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arXiv:cond-mat/0508

604v2

[cond-mat.s

tat-mech]28Aug

2005

Breaking of ergodicity and long relaxation times in systems with long-range

interactions

D. Mukamel 1, S. Ruffo 2, and N. Schreiber 11 Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel 76100

2 Dipartimento di Energetica Sergio Stecco, Universita di Firenze,INFN and CSDC, via s. Marta 3, 50139 Firenze, Italy.

(Dated: February 2, 2008)

The thermodynamic and dynamical properties of an Ising model with both short range and longrange, mean field like, interactions are studied within the microcanonical ensemble. It is found thatthe relaxation time of thermodynamically unstablestates diverges logarithmically with system size.This is in contrast with the case of short range interactions where this time is finite. Moreover, atsufficiently low energies, gaps in the magnetization interval may develop to which no microscopicconfiguration corresponds. As a result, in local microcanonical dynamicsthe system cannot moveacross the gap, leading to breaking of ergodicity even in finite systems. These are general featuresof systems with long range interactions and are expected to be valid even when the interaction isslowly decaying with distance.

PACS numbers: 05.20.Gg, 05.50.+q, 05.70.Fh

Systems with long range interactions are rather com-

mon in nature[1]. In such systems the inter-particle po-tential decays at large distance r as 1/r with d,in dimension d. Examples include magnets with dipolarinteractions, wave-particle interactions [2], gravitationalforces [3], and Coulomb forces in globally charged sys-tems[4]. It is well known that in such systems the vari-ous statistical mechanical ensembles need not be equiva-lent [5]. For example whereas the canonical ensemble spe-cific heat is always non-negative, it may become negativewithin the microcanonical ensemble when long range in-teractions are present[6]. It has recently been suggestedthat whenever the canonical ensemble exhibits a first or-der phase transition the canonical and the microcanoni-

cal phase diagrams may be different [7]. This has beendemonstrated by a detailed study of a spin-1 Ising modelwith long range, mean field like, interactions. While thethermodynamic behavior of such models is fairly wellunderstood, their dynamics, and the approach to equi-librium, has not been investigated in detail so far [8].The aim of the present Letter is to identify some generaldynamical features which distinguish systems with longrange interactions from those with short range ones.

One characteristic of systems with short range inter-actions is that the domain in the space of extensive ther-modynamic variables Xover which the model is defined

is convex for sufficiently large number of particles. Herethe components of the vector X are variables like theenergy, volume and possibly other extensive parameterscorresponding to the system under study. Convexity is adirect result of additivity. By combining two appropri-ately weighted sub-systems with extensive variables X1and X2, any intermediate value of X between X1 and X2may be realized. On the other hand systems with longrange interactions are not additive, and thus intermedi-ate values of the extensive variables are not necessarily

accessible. This feature has a profound consequence on

the dynamics of systems with long range interactions.Gaps may open up in the space of extensive variables andlead to breaking of ergodicity under local microcanoni-cal dynamics of such systems. An example of such gapsin a class of anisotropic XY models has recently beendiscussed in[9].

Another interesting feature of systems with long rangeinteractions is their relaxation time. It is well knownthat the relaxation time of metastable states grows ex-ponentially with the system size [10]. On the other handthe relaxation processes of thermodynamically unstablestates are not well understood. In systems with shortrange interactions the relaxation takes place on a finitetime scale. However previous studies of a mean field X Ymodel suggest that this relaxation time diverges with thesystem size[8].

