Noise driven dynamic phase transition in a a one dimensional Ising-like model

The dynamical evolution of a recently introduced one dimensional model in \cite{biswas-sen} (henceforth referred to as model I), has been made stochastic by introducing a parameter $\beta$ such that $\beta =0$ corresponds to the Ising model and $\beta \to \mathrm{\infty }$ to the original model I. The equilibrium behaviour for any value of $\beta$ is identical: a homogeneous state. We argue, from the behaviour of the dynamical exponent $z$,that for any $\beta \neq 0$, the system belongs to the dynamical class of model I indicating a dynamic phase transition at $\beta =0$. On the other hand, the persistence probabilities in a system of $L$ spins saturate at a value $P_{sat}(\beta, L) = (\beta/L)^{\alpha}f(\beta)$,where$\alpha$remainsconstantforall$\beta \neq 0$supportingtheexistenceofthedynamicphasetransitionat$\beta =0$. The scaling function $f(\beta )$ shows a crossover behaviour with $f(\beta) = \rm{constant} $for$\beta <<1$and$f(\beta) \propto \beta^{-\alpha}$for$\beta >>1$.