We have studied the dynamical decay of the autocorrelation function of the 3D Ising model for different sizes L = 20–52 of spin cluster-cubes. The behaviour of the longest, ergodic relaxation time, τ_e, of a finite domain below the phase transition temperature T_c was mostly considered for two types of phase transition dynamics. A study of the scaling properties of τe demonstrates a negligible difference between the types of dynamics used, but a considerable difference for different boundary conditions. In contrast to the known result for periodic boundary conditions (τ_e ~ L^z exp [const( Lє^ν)²], where z and ν are the dynamical and correlation length exponents, respectively, and є = 1 – T/ T_c), the ergodic relaxation time for open boundary conditions is proportional to L^z exp [const(Lє^ν) ^2k] with coeffcient k for lattices explored in this work slightly decreasing with L in between 1.65 and 1.58. This result implies that only the lattices of sizes close to or exceeding L = 300 with open boundary conditions might have ergodic relaxation times similar to those with perodic boundary conditions.

Keywords: Ising model, classical Monte Carlo simulations, finite size scaling, ergodic relaxation time
PACS: 64.60.an, 64.60.De, 64.60.Ht

APIE ERGODINĘ RELAKSACIJOS TRUKMĘ TRIMAČIAME ISINGO MODELYJE
R. Grigalaitis^a, S. Lapinskas^a, J. Banys^a , E. E. Tornau^b
aVilniaus universiteto Fizikos fakultetas, Vilnius, Lietuva
^bFizinių ir technologijos mokslų centro Puslaidininkių fizikos institutas, Vilnius, Lithuania

Tyrėme trimačio Isingo modelio autokoreliacijos funkcijos slopimo dinamiką skirtingų dydžių (kraštinės ilgis L = 20–52) kubiniams sukinių klasteriams. Ilgiausios – ergodinės relaksacijos trukmės τ_e elgsena baigtinio dydžio domene, žemesnėse nei fazinio virsmo T_c temperatūrose, buvo tiriama dviem fazinių virsmų dinamikos būdais. Trukmės τe skaliavimo savybių analizė parodė tik nedidelius skirtumus naudojant skirtingą fazinių virsmų dinamiką, tačiau pastebėti gana akivaizdūs pokyčiai naudojant kitokias kraštines sąlygas. Skirtingai nuo žinomo rezultato, gauto naudojant periodines kraštines sąlygas τ_e ~ L^z exp [const(Lє^ν)2], ergodinė relaksacijos trukmė, kai kraštinės sąlygos yra atviros, yra proporcinga L^z exp [const( Lє^ν)^2k ], o koeficientas k mažėja nuo 1,65 iki 1,58, didėjant gardelės dydžiui nuo L = 20 iki L = 52. Čia z ir ν yra atitinkamai krizinės dinaminė ir koreliacijos ilgio eksponentės, o є = 1- T/T_c. Daroma išvada, kad ergodinės relaksacijos trukmės esantatviroms ir periodinėms kraštinėms sąlygoms bus panašios, kai gardelių kraštinių ilgiai bus ne mažesni nei L = 300.