In the present Letter we study some of the issues dis-cussed above by considering a spin-1/2 Ising model withboth long range, mean field like, and short range near-est neighbor interactions on a ring geometry[11,12]. Westudy its thermodynamic and dynamical behavior in boththe canonical and microcanonical ensembles. It is foundthat the two ensembles result in different phase diagramsas was observed in other models [7]. Moreover, we findthat for sufficiently low energy, gaps open up in the mag-

netization interval 1 m 1, to which no microscopicconfiguration corresponds. Thus the phase space breaksinto disconnected regions. Within a localmicrocanonicaldynamics the system is trapped in one of these regions,leading to a breakdown of ergodicity even in finite sys-tems. In studying the relaxation time of thermodynam-ically unstable states, corresponding to local minima ofthe entropy, we find that unlike the case of short rangeinteractions where this time is finite, here it diverges log-arithmically with the system size. We provide a simple
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2

-0.5 -0.4 -0.3 -0.20

0.2

0.4

0.6

0.8

MTP

CTP

T

K

m=0

m0

FIG. 1: The canonical and microcanonical (K, T) phase di-agram. 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 micro-canonical ensemble the continuous transition coincides withthe canonical one at large K(bold line). It persists at lowerK (dotted line) down to the tricritical point MTP where itturns first order, with a branching of the transition line (solidlines). The region between these two lines (shaded area) isnot accessible.

explanation for this behavior by analyzing the dynamicsof the system in terms of a Langevin equation.

We start by considering an Ising model defined on aring ofNspinsSi= 1 with long and short range inter-actions. The Hamiltonian is given by

H= K2

Ni=1

(SiSi+1 1) J

2N

Ni=1

Si

2, (1)

where J > 0 is a ferromagnetic, long range, mean fieldlike coupling, andKis 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 hasbeen observed that in the (K, T) plane, where T is thetemperature, the model exhibits a phase transition lineseparating a disordered phase from a ferromagnetic one(see Fig. 1). The transition is continuous for large K,where it is given by = eK. Here = 1/T andJ= 1 is assumed for simplicity. Throughout this work wetake kB = 1 for the Boltzmann constant. The transitionbecomes first order below a tricritical point located atan antiferromagnetic coupling KCTP = ln 3/2

3

0.317.We now analyze the model within the microcanonical

ensemble. Let U =

1

2i(SiSi+1 1) be the number

of antiferromagnetic bonds in a given configuration char-acterized by N+ up spins and N down spins. Simplecounting yields that the number of microscopic configu-rations corresponding to (N+, N, U) is given, to leadingorder inN, by

(N+, N, U) =

N+U/2

NU/2

. (2)

ExpressingN+and Nin terms ofNand the magnetiza-tionM=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) = 1N ln is given in the ther-modynamic limit by

s(, m) = 1

2(1 + m)ln(1 + m) +

1

2(1 m) ln(1 m)

u ln u 12

(1 + m u) ln(1 + m u)

12

(1 m u) ln(1 m u), (3)

whereu satisfies

= J2

m2 + Ku . (4)

By maximizing s(, m) with respect to m one obtainsboth the spontaneous magnetization ms() and the en-tropy s() s(, ms()) of the system for a given energy.The temperature, and thus the caloric curve, is given by1/T =s()/. A straightforward analysis of (3) yieldsthe microcanonical (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 linegiven by the same expression as that of the canonicalensemble. However this line extends beyond the canoni-cal tricritical point, reaching a microcanonical tricriticalpoint at KMTP 0.359, which is computed analyti-cally. For K < KMTP the transition becomes first orderand it is characterized by a discontinuity in the temper-ature. The transition is thus represented by two linesin the (K, T) plane corresponding to the two tempera-tures at the transition point. These lines are obtained bynumerically maximizing the entropy (3).

We now consider the dynamics of the model. This is

done by using the microcanonical Monte Carlo dynamicssuggested by Creutz [13]. In this dynamics one samplesthe microstates of the system with energy E ED withED 0 by attempting random single spin flips. Onecan show that, to leading order in the system size, thedistribution of the energy ED takes the form

P(ED) eED/T . (5)

Thus, measuring this distribution, yields the temperatureTwhich corresponds to the energyE.

