Convex optimization on Landau Theory Models (2024)
Within the theoretical framework of phase transformation physics, one of the most studied and applied strategies to comprehend the universal behavior of many different physical systems is given by the Landau theory, which provides a phenomenological theory for the system's second order phase transformations, where the second derivative of the free energy with respect to temperature is discontinuous at the transition temperature. Furthermore, in many study cases, it is of utter importance to find global minima of the free energy density for the identification of such transition temperatures. In accordance with this problematic, due to the possible high complexity of the system's free energy polynomial, this work proposes an approach to locate its global minima by means of its convex envelope, considering the optimization problem given by the Method of Moments. The development of this approach utilizes semidefinite programming techniques to construct a self-contained algorithm that is applicable under the scope of applied mathematics, physics and mechanical engineering.