand dynamics

of such systems.1

In this paper we study the HMF-model, a model of planar spins on a d-dimensional lattice with couplingsthat decay as the inverse of the distances between spins raised to the power [4,5]. When is smaller thand,additivity does not hold and the system shows a series of anomalies. Even ensemble equivalence, the proof of

which is based on the possibility of separating the energy of a subsystem from that of the whole, might not be

guaranteed in that limit.

In particular we focus on = 0. In such a case the model reduces to a mean field model [613], since aspin interacts equally with all the others independently of their position on the lattice. This case is extremely

important since it has been proven that all the cases with /d 1 [4,5] can be reduced to it. Moreover thedynamical behavior of the model with = 0 is also a representative example of other nonextensive systems[4,5, 9,1417]. We will discuss the equilibrium phase and the dynamical anomalies found in an energy region

before the second-order critical point when one studies the relaxation towards equilibrium. The connections with

Tsallis generalized thermodynamics [18,9,15,11,19] and with glassy systems [20,21] will also be addressed.The paper is organized as follows. We introduce the model in Sect. 2, the equilibrium thermodynamics

is presented in Sect. 3, while the anomalous dynamical behavior and its possible theoretical interpretation is

discussed in Sect. 4. Conclusions and future perspectives are presented in Sect. 5.

2 The HMF model

The HMF- model was introduced in [4] and describes a system of classical bidimensional spins (XY spins)with massm= 1. The Hamiltonian is

H = K+V =N

i=1p2i

2 +

2N

N

i=j1 cos(i j)

rij, (1)

with= 1. TheNspins are placed at the sites of a genericd-dimensional lattice, and each one is representedby the conjugate canonical pair(pi, i), where thepis are the momenta and theis [0, 2)are the angles ofrotation on a family of parallel planes, each one defined at each lattice point. The interaction between rotators iandj decays as the inverse of their distancerij to the power 0.

Such a Hamiltonian is extensive if the thermodynamic limit (TL)N of the canonical partition function(ln Z)/Nexists and is finite. This is assured for each by the presence of the rescaling factor Nin front of thedouble sum of the potential energy. Nis a function of the lattice parameters , d, andN, and is proportional tothe rangeSof the interaction defined by [4,5]

N S=j=i

1

rij. (2)

The sum is independent of the origini because of periodic conditions. For each , V is proportional to N.

When > d, which we call here theshort-rangecase,Sis finite in the TL [5], and things go as if each rotatorinteracted with a finite number of rotators, those within the range S. On the contrary, when < d, which weconsequently call the long-rangecase,Sdiverges in the TL and the factor1/Nin (1) compensates for this.

For = 0, the model (1) reduces to the model introduced originally in [6] and called HMF model. TheHamiltonian of the HMF model is

H0 = K+V0 =Ni=1

p2i2

+

2N

Ni,j=1

[1 cos(i j)] . (3)

1 Very often, as in this case, the term "nonextensivity" is used to refer to the "non-additivity" property.

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Dynamics and thermodynamics of a model with long-range interactions 247

This model can be seen as classical X Y -spins with infinite-range couplings, or also as representing particlesmoving on the unit circle. In the latter interpretation, the coordinate iof particle i is its position on the circle andpi its conjugate momentum. For >0, particles attract each other or, equivalently speaking, spins tend to align(ferromagnetic case), while for

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0.0 0.2 0.4 0.6 0.8 1.0 1.2

U

0.0

0.2

0.4

0.6

0.8

1.0

M

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0

0.2

0.4

0.6

0.8

1.0

T Exact equilibrium solution

N=100N=1000

a

b

Fig. 1a,b. Temperature T and magneti-

zation M as functions of the energy perparticleU in the ferromagnetic case. Thesymbols refer to equilibrium molecular dy-

namic simulations for N = 102 and103,while the solid lines refer to the canoni-

cal equilibrium prediction obtained analyt-

ically (see text). The vertical dashed line

indicates the critical energy density located

atUc = 0.75andc = 1/Tc = 2

where x and y are two-dimensional vectors and is positive. We can therefore rewrite (9) as