In applying this dynamics to our model one should

note that to next order in Nthe energy distribution isgiven by P(ED) exp(ED/T E2D/2CVT2), whereCV = O(N) is the specific heat. In extensive systemswith short range interactions the specific heat is posi-tive and thus the correction term does not modify thedistribution function for large N. However in our sys-tem CVcan be negative in a certain range ofK and E.It even becomes arbitrarily small near the MTP point,making the next to leading term in the expansion arbi-trarily large. This could in principle qualitatively modify


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the energy distribution. However as long as the entropyis an increasing function ofE, namely for positive tem-peratures, the distribution function (5) is valid in thethermodynamic limit. This point is verified by our nu-merical studies, where an exponential distribution of theenergyED is clearly observed.

We now address the issue of the accessible magneti-zation intervals in this model. We find that in certain

regions in the (K, E) plane, the magnetization m can-not assume any value in the interval (1, 1). There existgaps in this interval to which no microscopic configura-tion could be associated. To see this, we take for simplic-ity the caseN+ > N. It is evident that the local energyUsatisfies 0 U 2N. The upper bound is achievedin microscopic configurations where all down spins areisolated. This implies that 0 u 1 m. Combiningthis with(4) one finds that for positive m the accessiblestates have to satisfy

m

2 , m m+ , m mwith m =

K

K2 2( K). (6)Similar restrictions exist for negativem. These restric-

tions yield the accessible magnetization domain shown inFig. 2for K=0.4. It is clear that this domain is notconvex. Entropy curves s(m) for some typical energiesare given in Fig. 3, demonstrating that the number ofaccessible magnetization intervals changes from one tothree, and then to two as the energy is lowered.

This feature of disconnected accessible magnetizationintervals, which is typical to systems with long range in-teractions, has profound implications on the dynamics.In particular, starting from an initial condition which lieswithin one of these intervals, local dynamics, such as the

one applied in this work, is unable to move the system toa different accessible interval. Thus ergodicity is brokenin the microcanonical dynamics even at finite N.

To demonstrate this point we display in Fig. 4 thetime evolution of the magnetization for two cases: one inwhich there is a single accessible magnetization interval,where one sees that the magnetization switches betweenthe metastable m = 0 state and the two stable statesm= ms. In the other case the metastable m = 0 stateis disconnected from the stable ones, making the systemunable to switch from one state to the other. Note thatthis feature is characteristic of the microcanonical dy-namics. Using local canonical dynamics, say Metropolisalgorithm [14] , would allow the system to cross the for-bidden region (by moving to higher energy states wherethe forbidden region diminishes), and ergodicity is re-stored in finite systems. However, in the thermodynamiclimit, ergodicity would be broken even in the canonicalensemble, as the switching rate between the accessibleregions decreases exponentially with N.

We conclude this study by considering the life time(N) of the m = 0 state, when it is not the equilib-

-1 -0.5 0 0.5 1m

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

FIG. 2: Accessible region in the (m, ) plane (shaded area)for K = 0.4. For energies in a certain range, gaps in theaccessible magnetization values are present.

-1 -0.5 0 0.5 1m

0

0.2

0.4

s-0.32-0.33-0.405

FIG. 3: The s(m) curves for K = 0.4, and for typicalenergy values, demonstrating that gaps in the accessible statesdevelop as the energy is lowered.

rium state of the system. In the case wherem = 0 is a

metastable state, corresponding to a local maximum ofthe entropy (such as in Fig. 4a) we find that the life timesatisfies eNs where s is the difference in entropyof the m = 0 state and that of the unstable magneticstate corresponding to the local minimum of the entropy(see Fig. 5a). Such exponential dependence on N hasbeen found in the past in Metropolis type dynamics ofthe Ising model with mean field interactions [10]. Simi-lar behavior has been found in theX Ymodel [15] and ingravitational systems[16] when microcanonical dynamicswas applied.

