Date:	Mon, September 28, 2009
Time:	17:15
Place:	Research II Lecture Hall

Abstract: Since Fermi, Pasta and Ulam investigated the (non-)equipartitioning of energy in a chain of 64 nonlinearly coupled atoms, lattice models have often provided mathematicians with puzzling effects. In the main part of the talk, we want to address one such effect: how can conservative microscopic (lattice) models generate dissipation on the macroscopic (continuum) scale? It is well-accepted by physicists that this is possible, but we seek to gain a clean mathematical understanding. To this behalf, we consider a model problem, namely that of a moving phase boundary in a solid (the talk will include a short survey on phase transitions in solids). A well-accepted microscopic model is then that of a one-dimensional chain of atoms with nearest neighbour interaction. To describe phase transitions, the elastic potential is chosen to be nonconvex; we will consider a piecewise quadratic energy with two wells. A simple solution class describing the motion of an interface is then the class of travelling waves. A solution which explores both wells of the energy will have an interface, moving with the speed of the wave. We show that for suitable fixed subsonic velocities, there is a family of "heteroclinic" travelling waves (heteroclinic means here that they connect both wells of the energy). Though the microscopic picture is Hamiltonian, we show that the macroscopic picture is dissipative. It turns out that the existence of asymmetric microscopic solutions leads here to macroscopic dissipation.

This is joint work with Hartmut Schwetlick (Bath)