Coarsening in the Potts model: out-of-equilibrium geometric properties

Transcription

1 Journal of Pysics: Conference Series Coarsening in te Potts model: out-of-equilibrium geometric properties To cite tis article: M P O Loureiro et al 2010 J. Pys.: Conf. Ser View te article online for updates and enancements. Related content - Geometric properties of two-dimensional coarsening wit weak disorder A. Sicilia, J. J. Arenzon, A. J. Bray et al. - Critical percolation in te dynamics of te 2D ferromagnetic Ising model Tibault Blancard, Leticia F Cugliandolo, Marco Picco et al. - A morpological study of cluster dynamics between critical points Tibault Blancard, Leticia F Cugliandolo and Marco Picco Tis content was downloaded from IP address on 31/01/2018 at 23:10

2 XI Latin American Worksop on Nonlinear Penomena Journal of Pysics: Conference Series 246 (2010) IOP Publising Coarsening in te Potts model: out-of-equilibrium geometric properties M P O Loureiro 1, J J Arenzon 1, L F Cugliandolo 2 and A Sicilia 3 1 Instituto de Física and INCT Sistemas Complexos, Universidade Federal do Rio Grande do Sul, CP 15051, Porto Alegre RS, Brazil 2 Université Pierre et Marie Curie Paris VI, LPTHE UMR 7589, 4 Place Jussieu, Paris Cedex 05, France 3 Department of Cemistry, University of Cambridge, Lensfield Road, CB2 1EW, Cambridge, United Kingdom arenzon@if.ufrgs.br Abstract. Geometric properties of polymixtures after a sudden quenc in temperature are studied troug te q-states Potts model on a square lattice, and teir evolution wit Monte Carlo simulations wit non-conserved order parameter. We analyse te distribution of ull enclosed areas for different initial conditions and compare wit exact and numerical findings for te q = 2 (Ising) case. 1. Introduction Curvature-driven ordering processes, in non conserved scalar order parameter systems, underlie many interesting cellular growt processes and are of bot teoretical and tecnological importance, applications including foams [1], cellular tissues [2], superconductors [3], magnetic domains [4, 5], liquid crystals [6], adsorbed atoms on surfaces, etc. Tese systems are spatially divided in many coexisting domains, in wic one of te q possible spin orientations dominates, separated by interfaces wose local velocity is ruled by te Allen-Can equation, tus being proportional to te local curvature, v = (λ/2π)κ, were λ is a temperature and q-dependent dimensional constant related wit te surface tension and mobility of a domain wall and κ is te local curvature. Te sign is suc tat te domain wall curvature is diminised along te evolution. In d = 2 te time dependence of te area contained witin any finite domain interface (te ull) on a flat surface is obtained by integrating te velocity around te ull and using te Gauss-Bonnet teorem: da n dt = λ, q = 2 λ 6 (n 6), q > 2 (1) were n is te number of sides (or, equivalently, vertices). At eac one of tese vertices te tangent vector to te surface as a turning angle, tat are absent for q = 2 (since in tis case eac domain is surrounded by a single neigbour). Notice tat in bot cases, te rate is size independent [7,8], wile for q = 2 it does not depend on te number of sides. For q = 2, te growt law is omogeneous and all ull enclosed areas decrease wit te same rate, wat as led us to exactly obtain [8] te number of ull-enclosed areas per unit system c 2010 IOP Publising Ltd 1