Z= C

dNi

dy exp[y2 +2M y] , (12)where = N. We now use definition (4) and exchange the order of the integrals in (12), factorizing theintegration over the coordinates of the particles. By introducing the rescaled variable y y

N/2, one ends

up with the following expression forZ:

Z= NC

2

dy exp

N

y2

2 ln(2I0(y))

, (13)

whereI0 is the modified Bessel function of order 0 and y is the modulus ofy . Finally, integral (13) can beevaluated by employing the saddle point technique in the thermodynamic limit, i.e. for N . In this limit,the Helmholtz free energy per particlefreads as

f= limN

ln Z

N =

1

2ln

2

+

2+ max

y

y2

2 ln(2I0(y))

, (14)

while the maximum condition leads to the following consistency equation:

y

=

I1(y)

I0(y), (15)

whereI1is the modified Bessel function of order1. The expression in (15) is analogous, in the XY model, to theCurieWeiss equation obtained by solving the Ising model in the mean-field approximation. For c = 2it presents only the solution y = 0, which is unstable . At=c, i.e. below the critical temperature Tc = 0.5(kB = 1), two new stable symmetric solutions appear through a pitchfork bifurcation, and a discontinuity inthe second derivative of the free energy is present, indicating a second-order phase transition. These results are

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-3 -2 -1 0 1 2 3p

0.001

0.01

0.1

1pdf

-3 -2 -1 0 1 2 3

0.001

0.01

0.1

1

101

102

103

104

105

106

107

t

0.4

0.425

0.45

0.475

0.5

2/N-3 -2 -1 0 1 2 3

p

0.001

0.01

0.1

1pdf

-3 -2 -1 0 1 2 3

0.001

0.01

0.1

1

U

T

0.5 0.6 0.7 0.8

0.4

0.5

U

T

0.5 0.6 0.7 0.8

0.4

QSS regime

N=500

BG regime

U=0.69

a b

c d

0.5

Teq

=0.476

Fig. 2ad.Microcanonical numerical simulations for N= 500and energy densityU= 0.69. In the main part of the figure we plottwice the average kinetic energy per particle (which gives the temperature) as a function of time (filled triangles). We can easily

distinguish a long metastable plateau (QSS regime) preceding the relaxation towards the BoltzmannGibbs equilibrium temperature

(BG regime). In the BG regime, one finds, as expected, very good agreement with the equilibrium thermodynamic value for the

temperature (paneld). In this regime the velocity pdfs reported in panelbare Gaussian. At variance, in the QSS region, we observe

strong deviations from the expected equilibrium temperature. Here the specific heat becomes negative (panel c) and the velocity

pdfs, reported in panela, are very different from the Gaussian equilibrium curves, reported as full lines for comparison (see text for

further details).

confirmed by an analysis of the order parameter2

M =I1(y)

I0(y). (16)

The magnetization Mvanishes continuously at c(see Fig. 1b). Thus, since Mmeasures the degree of clusteringof the particles, we have a transition from a clustered phase when > cto a homogeneous phase when < c.The exponent that characterizes the behavior of the magnetization close to the critical point is 1

2, as expected for

a mean field model [6]. One can also obtain the energy per particle vs temperature and magnetization

U =(f)

=

1

2+

1

2

1 M2

, (17)

the so calledcaloric curve, which is reported in Fig. 1a.

In Fig. 1 we report temperature and magnetization vs the energy density. The full curves are the exact results

obtained in the canonical ensemble, the BoltzmannGibbs (BG) equilibrium. The numerical simulations (sym-

bols) were performed at fixed total energy (microcanonical molecular dynamics) by starting the system close tothe equilibrium, i.e. with an initial Gaussian distribution of angles and velocities, and reproduce the theoretical

curves. This is already true for small system sizes such as N = 100, apart from some finite size effects in thehomogeneous phase [2]. As we shall see in the next section the scenario is very different when the system is

started with strong out-of-equilibrium initial conditions: in such a case the dynamics does have difficulties in

reaching the BG equilibrium and shows a series of anomalies.

