critical energy of the Ising model $E_c/N = - \sqrt{2}$. [@onsager; @yang]. Moreover, in Refs. [@hans; @goles] it is compared the magnetization as a function of the internal energy of the system showing a close agreement with the numerical values.

![ \[MvsE\] Magnetization curves as a function of initial energy for three different system sizes: $N=8\times 8$, $N=16\times 16$, and $N=256\times 256$, as indicated in the inset. Each point corresponds to a different initial condition, in this case we sample different energies. As it can be noticed, there is a finite system dependence on the critical behaviour of the system, indeed the critical behavior disappear for small system sizes $N=8\times 8$, $N=16\times 16$, whereas for large system the magnetization curve reach a critical behavior. The continuous line represents the well known statistical mechanics calculation for the Ising model $M/N \approx 2^{5/16} (\sqrt{2} + E/N)^{1/8}$.](MvsE_Curve.eps){width="7cm"}

The phase space. {#PhaseSpace}
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The configurational space of all states is defined through all possible values of the state $\{x,y\}$. The resulting space is composed by the $2^{2N}$ vertices of a $2N$-dimensional hypercube. The smallest possible system corresponds to a $N=2\times 2$ lattice. In this case there are $2^{2\times 4}=2^8=256$ states and the phase space 