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1 This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

Computer Physics Communications 180 (2009) 480 484 Contents lists available at ScienceDirect Computer Physics Communications www.elsevier.

de Oliveira Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, São Paulo, Brazil article info abstract Article history: Received 7 October 2008 Received in

2 Computer Physics Communications 180 (2009) Contents lists available at ScienceDirect Computer Physics Communications Glassy states in the stochastic Potts model Mário J. de Oliveira Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, São Paulo, São Paulo, Brazil article info abstract Article history: Received 7 October 2008 Received in revised form 18 December 2008 Accepted 12 January 2009 Availableonline15January2009 PACS: q Fs My We have studied by numerical simulations the relaxation of the stochastic seven-state Potts model after a quench from a high temperature down to a temperature below the first-order transition. For quench temperatures just below the transition temperature the phase ordering occurs by simple coarsening under the action of surface tension. For sufficient low temperatures however the straightening of the interface between domains drives the system toward a metastable disordered state, identified as a glassy state. Escaping from this state occurs, if the quench temperature is nonzero, by a thermal activated dynamics that eventually drives the system toward the equilibrium state Elsevier B.V. All rights reserved. Keywords: Dynamic metastability Potts model Glasses 1. Introduction Most liquids, when cooled below its melting temperature, become a supercooled liquid. Eventually the supercooled liquid turns into a crystalline solid by the process of nucleation, that is, by the generation and growth of nuclei. If, however, nucleation is avoided, by a further decrease in temperature, the supercooled liquid may instead turn into a glass [1 3]. The transformation of the supercooled liquid into the glass is continuous and occurs around a certain temperature about two thirds of the melting temperature [3]. Such quantities as the volume and enthalpy decrease continuously as the temperature is diminished in contrast to the ordinary liquid crystalline solid transition for which a jump in these quantities is observed. The glass transformation cannot be considered as an equilibrium phase transition because both the supercooled liquid and the glass are in metastable states that eventually decay to the equilibrium crystalline state. The time spent in the metastable state can however be, in some cases, so big that for all practical purposes the glass can be considered as stable. This point of view is summarized as follows by P.W. Anderson [4]: the glass must form from a supercooled liquid, which can in principle crystallize itself, the glass is never the equilibrium phase at any temperature and the ease of glass formation is probably almost entirely a kinetic question. The purpose of the present paper is to analyze models that may exhibit a glassy state, that are in accordance with the principles stated by Ander- address: oliveira@if.usp.br. son. Therefore, we will be concerned here with models that exhibit only one equilibrium phase at high temperatures, the liquid state, and one equilibrium phase at low temperatures, the crystalline state. The model must display a first-order transition between these phases, which is a necessary condition for the appearance of the supercooled liquid. In addition to the Anderson s principles we include another crucial ingredient: the crystalline state should be a coexistence of several equivalent crystalline phases connected by symmetry operations. The third principle stated by Anderson implies that the structural disorder of a glass should not be caused by any a priori quenched disorder as happens to spin-glasses. The glassy state should emerge, according to our point of view, out of the competition, induced by the dynamics, among the equivalent incipient crystalline phases, after a quench to a sufficient low temperature. In accordance to the point of view above we look for lattice models that have no a priori quenched disorder. There exists a number of such models, that in equilibrium display a first-order phase transition and many equivalent phases at low temperatures, which might exhibit a glassy behavior after a quench [5 17], in particular the stochastic Potts model [5,9,12 17]. Recently, it has been shown that after a quench from a high temperature to a sufficient low temperature a system described by the stochastic Potts model gets trapped into highly disordered metastable states that we identify as glassy states [9,12 17]. In this paper we will be concerned with the relaxation following a quench from the disordered single phase occurring at high temperatures down to a low temperature where in equilibrium many ordered phase may coexist. After the quench the sys /$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.cpc