2 This is obtained by adding to the Hamiltonian an external field and taking the derivative of the free energy with respect to this

field, evaluated at zero field.

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4 Dynamics

The dynamics of each particle obeys the following pendulum equation of motion:

i =Nj=1

sin(j i) = Msin(i ) , i= 1, .....,N , (18)

whereMandhave non-trivial time dependencies, related to the motion of all the other particles in the system.

Equations (18) can be integrated numerically (see, for example, [6,7] for technical details). In this section weshow that, in the energy range from U = 0.5up toUc = 0.75, when the system is started with strong out-of-equilibrium initial conditions, the model has a non-trivial dynamical relaxation to the BoltzmannGibbs (BG)

equilibrium. The class of out-of-equilibrium initial conditions we consider, calledwater baginitial conditions,

consists ofi = 0 i and uniformly distributed momenta (according to the total energy density U). In Fig. 2we report, for U = 0.69 and N = 500, the time evolution of2 < K > /N (where < K > denotes thetime-averaged kinetic energy), a quantity that coincides with the temperatureT. As expected, the system doesnot relax immediately to the BG equilibrium, but rapidly reaches a quasi-stationary state (QSS) corresponding

to a temperature plateau situated below the canonical prediction (dotted curve). TheTvsUplot (caloric curve),shown in inset c for the QSS regime, confirms a large disagreement with the equilibrium prediction (inset d).

Furthermore, it turns out that the system remains trapped in such a state for a time that diverges with the size Nofthe system [9]. This means that, if the thermodynamic limit is performed before the infinite-time limit, the QSS

becomes stable and the system never relaxes to the BG equilibrium, exhibiting different equilibrium properties

characterized by non-Gaussian velocity distributions (see inset a) [9]. Such velocity distributions have been fitted

in [9] and have been shown to be in agreement with the prediction of Tsallis generalized thermodynamics [18].

The latter is a theoretical formalism well suited to describing all situations in which long-range correlations,

weak mixing, and fractal structures in phase space are present [19], such as, for example, excited plasmas [22],

turbulent fluids [23,24], maps at the edge of chaos [2528], high-energy nuclear reactions [29], and cosmic rays

fluxes [30].

The characteristics of QSSs have been studied in several papers: vanishing Lyapunov spectrum [9,14,15], neg-

ative microcanonical specific heat [8], dynamical correlations in phase space [9,11], Levy walks and anomalous

diffusion [7]. Recently, the validity of the zeroth principle of thermodynamics has also been numerically demon-

strated for these metastable states [12].

In the rest of this paper we focus on the study of anomalous diffusion, long-range correlations, and the spin-glass

phase.

4.1 Anomalous diffusion

The link between relaxation to the BG equilibrium, and anomalous diffusion was studied in [7]. In that paper

the authors studied the variance of the angular displacement, defined as

2(t) = 1

N

Ni=1

[i(t) i(0)]2 , (19)

which typically depends on time as 2(t) t. The diffusion is anomalous when = 1and, in particular, itis calledsubdiffusionif0 < < 1 and superdiffusionif1 < < 2. In [7] the authors found the existence ofan anomalous superdiffusive behavior below the critical energy that was connected to the presence of the QSS.

Superdiffusion turns into normal diffusion after a crossover time crossover that coincides with the time relaxneeded for the QSS to relax to the BG equilibrium. We illustrate this behavior in Fig. 3. In particular, we show, for

the case ofN= 500and various energy densities, the time evolution for 2(t). No diffusion (= 0) is observedfor very low energy, while one gets ballistic motion in the overcritical energy region (= 2). On the other hand,superdiffusion is found in the energy regionU = 0.5 0.75with1.4. We plot the cases ofU = 0.6andU = 0.69. As expected, however, diffusion again becomes normal ( = 1) after complete relaxation. In thefigure we indicate with vertical dashed lines the relaxation time relax for U = 0.6and U = 0.69. Lines withdifferent slopes are also reported to indicate the different diffusion regimes.

