PhysRevLett.95.240604 Mukhamel 2005

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Breaking of Ergodicity and Long Relaxation Times in Systems with Long-Range Interactions

D. Mukamel,1,* S. Ruffo,2,† and N. Schreiber1

1 Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel2 Dipartimento di Energetica ‘‘Sergio Stecco,’’ Università di Firenze, INFN and CSDC, via s. Marta 3, 50139 Firenze, Italy

(Received 30 August 2005; published 9 December 2005)

The thermodynamic and dynamical properties of an Ising model with both short-range and long-range,

mean-field-like, interactions are studied within the microcanonical ensemble. It is found that therelaxation time of thermodynamically unstable states diverges logarithmically with system size. This isin contrast with the case of short-range interactions where this time is finite. Moreover, at sufficiently lowenergies, gaps in the magnetization interval may develop to which no microscopic configurationcorresponds. As a result, in local microcanonical dynamics the system cannot move across the gap,leading to breaking of ergodicity even in finite systems. These are general features of systems with long-range interactions and are expected to be valid even when the interaction is slowly decaying with distance.

DOI: 10.1103/PhysRevLett.95.240604 PACS numbers: 05.20.Gg, 05.50.+q, 05.70.Fh

Systems with long-range interactions are rather commonin nature [1]. In such systems the interparticle potentialdecays at large distance r as 1=r with d, in dimen-sion d. Examples include magnets with dipolar interac-

tions, wave-particle interactions [2], gravitational forces[3], and Coulomb forces in globally charged systems [4]. Itis well-known that in such systems the various statisticalmechanical ensembles need not be equivalent [5]. Forexample, whereas the canonical ensemble specific heat isalways non-negative, it may become negative within themicrocanonical ensemble when long-range interactions arepresent [6]. It has recently been suggested that wheneverthe canonical ensemble exhibits a first order phase transi-tion the canonical and the microcanonical phase diagramsmay be different [7]. This has been demonstrated by adetailed study of a spin-1 Ising model with long-range,

mean-field-like, interactions. While the thermodynamicbehavior of such models is fairly well understood, theirdynamics, and the approach to equilibrium, has not beeninvestigated in detail so far [8]. The aim of this Letter is toidentify some general dynamical features which distin-guish systems with long-range interactions from thosewith short-range ones.

One characteristic of systems with short-range interac-tions is that the domain in the space of extensive thermo-

dynamic variables ~X over which the model is defined isconvex for sufficiently large numbers of particles. Here the

components of the vector ~X are variables like the energy,volume, and possibly other extensive parameters corre-

sponding to the system under study. Convexity is a directresult of additivity. By combining two appropriately

weighted subsystems with extensive variables ~X 1 and ~X 2,

any intermediate value of ~X between ~X 1 and ~X 2 may berealized. On the other hand, systems with long-range in-teractions are not additive, and thus intermediate values of the extensive variables are not necessarily accessible. Thisfeature has a profound consequence on the dynamics of systems with long-range interactions. Gaps may open up in

the space of extensive variables and lead to breaking of ergodicity under local microcanonical dynamics of suchsystems. An example of such gaps in a class of anisotropicXY models has recently been discussed in Ref. [9].

Another interesting feature of systems with long-rangeinteractions is their relaxation time. It is well-known thatthe relaxation time of metastable states grows exponen-tially with the system size [10]. On the other hand therelaxation processes of thermodynamically unstable statesare not well understood. In systems with short-range inter-actions the relaxation takes place on a finite time scale.However, previous studies of a mean field XY modelsuggest that this relaxation time diverges with the systemsize [8].