3 M.J. de Oliveira / Computer Physics Communications 180 (2009) tem does not order instantaneously because the incipient ordered phases, or domains, compete with each other. Small domains coalesce into larger domains which eventually become macroscopic ordered phases. The phase ordering of systems with only two ordered phases after a quench is dominated by a simple coarsening dynamics under the action of the interface surface tension, in which the average linear size of a domain growths with time according to the Lifshitz Allen Cahn law l t [18 22]. Forsystems with many ordered phases, such as the seven-state Potts model studied here, the simple coarsening dynamics also occurs but a distinct scenario takes place for sufficient low temperatures. In this regime, the straightening of the interface by simple coarsening drives the system towards a trapped disordered state, identified as a glassy state, that survives by a certain period of time, after which the system decays by thermal activated dynamics to equilibrium state. 2. The stochastic Potts model A model that may display a first-order phase transition and several equivalent states at low temperatures is the Potts model with q states or colors [23,24]. This model is defined on a lattice in which each site can be in one of q states or colors. Two nearest neighbor sites with distinct colors contribute an amount ɛ > 0to the total energy. Sites with the same color do not contribute. Here, we will be concerned with the ordinary Potts model in two dimensions which is known to exhibit a first-order phase transition when q > 4. The Hamiltonian of the Potts model is defined by H(η) = ε [ 1 δ(ηi, η j ) ], (1) ij where the sum is over all pairs of nearest neighbor sites of a square lattice and δ(x, y) is the Kronecker delta. The dynamical variables η i take the values 1, 2, 3,...,q. The equilibrium properties of the model are described by the Gibbs probability distribution P(η) = 1 Z e H(η)/kT, (2) where T is the temperature. Above a transition temperature T 0 the model exhibits one phase, which we identify as the liquid state; below T 0, it exhibits q coexisting ordered state, which we identify as the crystalline state. The transition temperature is exactly known [24] and is given by kt 0 /ε = 1/ ln(1 + q). The third principle stated by Anderson declares that the glass state should emerge out of a kinetic approach. Since the Potts model is defined in a static way by Eqs. (1) and (2), a dynamics has to be assigned to the model. Here we choose a Markov stochastic dynamics defined by the Metropolis algorithm, whose stationary probability distribution is the Gibbs distribution (2). If the temperature is nonzero, this stochastic dynamics guarantees that for sufficient long times the equilibrium phases will emerge so that the system can in principle crystallize itself. The glass state will come out as a state that survives in a limited period of time. Of course,wedesiretheperiodtobemuchlargerthanthetimeof observation. The stochastic Potts model is here defined by the following rules. A site of the lattice is pick out at random and a trial color for this site is randomly chosen. This defines a trial state η for the whole lattice. The energy difference H = H(η ) H(η) between the trial state η and the present state η is determined. If H 0, then the trial state becomes the new state. If H > 0 then the trial state is accepted with probability p = e H/k B T. The dynamics so defined is a Markovian stochastic process whose stationary probability distribution is the equilibrium distribution given by Eq. (2). Therefore, if one waits enough time the system will eventually reach an equilibrium state, as long as T 0. Fig. 1. Log log plot of the excess energy per site u u e versus t for the stochastic seven-state Potts model, for several working temperatures. The asymptotic behavior is linear with a slope 1/2. 3. Numerical results We have performed Monte Carlo simulations by the use of the Metropolis algorithm just mentioned on a square lattice with periodic boundary condition starting from a complete random configuration. In this paper we have chosen to study the seven-state Potts model, which displays a first-order phase transition at kt 0 /ε = 1/ ln(1 + 7) = , and a square lattice with sites. Since the Monte Carlo dynamics mimics the contact of a system to a thermal reservoir, the simulation corresponds to a quench from an infinite temperature down to the temperature T of the reservoir. The relaxation is observed by measuring the energy per site u(t) as a function of time t in units of Monte Carlo steps. In the present simulations we have set ε = 1 and used a scale of temperature such that k = 1. We firstly analyze the relaxation after a quench to temperatures just below the transition temperature. Afterwards we examine the relaxation to temperatures low compared to the transition temperature. We distinguish two types of relaxation toward the equilibrium crystalline state. For temperatures between T and the transition temperature T 0, where T 0.35, the relaxation is dominated by a simple coarsening dynamics. For temperatures below T, the relaxation is dominated by a thermal activated dynamics to be explained in the next section. By a simple coarsening we mean a relaxation in which the growth of a domain is caused by the straightening of interface under the action of the surface tension, which follows the Lifshitz Allen Cahn law l t, the mean linear size l of a domain increasing with the square root of time t [18 22]. Since the excess energy per site u = u u e, where u e is the equilibrium energy per site, comes from the interface, it follows that, in a two-dimensional system, u l/l 2 = 1/l 1/ t. In Figs. 1 and 2 we show the excess energy u u e as a function of t for quenches to several temperatures in the interval T T < T 0. The log log plot shown in Fig. 1 confirms that for sufficient long times the excess energy indeed decays with the square root of time, as expected from the Lifshitz Allen Cahn law, a result that we write in the form u u e = 1, (3) λt where 1/ λ is to be understood as the slope at the origin of the excess energy versus 1/ t.fromfig. 2 we see that this slope increases and diverges as one approaches the transition temperature.