3 XI Latin American Worksop on Nonlinear Penomena Journal of Pysics: Conference Series 246 (2010) IOP Publising area, in te interval (A,A + da). For example, if one takes as initial states (t = 0), equilibrium configurations at T c, wose distribution is exactly known for q = 2 [9], n (A,0) = c (2) /A2 wit c (2) = 8π 3, te distribution at any t > 0 is n (A,t) = c (2) (A + λ t) 2, (2) tat compares extremely well wit simulation data for te Ising model on a square lattice [8, 10, 11]. Moreover, te above equation scales as n(a,t) = t 2 f(a/t), in accordance wit te scaling ypotesis. Altoug tis is usually only a ypotesized, but well-establised feature, for te q = 2 model it can be obtained from first principles [8]. For a quenc from infinite temperature, on te oter and, te initial distribution corresponds to te one of te critical random continuous percolation [8, 10]. Tis observation is an essential ingredient to understand te fact tat, and obtain te probability wit wic, te system attains a striped frozen state at zero temperature (see [12] and references terein). Interestingly, on a square lattice, te initial state does not correspond to te random percolation critical point but te coarsening evolution gets very close to it after one or two Monte Carlo steps. Terefore, altoug te analytical results of Ref. [8] were obtained wit a continuous description, a priori not guaranteed to apply on a lattice, tey do describe te coarsening dynamics of te discrete Ising model wit remarkable accuracy. For q > 2, weter a cell grows, srinks or remains wit constant area depends on its number of sides being, respectively, larger tan, smaller tan or equal to 6, wat is known as te von Neumann s law. Terefore, one cannot write a simple relation to link te area distribution at time t to te one at te initial time t = 0 and te distribution migt get scrambled in a nontrivial way during te coarsening process (for example, wen a domain disappears, te number of sides of te neigboring domains canges, along wit teir growt rate) [13]. Noneteless, te system again evolves to a scaling state in wic te domain morpology is statistically te same at all times wen lengts are measured in units of R(t), a single caracteristic growing lengt scale, in accordance wit te scaling ypotesis. In tis paper we briefly compare te cases q = 2 and q > 2, in particular wen te initial state was equilibrated at T = T c. We first present te d = 1 case and ten te muc ricer two dimensional one, reviewing some of te results already publised in Refs. [8, 13]. Besides completeness, it is interesting to compare te uni and two dimensional cases as tey strongly differ. 2. Coarsening in d = 1 Te Hamiltonian in tis case is H = J i δ si,s i+1, (3) were s i = 0,...,q 1 and J > 0. We measure te domain lengt distribution at time t, n(l,t) (1 l L) after quencing te system from T 0 to te final, working temperature, T = 0. Tere is no finite static critical temperature and te system orders ferromagnetically only at T = 0. Te completely disordered initial condition starts evolving in a coarsening regime in wic regions of finite lengt order. Differently from te iger dimensional case were domain growt is driven by interfacial tension, in d = 1 coarsening is driven by te diffusion of domain walls and anniilation wen tey meet. At tis point wen a domain disappears, te two neigboring domains always coalesce if q = 2. Tis may not be te case for q > 2 and becomes less probable as q increases. Tus, te average lengt size of te domains, l, grows faster te smaller te value of q. Indeed, te growt law is l = q πt/(q 1) [14, 15], were te t 1/2 beaviour is expected from te diffusive interface motion. 2

4 XI Latin American Worksop on Nonlinear Penomena Journal of Pysics: Conference Series 246 (2010) IOP Publising 10 0 l n(l/ l ) q = q = q = 8 q = l/ l Figure 1. Cluster lengt distribution for te unidimensional Potts model wit q = 2, 3 and 8 versus te rescaled lengt, l/ l. Te qualitative beaviour is te same, linear for small lengts and exponential at large ones. Te (red) lines are analytical predictions of Derrida and Zeitak [15], eq. (5) wit (A q,b q ) equal to (1.306,0.597), (1.500,0.963) and (1.993,1.876) for q = 2, 3 and 8, respectively. Te system is 10 5 large and averages were taken over 1000 samples. Data corresponds to times from t = 2 2 to 2 9. During te coarsening regime, te distribution of domain sizes obeys te scaling beaviour n(l,t) = l(t) 1 f ( ) l. (4) l(t) At zero working temperature te universal function f(x) is given by [14 17] πqx x 1, f(x) = 2(q 1) exp( A q x + B q ) x 1, (5) were te constants are known exactly [15]. In Fig. 1 we sow, for q = 2, 3 and 8, tat te above equations fully agree wit numerical simulations (te q = 2 case was previously presented in Ref. [10]). In all cases te initial state as null correlations, all spins being fully uncorrelated, wat seems to be te key ingredient for te exponential tail in te distribution. As we will see in te next section, te q = 2 case, starting from infinite temperature, is an exception to suc beaviour. 3. Coarsening in d = 2 Te detailed evolution of te system during te coarsening dynamics depends on te correlations already present in te initial state. Below we sow te beaviour wen suc correlations are eiter absent (T 0 ) or long-ranged (T 0 = T c for q 4). In te latter case, since te termodynamic transition also corresponds to a percolation transition in 2d, te initial state already presents one spanning cluster at t = 0. On te oter and, for sort-range initial correlations, suc spanning domains are eiter formed very fast (e.g. in te case q = 2 wit T 0 ), or not formed at all. We sould empasize te exceptionality of te q = 2 case: aving te igest concentration of a single species at T 0, te proximity wit te 3