In this Letter we study some of the issues discussedabove by considering a spin-1=2 Ising model with bothlong-range, mean-field-like, and short-range nearest neigh-

bor interactions [11,12]. For simplicity we consider a ringgeometry although the general features found in this studyare valid for higher dimensions. We study the thermody-namic and dynamical behavior of the model in both thecanonical and microcanonical ensembles. It is found thatthe two ensembles result in different phase diagrams aswas observed in other models [7]. Moreover, we find thatfor sufficiently low energy, gaps open up in the magneti-zation interval 1 m 1, to which no microscopicconfiguration corresponds. Thus the phase space breaksinto disconnected regions. Within a local microcanonicaldynamics the system is trapped in one of these regions,leading to a breakdown of ergodicity even in finite systems.In studying the relaxation time of thermodynamically un-stable states, corresponding to local minima of the entropy,we find that unlike the case of short-range interactionswhere this time is finite, here it diverges logarithmicallywith the system size. We provide a simple explanation forthis behavior by analyzing the dynamics of the system interms of a Langevin equation.

We start by considering an Ising model defined on a ringof N spins Si 1 with long and short-range interactions.

PRL 95, 240604 (2005) P H Y SI C A L R E V I EW L E T T E RS week ending

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0031-9007=05=95(24)=240604(4)$23.00 240604-1 © 2005 The American Physical Society

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The Hamiltonian is given by

H K 2

XN i1SiSi1 1

J

2N

XN i1

Si

2

; (1)

where J > 0 is a ferromagnetic, long-range, mean-field-like coupling, and K is a nearest neighbor coupling whichmay be of either sign. The canonical phase diagram of this

model has been derived in the past [11,12]. It has beenobserved that in the K; T plane, where T is the tempera-ture, the model exhibits a phase transition line separating adisordered phase from a ferromagnetic one (see Fig. 1).The transition is continuous for large K , where it is givenby eK . Here 1=T and J 1 is assumed forsimplicity. Throughout this work we take kB 1 for theBoltzmann constant. The transition becomes first orderbelow a tricritical point located at an antiferromagnetic

coupling K CTP ln3=23

p ’ 0:317.We now analyze the model within the microcanonical

ensemble. Let U 12

PiSiSi1 1 be the number of

antiferromagnetic bonds in a given configuration charac-

terized by N up spins and N down spins. Simple count-ing yields that the number of microscopic configurationscorresponding to N ; N ; U is given, to leading order inN , by

N ; N ; U N U=2

N U=2

: (2)

Expressing N and N in terms of N and the magnetiza-tion M N N , and denoting m M=N , u U=N ,and the energy per spin E=N , one finds that theentropy per spin, s; m 1

N ln is given in the thermo-

dynamic limit by

s; m 121 m ln1 m 1

21 m ln1 m

u lnu 121 m u ln1 m u

121 m u ln1 m u; (3)

where u satisfies

J 2

m2 Ku: (4)

By maximizing s; m with respect to m one obtains boththe spontaneous magnetization ms and the entropys s; ms of the system for a given energy. Thetemperature, and thus the caloric curve, is given by 1=T @s=@. A straightforward analysis of (3) yields themicrocanonical K; T phase diagram of the model(Fig. 1), where it is also compared with the canonicalone. It is found that the model exhibits a critical line givenby the same expression as that of the canonical ensemble.However, this line extends beyond the canonical tricritical

point, reaching a microcanonical tricritical point at K MTP ’0:359, which is computed analytically. For K < K MTPthe transition becomes first order and it is characterized bya discontinuity in the temperature. The transition is thusrepresented by two lines in the K; T plane correspondingto the two temperatures at the transition point. These linesare obtained by numerically maximizing the entropy (4).

We now consider the dynamics of the model. This isdone by using the microcanonical Monte Carlo dynamicssuggested by Creutz [13]. In this dynamics one samples themicrostates of the system with energy E ED with ED 0 by attempting random single spin flips. One can showthat, to leading order in the system size, the distribution of

the energy ED takes the form

PED eED=T : (5)Thus, measuring this distribution yields the temperature T which corresponds to the energy E.

