Short-time dynamic in the Majority vote model: The ordered and disordered initial cases Francisco Sastre 1 $^{1}$Departamento de Ingenier´ıa F´ısica, Divisi´ on de Ciencias e Ingenier´ıa, Campus Le´ on de la Universidad de Guanajuato, AP E-143, CP 37150, Le´ on, Guanajuato, M´ exico ∗ This work presents short-time Monte Carlo simulations for the two dimensional Majority-vote model starting from ordered and disordered states. It has been found that there are two pseudo- critical points, each one within the error-bar range of previous reported values performed using fourth order cumulant crossing method. The results show that the short-time dynamic for this model has a dependence on the initial conditions. Based on this dependence a method is proposed for the evaluation of the pseudo critical points and the extraction of the dynamical critical exponent z and the static critical exponent β/ν for this model. PACS numbers: 64.60.Ht,75.40.-s,05.70.Ln Keywords: reference [9] for a review of these results). Concerning the critical dynamic in systems without detailed balance, there are some works that evaluate the critical dynamic exponent z [11, 12] and the fluctuation-dissipation ra- tio X ∞ [13] for the Majority model, but always starting from a disordered state. As expected the results were compatibles with the Ising ones. Given all these results one should expect that the basic assumption for the dy- namic relaxation of the k-th moment of the order param- eter starting from an completely ordered state will be the same as in the Ising model, but this has not been proved yet. The aims of this work are: a) to evaluate the critical point using short time dynamic starting from ordered and disordered initial conditions and b) evaluate the dynamic critical exponent z and the static critical exponent β/ν . This will test if the Grinstein prediction holds for the short time dynamic in the majority vote model. Critical phenomena in equilibrium statistical systems is one of the most important topics in physics. Much of the attention has been focused on the universality of the critical exponents, with several universality classes al- ready characterized in equilibrium systems. On the other hand the critical behavior of non-equilibrium statistical systems has been receiving a lot of attention in recent years, but the characterization of the different universal- ity classes is far from be complete. One of the simplest non-equilibrium models is the two dimensional majority vote model, an Ising-like system (up-down symmetry and spin-flip dynamic) with a continuous order-disorder tran- sition with the same critical exponents that the two di- mensional Ising model [1–3], as expected from the predic- tion of Grinstein et al [4]: every spin system with spin-flip dynamic and up-down symmetry falls in the Ising model universality class. However, there is some controversy about the universality class for higher dimensions. A re- cent work claims [5] that the upper critical dimension for the majority vote model is 6 instead of 4, based on nu- merical calculations. Another discrepancy in the critical exponents have been found in simulations on non-regular lattices [6, 7]. It must be mentioned that all of the results mentioned above were performed using standard ”Monte Carlo” simulations and Finite Size Scaling approaches for the evaluation of the static critical exponents. On the other hand the time evolution can gives important information about the universality of a given system. It has been shown by Janssen et al. [8] that when systems with relaxation dynamics are quenched from high tem- peratures to the critical temperature there is a short crit- ical universal behavior. Numerical simulations have con- firmed this behavior in the Ising and the Pott models (see here H$_{i}$ is the local field ∑ $_{nn}$σ$_{j}$ = , 0 , ± 2 , ± 4 produced by its nearest neighbors and f is a function with up down symmetry that depends on two control parameters f (0) = 0, f (2) = − f ( − 2) = x and f (4) = − f (4) = y . The parameters x and y can be associated with inter- face and bulk temperatures respectively [15] using the relations In the Majority vote model each lattice site has a spin whose values are σ = ± 1 and its dynamic can be grouped by the spin flip rule The Majority model is obtained setting x = y ( β$_{4}$ < β$_{2}$ )and the critical point is at x$_{c}$ = 0 . 850(2) [2, 3]. II. MODELS AND DEFINITIONS arXiv:1001.0163v1 [cond-mat.stat-mech] 31 Dec 2009 ∗ Electronic address: sastre@fisica.ugto.mx x = tanh 2 β$_{2}$, y = tanh 4 β$_{4}$. (2) W$_{i}$ = 1 2 [1 − σ$_{i}$f ( H$_{i}$ )] , (1) I. INTRODUCTION