5 XI Latin American Worksop on Nonlinear Penomena Journal of Pysics: Conference Series 246 (2010) IOP Publising percolation critical point strongly affects te system s evolution [8]. Indeed, te ull-enclosed area distribution (geometric domains present an analogous beaviour [10]) becomes power-law in a couple of steps, and te effective initial condition is given by n (A,0) = 2c (2) /A2 [9]. Powerlaws are also te initial distributions for T 0 = T c for q 4 and generally given by (te prefactor is different from te previous case) n (A,0) = (q 1)c(q) A 2. (6) Notice tat, unless for q = 2, te value of c (q) is not known exactly, altoug tey are all close to eac oter. For 2 q 4, te Potts model presents a continuous transition, and one could imagine tat teir non equilibrium coarsening penomenology would be similar as well. Indeed, te equal time correlation function seems to sare te same universal scaling function and te related correlation lengt grows wit te same power of time, t 1/2. Tere are, owever, fundamental differences. Differently from te q = 2 case, tat presents a percolating domain wit probability almost one as early as t = 2 after te quenc [8, 10], te q > 2, T 0 initial condition is sufficiently far from critical percolation tat te system remains, at least in te time window of our simulations, distant from te percolation tresold (in spite of te largest domain steadily, but slowly, increasing wit time). In tis regard, te q = 2 case is te exception, wile te q = 3 and 4 are similar. As te system evolves after te quenc, te distribution keeps memory of te initial state, tat corresponds to random percolation wit occupation probability p = 1/q. And, by not getting close to a critical point, te distributions do not become critical and, as a consequence, do not develop a power law tail, as illustrated in Fig. 2 for q = 3 and T 0. Tere is, owever, an A 2 envelope tat is a direct consequence of dynamical scaling, present also for oter values of q [13]. n(a, t) A 2 t = A Figure 2. Hull-enclosed area distribution at several times (given in te key) after a quenc from equilibrium at T 0 to T f = T c /2 in te q = 3 case. Analogous distributions are obtained for q > 4 and T 0 (not sown). Te declivity of te envelope is 2 as a consequence of te scaling obeyed by te distribution. Figure from Ref. [13]. Te collapsed distribution of ull-enclosed areas after a quenc from T 0 = T c to T f = T c /2 for q = 3 is sown in Fig. 3. Altoug for q = 2 tis distribution conserves its form during 4

6 XI Latin American Worksop on Nonlinear Penomena Journal of Pysics: Conference Series 246 (2010) IOP Publising te coarsening regime due to an overall advection to te rigt as domains increase in size, for q > 2 tere is a dependence on te number of sides in von Neumann s equation, some domains increase wile oter decrease in size, and tus tere is no obvious reason for te distribution at t > 0 to keep its initial power-law form, eq. 6. Remarkably, te general beavior is similar to te q = 2 case, in wic te collapse of curves for different times onto a single universal function demonstrates te existence of a single lengt scale tat, moreover, follows te Allen-Can growt law, R(t) t 1/2. We can make a mean-field-like approximation and replace te number of sides in te von Neumann equation (1) by a constant mean, n n. Using Eq. (6) and te results in Refs. [8,10], te ull-enclosed area distribution for q 4 becomes n (A,t) = (q 1)c(q) ( A + λ (q) t ) 2, (7) tat fits very accurately te data using λ (3) 1.4. Te deviations present at small values of A/t are due to termal fluctuations tat are visible in te inset (see [10] for details). In order to test te above approximation, we measured te average cange in area, da/dt, tat is, te number of spins included or excluded in tose domains tat survived during a given time interval. Te above rate, from von Neumann s equation, depends on n 6 (bot in absolute value and sign), and is different for domains wit different numbers of neigbours (except for q = 2 were it is constant). However, upon average, tere is an effective, constant λ eff. Similarly, only tose tat ave λ eff 0 present a power law distribution. Tus, tere seems to be a net difference between cases tat give a power law distribution function from te ones tat do not. t 2 n(a, t) t = A n(a, t) A/t Figure 3. Collapsed ull-enclosed area distributions at several times after a quenc from equilibrium at T 0 = T c to T f = T c /2, for q = 3. Te line is Eq. (7) wit λ (3) 1.4. Te points at A/t 1 tat deviate from te scaling function are due to termal fluctuations. Tese fluctuations ave been independently measured and are depicted as a continuous black line in te inset (notice tat teir distribution is time independent). Figure from Ref. [13]. 4. Conclusions We studied te Potts model during te coarsening dynamics after a sudden quenc in temperature, empasizing te cases q = 2 and 3 tat present a continuous transition. Altoug 5

