Phase transitions in solids: from atomistic waves to a continuum picture Seminar

The equations of elasticity in one space dimension, $u_{tt} =
\sigma(u_x)_x$, become ill-posed if the potential energy density is
nonconvex, or equivalently if $\sigma$ is non-monotone. This
complication necessarily arises in the theory of so-called martensitic
phase transitions, which are diffusionless solid-solid transformations
where several stable phases can coexist.

Different regularisations of this ill-posed problem have been proposed;
we will here focus on so-called kinetic relations, which relate the
velocity of a moving interface to a driving force. Phenomenological
kinetic relations have been proposed, but a natural question is whether
they can in simple situations be derived from first principles, namely
atomistic considerations.

To investigate this question, we study the simplest one-dimensional
chain model of martensitic materials, where neighbouring atoms are
coupled by a spring with bi-quadratic potential. We present existence
results for travelling waves and discuss non-uniqueness of microscopic
solutions. This non-uniqueness will be discussed in light of the
macroscopic kinetic relation.