content_image/420477.jpg content_image/420479.jpg content_image/420476.jpg content_image/420478.jpg content_image/420480.jpg content_image/420469.jpg
The equilibrium case can be obtained along the line y = 2 x/ (1 + x $^{2}$) (Glauber dynamic), where the temper- atures are equal ( β$_{2}$ = β$_{4}$ = β ) and the critical point is β$_{c}$ = 1 2 $_{log(1 +}$√$_{2).}$ 

The order parameter is the standard for Ising-like sys- tems, defined by 

where N = L 2 is the total number of lattice sites ( L is the lateral size. 

Starting from a disordered state the dynamic for the k - th moment of the order parameter was derived by Janssen et al [8], the mathematical expression is given by 

<img src='content_image/420477.jpg'>

here τ = ( T − T$_{c}$ ) /T$_{c}$ is the reduced temperature, or the reduced control parameter for non-equilibrium systems, t is the dynamic time variable, b is the re-scaling factor, β and ν are static critical exponents, z is the dynamic critical exponent, x$_{0}$ is the scaling dimension of the ini- tial (small) order parameter m$_{0}$ (see [9] and reference therein). 

At the critical point for sufficiently small m$_{0}$ and large systems ( L → ∞ ) the order parameter follows a power law dynamic 

where θ is defined by ( x$_{0}$ − β/ν ) /z . One must remark that there is a strong dependence on the initial value of the order parameter. This dependence does not affect the power law behavior at the critical point, what is affected is the exponent θ that tends to the real value in the limit m$_{0}$ → 0 For the dynamic starting, from the ordered state we 

have the assumption that the scaling dynamic form is given by 

<img src='content_image/420479.jpg'>

again it can be obtained the scaling form at the critical point taking the limit L → ∞ . 

This has been proved in equilibrium systems, like the Ising or the Potts model. 

For the evaluation of the critical point it has been used the fact that theoretically the order parameter evolves as a power law at the critical point If it is evaluated the difference between a power law and the time evolution for different values of the control parameter x we will expect a minimum in those differences at the critical point. 

The simulations were performed choosing the lattice site randomly starting from both, a completely ordered state ( m = 1 . 0) and a carefully prepared disordered state 

<img src='content_image/420476.jpg'>

<img src='content_image/420478.jpg'>

<img src='content_image/420480.jpg'>

<img src='content_image/420469.jpg'>

with small magnetization values m$_{0}$ < 0 . 1. The or- der parameter is evaluated as a function of the time t , with Δ t = 1 corresponding to a Monte Carlo time step (MCTS). In order to avoid finite size effects we used lat- tices with lateral size L = 2 9 ( N = L $^{2}$). The average were taken with at least 10 5 independent simulations and 250 and 1000 MCTS for the evaluation of the critical point and dynamical exponents respectively. 

Given that the power law behavior of the order pa- rameter as function of t (decay from an ordered state, increase for a disordered one) one can performed simula- tions for different values of x around the expected critical point. In Fig. 1 we can observe that above and below a certain value of x we have dynamic that differ from a power law (illustrated by the dashed line). 

From here one must define the criterion that will mea- sure which is the best power law curve, in this work it was used the measurement of the χ 2 for each curve with respect to a power law behavior. I must remark that pre- vious results for equilibrium systems give the same result for the critical point for both initial states. However, two different values were found for the majority vote model, see Fig. 2. 

The critical point values obtained were x$_{c}$ = 0 . 85007(6) and x$_{c}$ = 0 . 85147(2) for the order and disorder state respectively, both results are in perfect agreement with the obtained in the static case (references [1–3]) and are quite close. 

We have a clearly dependence on the initial condition in short time dynamic of this model, which is not the 

## III. RESULTS 