In applying this dynamics to our model one should notethat to next order in N the energy distribution is given byPED expED=T E2D=2CV T 2, where CV ON is the specific heat. In extensive systems with short-rangeinteractions the specific heat is positive and thus the cor-rection term does not modify the distribution function forlarge N . However in our system CV can be negative in acertain range of K and E. It even becomes arbitrarily smallnear the MTP point, making the next to leading term in theexpansion arbitrarily large. This could, in principle, quali-tatively modify the energy distribution. However, as longas the entropy is an increasing function of E, namely, forpositive temperatures, the distribution function (5) is validin the thermodynamic limit. This point is verified by ournumerical studies, where an exponential distribution of theenergy ED is clearly observed.

We now address the issue of the accessible magnetiza-tion intervals in this model. We find that in certain regions

-0.5 -0.4 -0.3 -0.20

0.2

0.4

0.6

0.8

MTP

CTP

T

K

m=0

m≠0

FIG. 1. The canonical and microcanonical K; T phase dia-gram. In the canonical ensemble the large K transition iscontinuous (bold solid line) down to the trictitical point CTPwhere it becomes first order (dashed line). In the microcanonicalensemble the continuous transition coincides with the canonicalone at large K (bold line). It persists at lower K (dotted line)down to the tricritical point MTP where it turns first order, with abranching of the transition line (solid lines). The region betweenthese two lines (shaded area) is not accessible.


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in the K; E plane, the magnetization m cannot assume anyvalue in the interval 1; 1. There exist gaps in thisinterval to which no microscopic configuration could beassociated. To see this, we take for simplicity the caseN > N . It is evident that the local energy U satisfies 0 U 2N . The upper bound is achieved in microscopicconfigurations where all down spins are isolated. Thisimplies that 0 u 1 m. Combining this with (4)one finds that for positive m the accessible states have tosatisfy

m 2

p ; m m; m m

with m

K

K 2

2

K

q :

(6)

Similar restrictions exist for negative m. These restrictionsyield the accessible magnetization domain shown in Fig. 2for K 0:4. It is clear that this domain is not convex.Entropy curves sm for some typical energies are given inFig. 3, demonstrating that the number of accessible mag-netization intervals changes from one to three, and then totwo as the energy is lowered.

This feature of disconnected accessible magnetizationintervals, which is typical to systems with long-rangeinteractions, has profound implications on the dynamics.In particular, starting from an initial condition which lieswithin one of these intervals, local dynamics, such as theone applied in this work, is unable to move the system to adifferent accessible interval. Thus ergodicity is broken inthe microcanonical dynamics even at finite N .

To demonstrate this point we display in Fig. 4 the timeevolution of the magnetization for two cases: one in whichthere is a single accessible magnetization interval, whereone sees that the magnetization switches between themetastable m 0 state and the two stable states m ms. In the other case the metastable m 0 state isdisconnected from the stable ones, making the system

unable to switch from one state to the other. Note thatthis feature is characteristic of the microcanonical dynam-ics. When local canonical dynamics, say, a Metropolisalgorithm [14], is applied, the system may cross the for-bidden region (by moving to higher energy states where theforbidden region diminishes). However, the breakdown of ergodicity is manifested in the fact that the switching ratebetween the accessible regions is exceedingly small, de-creasing exponentially with N .

We conclude this study by considering the lifetime N of the m 0 state, when it is not the equilibrium state of the system. In the case where m 0 is a metastable state,corresponding to a local maximum of the entropy [such asin Fig. 4(a)] we find that the life time satisfies eN s

-1 -0.5 0 0.5 1m

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

ε

FIG. 2. Accessible region in the m; plane (shaded area) forK 0:4. For energies in a certain range, gaps in the accessiblemagnetization values are present.

-1 -0.5 0 0.5 1m

0

0.2

0.4

s-0.32-0.33-0.405

ε

FIG. 3. The sm curves for K 0:4, and for typical energyvalues, demonstrating that gaps in the accessible states developas the energy is lowered.