Notice that diffusion starts around U = 0.3when particles start to be evaporated from the main cluster [6].

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103

104

105

t

10-2

100

102

104

106

108

1010

1012

2

=0

=2

=1

N=500

=1.38

U=0.2

U=0.69

U=0.6

U=5.0

Fig. 3. Time evolution of the variance for

theangulardisplacement (see(18))for N =500and various energy densities. A ballis-tic diffusion behavior ( = 2) is found forovercritical energies. No diffusion (= 0)is of course observed at very small ener-

gies (U

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101

102

103

t

10-3

10-2

10-1

100

Cp

(t)

q-exp: q=1.55 =245 A=0.72

0 2000 4000

t

-20

-10

0

lnq

[C

p(t)]

U=0.69 N=1000

a b

Fig. 4. a Velocity correla-

tion function Cp(t) vs timet for the case of U = 0.69and N = 500 in the QSSregion (open symbols). The

curve plotted as a full line

is a q-exponential fit, withq = 1.55.b Theq-logarithmof the curves and points re-

ported in a (see text for fur-

ther details)

100

101

102

t /(tw)1/4

0.1

1.0

Cp

(t+tw,

tw)

q-exp: q=1.65 =60 A=0.7

101

102

103

t

0.1

1.0

Cp

(t+tw,

tw)

tw=500

tw=1000

tw=2000tw=5000

0 100 200

t /(tw)1/4

-3.0

-2.0

-1.0

0.0

lnq[

Cp

(t+tw,

tw)]

bU=0.69 N=1000 a

c

Fig. 5. a Two-time velocity

autocorrelation function de-

fined by (25) for U = 0.69andN= 1000. Different de-lay times tw were considered(open symbols). Aging, i.e.

a marked dependence of the

correlation function ontw is

clearly evident.bThe curvescollapse onto a single one if

an opportune scaling is per-

formed (see text). This be-

havior can be reproduced by

the q-exponential fit that isalso shown. We also plot in

cthe q-logarithm of the datadrawn inb

If we want to take into account several dynamicalrealizations (events)in order to obtainbetter averagedquantities,

it is possible to use the following alternative definition of the autocorrelation function:

Cp(t) =

< P(t) P(0)> < P(t)> < P(0)>

p(t)p(0) , (22)

where P = (p1, p2, ...pN) is theN-component velocity vector and the brackets< ... > indicate the averageover different events, while p(t) and p(0) are the standard deviations at time t and at the initial time. Sucha function has been numerically evaluated [11] in the QSS region. The results are shown in Fig. 4 for the case

U = 0.69and N= 1000: a power-law tail is observed, this being a signature of long-range correlations. Thepower-law tail and the initial saturation can be fitted by means of the q-exponential relaxation function of Tsallis

generalized thermodynamics [18],

eq(z) = [1 + (1 q)z]1

(1q) , (23)

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equilibrium properties of the HMF model can be derived analytically, though the model dynamics present very

interesting anomalies like metastability, superdiffusion, non-Gaussian velocity pdfs, long-range correlations,

vanishing Lyapunov exponents, and weak ergodicity breaking in an energy region below the critical point. These

anomalies are common to many long-range systems. Recently, it has been found that such an anomalous regime

can be characterized as a glassy phase. Therefore, the model also represents a very interesting bridge between

nonextensive systems and glassy systems and in this respect it deserves more detailed investigations in the

future. The link with the nonextensive thermostatistics formalism proposed by Constantino Tsallis is also a very

promising and intriguing line of research. All the anomalies point in that direction and the future years will be

crucial in order to establish this connection in a firm and rigorous way as has recently been done, for example,

with unimodal maps.

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