4 482 M.J. de Oliveira / Computer Physics Communications 180 (2009) Fig. 2. Excess energy u u e versus 1/ t for the stochastic seven-state Potts model. The linear behavior at the origin confirms the asymptotic behavior (3). Fig. 4. Excess energy u u e versus 1/ t for the stochastic seven-state Potts model. The linear behavior at the origin confirms the asymptotic behavior (3). Fig. 3. Log log plot of the excess energy per site u u e versus t for the stochastic seven-state Potts model, for several working temperatures. The asymptotic behavior is linear with a slope 1/2. In Figs. 3 and 4 we show the excess energy u u e as a function of time for quenches to working temperatures in the interval 0 T < T. The log log plot of the excess energy versus time shown in Fig. 3 shows that asymptotically the excess energy behaves also as 1/ t. Although this behavior is similar to that following from the Lifshitz Allen Cahn law, the late stage time behavior is a consequence of a thermal activated dynamics as we shall see. Again we may write the late stage time behavior in the form of Eq. (3). From Fig. 4 we see that the slope at the origin increases and diverges as one approaches zero temperature. The slope 1/ λ at the origin of the excess energy versus 1/ t can be measured directly from the plot shown in Figs. 2 and 4. The quantity λ, obtained by this technique, plotted as a function of temperature is shown in Fig. 5. As one approaches the transition temperature, from below, the slope diverges so that λ vanishes. We find numerically that λ vanishes as λ (T 0 T ) θ, (4) where θ = 1, that is, λ decreases linearly with temperature. A similar behavior, that is, the vanishing of λ at the transition point, has Fig. 5. Quantity λ versus temperature T as obtained by fitting Eq. (3) to the data points of the late stage time behavior (squares) and by fitting Eq. (8) to the data points and using Eq. (9) (circles). The dashed straight line is a fit to the data points around the transition temperature T 0 = been observed also for the two-dimensional Ising model (or twostate Potts model) by Sicilia et al. [25]. However,theyhavefound for the two-dimensional Ising model an exponent θ = Notice that in this case the transition is second order distinct from the first order observed in the seven-state Potts model. As one approaches zero temperature the quantity λ also vanishes. We observe however that the vanishing is very rapid suggesting a very singular behavior. We will see that the behavior of λ versus T follows an Arrhenius law, reflecting the thermal activated dynamics occurring at low temperatures. We remark that for the two-dimensional Ising model the quantity λ does not vanish when T 0 but approaches a nonzero constant [25]. 4. Thermal activated dynamics In the interval T T < T 0, the simple coarsening is the dominant mechanism that drives the system toward equilibrium even in the late stage of time relaxation. If the temperature is low enough, in the present case smaller than T, the simple coarsening is dom-

M.J. de Oliveira / Computer Physics Communications 180 (2009) 480 484 483 to a T defect. The generation of such a lateral dent costs an energy ε and therefore occurs with a probability p = e ε/kt.

After the formation of a lateral dent, the line joining the T defects moves without cost of energy until a dent is annihilated as shown in Fig. 7.

