Colloquium Schedule for 2000-1 |

The Mathematics of Phase Transitions


Professor Peter Bates
Brigham Young University


(Candidate for Mathematics Chair at MSU)




Starting at the atomic level, a free energy is postulated along the lines of Van der Waals'original formulation. If, consistent with thermodynamic principles, free energy is decreasedas a gradient flow along trajectories of the order parameter field, one may derive models forthe macroscopic evolution of phase states. If the inter-atomic interaction is taken to be short-ranged, the evolution equation is of reaction-diffusion type. If the interactionenergy is long-ranged, then a nonlocal dissipative equation results. Other issues such as conservation of species lead to interesting modifications in the field equations. If one isonly interested in the evolution of phase boundaries, then asymptotically correct equationsfor the evolution of such surfaces can be derived. The qualitative behavior of solutionsto these equations will be compared.