7 XI Latin American Worksop on Nonlinear Penomena Journal of Pysics: Conference Series 246 (2010) IOP Publising te teory developed in Refs. [8,10] for te Ising model (q = 2) cannot be easily extended to q > 2, our numerical results sow te existence of similarities between bot cases. Besides te equilibrium continuous transition, tere is a dynamical lengt scale tat grows wit t 1/2 after a sudden quenc in temperature and te same universal scaling function for te rescaled correlation function in bot cases. Noneteless, very different are te distributions of dynamic ull-enclosed areas (and geometric domains) evolved from initial states wit zero correlation lengts, as tose obtained in equilibrium at T 0. One expects tat te scaling functions of te dynamic distributions will be reminiscent of te disordered state at te initial temperature, wit an exponential tail. Tis is indeed wat appens in d = 1 and, in d = 2, for q > 2. Te remarkable exception is q = 2 (in d = 2): te proximity from te percolation critical point, and te subsequent flow to it after te temperature quenc, develops a power-law tail in te distribution. For q = 3 te initial state is no longer close to any critical point and tis does not occur. A furter surprise occurs for te distributions after a quenc from T 0 = T c. Despite von Neumann s law sowing tat differently sided domains could eiter grow or srink, scrambling te distribution, it still sows a power law, in analogy wit te Ising case but witout a complete explanation up to now besides a simple mean-field-like argument. Alongside, many questions are still unanswered for tis long studied, classic problem and surely many surprises are still awaiting. Acknowledgments Work partially supported by CAPES/Cofecub grant 667/10. JJA and MPOL are partially supported by te Brazilian agency CNPq. References [1] Glazier J A, Anderson M P and Grest G S 1990 Pil. Mag. B [2] Mombac J C M, de Almeida R M C and Iglesias J R 1993 Pys. Rev. E [3] Prozorov R, Fidler A F, Hoberg J R and Canfield P C 2008 Nature Pysics [4] Babcock K L, Sesadri R and Westervelt R M 1990 Pys. Rev. A [5] Jagla E A 2004 Pys. Rev. E [6] Sicilia A, Arenzon J J, Dierking I, Bray A J, Cugliandolo L F, Martinez-Perdiguero J, Alonso I and Pintre I C 2008 Pys. Rev. Lett [7] Mullins W W 1956 J. Appl. Pys [8] Arenzon J J, Bray A J, Cugliandolo L F and Sicilia A 2007 Pys. Rev. Lett [9] Cardy J and Ziff R M 2003 J. Stat. Pys [10] Sicilia A, Arenzon J J, Bray A J and Cugliandolo L F 2007 Pys. Rev. E [11] Sicilia A, Sarrazin Y, Arenzon J J, Bray A J and Cugliandolo L F 2009 Pys. Rev. E [12] Barros K, Krapivsky P L and Redner S 2009 Pys. Rev. E (R) [13] Loureiro M P O, Arenzon J J, Cugliandolo L F and Sicilia A 2010 Pys. Rev. E [14] Amar J G and Family F 1990 Pys. Rev. A [15] Derrida B and Zeitak R 1996 Pys. Rev. E [16] Alemany P A and ben Avraam D 1995 Pys. Lett. A [17] Krapivsky P L and Ben-Naim E 1997 Pys. Rev. E