5 M.J. de Oliveira / Computer Physics Communications 180 (2009) to a T defect. The generation of such a lateral dent costs an energy ε and therefore occurs with a probability p = e ε/kt.the formation of an isolated dent needs twice as more energy and is negligible at low enough temperatures when compared to the formation of a lateral dent. After the formation of a lateral dent, the line joining the T defects moves without cost of energy until a dent is annihilated as shown in Fig. 7. A bounded pair of T defects performs a random walk with a jump probability p = e ε/kt, the probability of forming a lateral dent. From this random walk it follows that the average size l of a domain increases as l pt resulting in the following asymptotic behavior of the excess energy u u e 1 pt. (5) For temperatures in the interval 0 < T < T, we expect to observe the result (5) and indeed this can be inferred from Figs. 3 and 4. Comparison of Eqs. (3) and (5) gives the following result λ p so that λ depends on temperature as λ = Ae ε/kt, (6) Fig. 6. A trapped state obtained quenching from infinite to zero temperature of the seven-state Potts model defined on a square lattice with periodic boundary conditions. inant only on the first stage of relaxation. The straightening of the interface drives the system toward configurations with local minimum of energy of the type shown in Fig. 6 and not to the equilibrium state. This glassy state is characterized by the presence of a large number of T defects that are locally stable. These defects are bounded in pairs joined by a line separating two domains. In the late stage of relaxation, the simple coarsening plays no role, and another mechanism takes place, namely the thermal activated dynamics which acts in such a way as to annihilate the bounded pairs of T defects. This dynamics eventually drives the system toward the equilibrium state if the temperature is nonzero. At zero temperature, the T defects cannot disappear and the system becomes trapped into a glassy state of the type shown in Fig. 6. Atnonzerobutlowtemperatures,aboundedpairof T defects performs a random walk and is eventually annihilated when meeting with another bounded pair of T defects which is also performing a random walk as illustrated in Fig. 7. The random walk of a bounded T defects begins by the formation of a dent next where A may depend on temperature. The basic mechanism of forming a dent corresponds to a escape from a metastable estate with a barrier of energy ε. The probability of leaving the metastable state thus follows an Arrhenius law. We therefore may permit the coefficient A to depend on temperature. We assume that A depends on T as A 1/T. In order to check the result (6) we have plotted the quantity λt versus 1/T, shown in Fig. 8. At zero working temperature the energy does not approach the ground state (zero) energy but approaches a finite value u 0 as can be seen in Figs. 3 and 4. This behavior has been found before [9, 12 14] and is identified as the energy per site of the glassy state of the type shown in Fig. 6. It has also been found [9,12 14] that the energy u approaches this value according to u u 0 1 t, (7) valid for a quench to zero temperature. For a nonzero temperature we assume the following interpolation formula for the time dependence of the excess energy u u e = b λco t + b λac t + 1, (8) where λ ac = 0 and b = u 0 u e when T = 0, so that 1 λ = b λco + b λac. (9) Fig. 7. A bounded pair of T defects performing one step of a random walk (upper panel) and being annihilated when meeting with another bounded pair (lower panel). In this process the energy decreases. The values on top of the arrows indicate the change in energy.

