Phase transition versus disorder: A criterion derived from a two-dimensional dynamic ferromagnetic model

The dynamical equations for spin correlation functions for the two-dimensional Ising model are discussed for rapid-quench conditions. The temperature of the bath in which the Ising spins are immersed is postulated to be driven by the hyperbolic cooling program T(-1) = T(0) (-1) + bt (T(0) being the initial temperature, t the time, and b a constant). It is shown that for sufficiently large b the term in the dynamical equations that would normally drive the system into an ordered state would not have time to become effective so that the system freezes into a disordered state. The correlation length for the disordered state is characterized by the distance a random walker on a two-dimensional lattice would have been able to diffuse in the time required for the quench to reduce the temperature to some preassigned value.