References / Nuorodos

[1] K. Binder and E. Luijten, Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models, Phys. Rep. 344, 179–253 (2001),
http://dx.doi.org/ 10.1016/S0370-1573(00)00127-7
[2] A. Pelisseto and E. Vicari, Critical phenomena and renormalization-group theory, Phys. Rep. 368, 549–727 (2002),
http://dx.doi.org/ 10.1016/S0370-1573(02)00219-3
[3] Z.-D. Zhang, Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices, Philos. Mag. 87, 5309–5419 (2007),
http://dx.doi.org/10.1080/14786430701646325
[4] K. Binder and D.W. Heermann, Monte Carlo Simulations in Statistical Physics (Springer-Verlag, Berlin, Heidelberg, 1988),
http://dx.doi.org/10.1007/978-3-662-08854-8
[5] K. Binder, Finite size scaling analysis of Ising model block distribution functions, Z. Phys. B 43, 119–140 (1981),
http://dx.doi.org/10.1007/BF01293604
[6] K. Binder, Monte Carlo calculation of the surface tension for two- and three-dimensional lattice-gas models, Phys. Rev. A 25, 1699–1709 (1982),
http://dx.doi.org/10.1103/PhysRevA.25.1699
[7] S. Wansleben and D.P. Landau, Monte Carlo investigation of critical dynamics in three-dimensional Ising model, Phys. Rev. B 43, 6006–6014 (1991),
http://dx.doi.org/10.1103/PhysRevB.43.6006
[8] N. Ito, Non-equilibrium relaxation and interface energy of the Ising model, Physica A 196, 591–614 (1993) ,
http://dx.doi.org/ 10.1016/0378-4371(93)90036-4
[9] F.-G. Wang and C.-K. Hu, Universality in dynamic critical phenomena, Phys. Rev. E 56, 2310–2313 (1997)
http://dx.doi.org/10.1103/PhysRevE.56.2310
[10] M. Collura, Off-equilibrium relaxational dynamics with an improved Ising Hamiltonian, J. of Stat. Mech. Theor. Exp. P12036, 1–14 (2010),
http://dx.doi.org/10.1088/1742-5468/2010/12/P12036
[11] A.S. Krinitsyn, V.V. Prudnikov, and P.V. Prudnikov, Calculations of the dynamical critical exponent using the asymptotic series summation method, Theor. Math. Phys. 147, 561–575 (2006),
http://dx.doi.org/10.1007/s11232-006-0063-z
[12] W. Koch, V. Dohm, and D. Stauffer, Order-parameter relaxation times of finite three-dimensional Ising-like systems, Phys. Rev. Lett. 77, 1789–1792 (1996),
http://dx.doi.org/ 10.1103/PhysRevLett.77.1789
[13] W. Koch and V. Dohm, Finite-size effects on critical diffusion and relaxation towards metastable equilibrium, Phys. Rev. E 58, R1179–R1182 (1998),
http://dx.doi.org/10.1103/PhysRevE.58.R1179
[14] R. Grigalaitis, S. Lapinskas, J. Banys, and E.E. Tornau, Simulation of relaxation times distribution for relaxors using distribution of three-dimensional Ising-type clusters, Ferroelectrics 415 (1), 40–50 (2011),
http://dx.doi.org/ 10.1080/00150193.2011.577370
[15] M.E.J. Newman, and G.T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon, Oxford, 1999),
http://ukcatalogue.oup.com/product/9780198517979.do
[16] H. Jang, M.J. Grimson, and T.B. Woolf, Stochastic dynamics and the dynamic phase transition in thin ferromagnetic films, Phys. Rev. E 70, 047101 (2004),
http://dx.doi.org/10.1103/PhysRevE.70.047101
[17] R.J. Glauber, Time dependent statistics of the Ising model, J. Math. Phys. 4, 294–307 (1963),
http://dx.doi.org/10.1063/1.1703954
[18] A.B. Bortz, M.H. Kalos, and J.L. Lebowitz, A new algorithm for Monte Carlo simulation of Ising spin systems, J. Comput. Phys. 17, 10–18 (1975),
http://dx.doi.org/10.1016/0021-9991(75)90060-1
[19] A.M. Ferrenberg and D.P. Landau, Critical behavior of the three-dimensional Ising model: a high resolution Monte Carlo study, Phys. Rev. B 44, 5081–5091 (1991),
http://dx.doi.org/10.1103/PhysRevB.44.5081
[20] R. Häggkvist, A. Rosengren, P.H. Lundow, K. Markström, D. Andrén, and P. Kundrotas, On the Ising model for the simple cubic lattice, Adv. Phys. 56, 653–755 (2007),
http://dx.doi.org/10.1080/00018730701577548
[21] A. Miyashita and H. Takano, Dynamical nature of the phase transition of the two-dimensional kinetic Ising model, Prog. Theor. Phys. 73, 1122–1140 (1985),
http://dx.doi.org/10.1143/PTP.73.1122