We now turn to the case where the m = 0 state is ther-modynamicallyunstable, where it corresponds to a localminimum of the entropy. In systems with short range in-teractions, the relaxation time of this state is finite. Herewe find unexpectedly that the life time diverges weaklywith N, (N) log(N) (see Fig. 5b). This is to becompared with studies of the life time of the zero magne-tization state in the X Y model under similar conditionswhich show that(N) N with 1.7[8].

In order to understand this behavior we consider theLangevin equation corresponding to the dynamics of the


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4

-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 non-ergodicregion ( = 0.325). The corresponding entropy curves areshown in the inset.

0 200 400N

100

102

104

a

102

103

104

105

N

6

8

10

12

b

FIG. 5: Relaxation time of them = 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).

system. The evolution ofm is given by

mt

= sm

+ (t) , < (t)(t)>=D(t t) (7)

where (t) is the usual white noise term. The diffusionconstant Dscales asD 1/N. This can be easily seen byconsidering the non-interacting case in which the magne-tization evolves by pure diffusion where the diffusion con-stant is known to scale in this form. Takings(m) = am2

witha >0, making the m= 0 state thermodynamicallyunstable, and neglecting higher order terms, the distri-bution function of the magnetization, P(m, t), may becalculated. This is done by solving the Fokker-Planckequation corresponding to (7). With the initial condi-

tion for the probability distribution P(m, 0) = (m), thelarge time asymptotic distribution is found to be [17]

P(m, t) expae

atm2

D

. (8)

This is a Gaussian distribution whose width grows withtime. The relaxation time corresponds to the widthreaching a value of O(1), yielding ln D ln N.Similar analysis and simulations can be carried out forthe canonical 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 mag-netization interval and breaking of ergodicity in finitesystems have been demonstrated. We also find that therelaxation time of an unstable m = 0 state, correspond-ing to a local minimum of the entropy, is not finite but

rather diverges logarithmically withN. We expect thesephenomena to appear in other systems with long rangeinteractions which are not necessarily of mean field type.This study is thus of relevance to a wide class of physi-cal systems, 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.This study was supported by the Israel Science Foun-dation (ISF), the PRIN03 project Order and chaos innonlinear extended systems and INFN-Florence . D.M.

and S.R. thank ENS-Lyon for hospitality and financialsupport.

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[2] J. Barre, T. Dauxois, G. De Ninno, D. Fanelli and S.Ruffo, Phys. Rev. E, 69, R045501 (2004).

[3] T. Padmanabhan, Phys. Rep., 188, 285 (1990).[4] D. R. Nicholson,Introduction to Plasma Physics, Krieger

Pub. Co. (1992).[5] P. Hertel, W. Thirring, Annals of Physics, 63, 520 (1971).[6] V. A. Antonov, Leningrad Univ. 7, 135 (1962); Transla-

tion in IAU Symposium 113, 525 (1995); D. Lynden-Bell,R. Wood, Monthly Notices of the Royal Astronomical So-ciety, 138, 495 (1968).

[7] J. Barre, D. Mukamel, S. Ruffo, Phys. Rev. Lett., 87,030601 (2001).

[8] Y. Y. Yamaguchi, J Barre, F. Bouchet, T. Dauxois, S.Ruffo, Physica A, 337, 36 (2004).

[9] F. Borgonovi, G.L. Celardo, M. Maianti and E. Pedersoli,J. Stat. Phys., 116, 1435 (2004).

[10] R.B. Griffiths, C.Y. Weng and J.S. Langer, Phys. Rev.,149, 1 (1966).

[11] J. F. Nagle, Phys. Rev. A 2, 2124 (1970); J. C. Bonner

and 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. Teller and E. Teller, J. Chem. Phys. 21, 1087 (1953).[15] M. Antoni, S. Ruffo and A. Torcini, Europhys. Lett., 66,

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physics, 412, 1 (2003); P.H. Chavanis, astrph/0404251.[17] H. Risken The Fokker-Planck Equation, Springer-Verlag,

Berlin (1996), p. 109.

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