6 484 M.J. de Oliveira / Computer Physics Communications 180 (2009) Eq. (3). Notice that while λ ac vanishes as T 0, the quantity λ co approaches a nonzero value. We remark that at low temperatures the second term in Eq. (9) dominates over the first so that in this regime λ ac = λ which in turn depends on temperature according to Eq. (6). 5. Conclusion Fig. 8. Quantity λ as a function of T as obtained by fitting Eq. (3) to the data points of the late stage time behavior (squares) and by fitting Eq. (8) to the data points and using Eq. (9) (circles).theslopeofthedashedlineequals 1. We have studied the relaxation of the seven-state stochastic Potts model after a quench to temperatures below the first-order transition temperature. We distinguish two regimes according to whether the quench temperature is above or below T When the temperature is above T, the relaxation is dominated by a simple coarsening dynamics and the energy decreases as 1/ t, in accordance to the Lifshitz Allen Cahn law. For temperatures below T, this dynamics is dominant only on the first stage of the relaxation. The straightening of the interface by simple coarsening drives the system toward a glassy state with local minimum of energy. The escaping from this metastable glassy state is realized by a thermal activated dynamics which dominates the late stage of the relaxation and eventually drives the system toward equilibrium. The asymptotic time behavior of the energy follows also a 1/ t law. Finally, we remark that the present scenario for phase ordering is consistent to that advanced by Ferrero and Cannas [17]. References Fig. 9. Quantities λ co and λ ac as a function of temperature. The first and the second terms in (8) may be related to the time behavior of the coarsening dynamics and thermal activated dynamics, respectively. We have fitted the data points shown in Figs. 2 and 4 to Eq. (8) from which we have obtained the two quantities λ co and λ ac, shown in Fig. 9 as a function of temperature, and the quantity λ by means of the result (9). Thevalues of λ so obtained are ploted in Figs. 5 and 8 together with those obtained by fitting the last stage time behavior of relaxation with [1]W.Kauzmann,Chem.Rev.43(1948)219. [2] P.G. Debenedetti, Metastable Liquids, Princeton University Press, Princeton, [3] P.G. Debenedetti, F.H. Stillinger, Nature 410 (2001) 259. [4] P.W. Anderson, in: R. Balian, R. Maynard, G. Toulouse (Eds.), Ill-Condensed Matter, North-Holland, Amsterdam, [5] J. Viñals, M. Grant, Phys. Rev. B 36 (1987) [6] A. Lipowski, D. Johnston, Phys. Rev. E 61 (2000) [7] A. Lipowski, D. Johnston, D. Espriu, Phys. Rev. E 62 (2000) [8] M.R. Swift, H. Bokil, R.D.M. Travasso, A.J. Bray, Phys. Rev. B 62 (2000) [9] M.J. de Oliveira, A. Petri, Phil. Mag. 82 (2002) 617. [10] A. Cavagna, I. Giardina, T.S. Grigera, J. Chem. Phys. 118 (2003) [11] A. Cavagna, I. Giardina, T.S. Grigera, Eurohys. Lett. 61 (2003) 74. [12] A. Petri, Braz. J. Phys. 33 (2003) 521. [13] M.J. de Oliveira, A. Petri, T. Tomé, Physica A 342 (2004) 97. [14] M.J. de Oliveira, A. Petri, T. Tomé, Europhys. Lett. 65 (2004) 20. [15] M. Ibáñez de Berganza, V. Loreto, A. Petri, Phil. Mag. 87 (2007) 779. [16] M. Ibáñez de Berganza, E.E. Ferrero, S.A. Cannas, V. Loreto, A. Petri, Eur. Phys. J. (Special Topics) 143 (2007) 273. [17] E.E. Ferrero, S.A. Cannas, Phys. Rev. E 76 (2007) [18] I. Lifshitz, Sov. Phys. JETP 15 (1962) 939. [19] S.M. Allen, J.W. Cahn, Acta Metall. 27 (1979) [20] P.S. Sahni, D.J. Srolovitz, G.S. Crest, M.P. Anderson, S.A. Safran, Phys. Rev. B 28 (1983) [21] J.S. Langer, in: C. Godrèche (Ed.), Solids Far from Equilibrium, Cambridge University Press, Cambridge, 1992, p [22] A.J. Bray, Adv. Phys. 43 (1994) 357. [23] R.B. Potts, Proc. Camb. Phil. Soc. 48 (1952) 106. [24] F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235. [25] A. Sicilia, J.J. Arenzon, A.J. Bray, L. Cugliandolo, Phys. Rev. E 76 (2007)

Quasi-Stationary Simulation: the Subcritical Contact Process

Brazilian Journal of Physics, vol. 36, no. 3A, September, 6 685 Quasi-Stationary Simulation: the Subcritical Contact Process Marcelo Martins de Oliveira and Ronald Dickman Departamento de Física, ICEx,