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Well known examples of far from equilibrium systems are provided by phase separating systems, initially prepared in a state of equilibrium, and render unsable by modifying a control parameter such as temperature, pressure or magnetic field. To restore stability they evolve towards a different equilibrium state determined by the final value of the controlling field. Such evolution can be very slow and is often characterized by nonuniform, complex structures in both space and time. The phase field model has been introduced in the literature in order to understand kinetic ordering phenomena. It describes the dynamics of an ordering non-conserved field φ (e.g. solid-liquid order parameter) coupled to a conserved field (e.g. thermal field).

The model can be cast in the form of coupled differential equations for a nonconserved order parameter interacting with the time- dependent conserved field. After a rapid initial evolution one can observes an intermediate stage in which the growth is curvature driven and an asymptotic regime during which diffusion-limited behaviour is seen. The model provides a theoretical framework to study the relaxation dynamics of the order parameter, associated with the presence of a liquid or a solid, is coupled to the diffusion of heat released during the change of state.

An example is the growth of a solid nucleus from its undercooled melt. In collaboration with A. Crisanti and U.Marini Bettolo, we studied the kinetics of an initially undercooled solid-liquid melt by means of a spherical version of the Phase Field model. After obtaining the rules governing the evolution process by means of analytical arguments, we present a discussion of the asymptotic time-dependent solutions.

The full solutions of the exact self-consistent equations for the model were also obtained and compared with computer simulation results. In addition, in order to check the validity of the model we compare its prediction with those of the standard phase field model and found reasonable agreement. Interestingly, we found that the system relaxes towards a mixed phase, depending on the average value of the conserved field, i.e. on the initial condition. Such a phase is characterized by large fluctuations of the φ field.