-0.5

0

0.5

1

1.5

m -1 0 1

0 25 50 75 100(MC sweeps)/1000

0

0.5

1

1.5

m

-1 0 1

a

b

FIG. 4. Time evolution of the magnetization for K 0:4(a) in the ergodic region ( 0:318) and (b) in the nonergodicregion ( 0:325). The corresponding entropy curves areshown in the insets.


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where s is the difference in entropy of the m 0 stateand that of the unstable magnetic state corresponding to thelocal minimum of the entropy [see Fig. 5(a)]. Such expo-nential dependence on N has been found in the past inMetropolis-type dynamics of the Ising model with mean-field interactions [10]. Similar behavior has been found inthe XY model [15] and in gravitational systems [16] whenmicrocanonical dynamics was applied.

We now turn to the case where the m

0 state is

thermodynamically unstable, where it corresponds to alocal minimum of the entropy. In systems with short-rangeinteractions, the relaxation time of this state is finite. Herewe find unexpectedly that the life time diverges weaklywith N , N logN [see Fig. 5(b)]. This is to be com-pared with studies of the life time of the zero magnetizationstate in the XY model under similar conditions which showthat N N with ’ 1:7 [8].

In order to understand this behavior we consider theLangevin equation corresponding to the dynamics of thesystem. The evolution of m is given by

@m

@t @s

@m t; htt0i Dt t0; (7)where t is the usual white noise term. The diffusionconstant D scales as D 1=N . This can be easily seen byconsidering the noninteracting case in which the magneti-zation evolves by pure diffusion where the diffusion con-stant is known to scale in this form. Taking sm am2with a > 0, making the m 0 state thermodynamicallyunstable, and neglecting higher order terms, the distribu-tion function of the magnetization, Pm; t, may be calcu-lated. This is done by solving the Fokker-Planck equationcorresponding to (7). With the initial condition for theprobability distribution P

m;0

m

, the large time

asymptotic distribution is found to be [17]

Pm; t expae

atm2

D

: (8)

This is a Gaussian distribution whose width grows withtime. The relaxation time corresponds to the width reach-ing a value of O1, yielding lnD lnN . Similaranalysis and simulations can be carried out for the canoni-cal ensemble yielding the same divergence.

In summary, some general features of the dynamicalbehavior of systems with long-range interactions werestudied using the microcanonical local dynamics of anIsing model. Properties like gaps in the accessible magne-tization interval and breaking of ergodicity in finite sys-tems have been demonstrated. We also find that therelaxation time of an unstable m 0 state, correspondingto a local minimum of the entropy, is not finite but rather

diverges logarithmically with N . We expect these phe-nomena to appear in other systems with long-range inter-actions which are not necessarily of mean-field type. Thisstudy is thus of relevance to a wide class of physicalsystems, such as dipolar systems, self gravitating andCoulomb systems, and interacting wave-particle systems.

We have benefited from discussions with F. Bouchet,A. Campa, T. Dauxois, A. Giansanti, and M. R. Evans. Thisstudy was supported by the Israel Science Foundation(ISF), the PRIN03 project Order and chaos in nonlinear extended systems and INFN-Florence. D.M. and S. R.thank ENS-Lyon for hospitality and financial support.

*Electronic address: [email protected]†Electronic address: [email protected]

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J. F. Nagle, J. Appl. Phys. 42, 1280 (1971).[12] M. Kardar, Phys. Rev. B 28, 244 (1983).[13] M. Creutz, Phys. Rev. Lett. 50, 1411 (1983).[14] N. Metropolis, A. W. Rosenbluth, M.N. Rosenbluth, A.H.

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Berlin, 1996), p. 109.

0 200 400N

100

102

104

τ

(a)

102

103

104

105

N

6

8

10

12

τ

(b)

FIG. 5. Relaxation time of the m 0 state when this state is(a) a local maximum K 0:4; 0:318 and (b) a localminimum of the entropy K 0:25; 0:2.


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