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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: andrea.rapisarda@ct.infn.it)
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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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248 A. Pluchino et al.
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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Dynamics and thermodynamics of a model with long-range interactions 249
-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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250 A. Pluchino et al.
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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Dynamics and thermodynamics of a model with long-range interactions 251
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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252 A. Pluchino et al.
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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254 A. Pluchino et al.
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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