online). (a) The transition probability $\tau=\tau(1|h)$, Eq.  as a function of the local field $h$ for three values of the coupling constant $J$. (b) Graph of the magnetization $m'$ at time $t+1$ as a function of the magnetization $m$ at time $t$, Eq. , with some iterates for $J=-7.5$. ](PI00-o1-k20-J-10p00-W15p0 "fig:"){width="0.45\columnwidth"}   ![\[fig:tau\] (Color online). (a) The transition probability $\tau=\tau(1|h)$, Eq.  as a function of the local field $h$ for three values of the coupling constant $J$. (b) Graph of the magnetization $m'$ at time $t+1$ as a function of the magnetization $m$ at time $t$, Eq. , with some iterates for $J=-7.5$. ](m-m-I00-o1-k20-J-7p50 "fig:"){width="0.45\columnwidth"}
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The parallel version of the linear ($W=0$) Ising model does not show many differences with respect to the standard serial one [@Derrida]. The observables that depend only on single-site properties take the same values in parallel or sequential dynamics [@NewmannDerrida], although differences arise for two-site correlations [@SS-meta]. In general the resulting dynamics is no more reversible with respect to the Gibbs measure induced by any Hamiltonian [@Cirillo].

In the following, unless otherwise specified, we always use $W=15$ and $K=20$.

Mean-field approximation {#sec:meanfield}
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The mean-field approximation for the magnetization $m$ of the fully parallel 