A Study of a Hamiltonian Model for Phase Transformations Including Microkinetic Energy

How can a system in a macroscopically stable state explore energetically more favorable states, which are far away from the current equilibrium state? Based on continuum mechanical considerations we derive a Boussinesq-type equation ρü = ∂xσ(∂xu) + β∂ 2 x ü, x ∈ (0, 1), β > 0, which models the dynamics of martensitic phase transformations. Here ρ > 0 is the mass density, β∂ x ü is a regularization term which models the inertial forces of oscillations within a representative volume of length √ β and σ is a nonmonotone stress-strain relation. The solutions of the system, which we refer to as microkinetically regularized wave equation, exhibit strong oscillations after times of order √ β, thermalization can be confirmed. That means that macroscopic fluctuations of the solutions decay at the benefit of microscopic fluctuations. The mathematical analysis for the microkinetically regularized wave equation consists in two parts. First we present some analytical and numerical results on the propagation of phase boundaries and thermalization effects. Despite the fact that model is conservative, it exhibits the hysteretic behavior. Such a behavior is usually interpreted in macroscopic models in terms of dissipative threshold which the driving force has to overcome to ensure that the phase transformation proceeds. The threshold value depends on the amount of the transformed phase as it is observed in known experiments. Secondly we investigate the dynamics of oscillatory solutions. Our mathematical tool are Young measures, which describe the one-point statistics of the fluctuations. We present a formalism which allows us to describe the effective dynamics of rapidly fluctuating solutions. The extended system has nontrivial equilibria which are only visible when oscillatory solutions are considered. The new method enables us to derive a numerical scheme for oscillatory solutions based on particle methods.