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Travelling waves in atomistic chains and kinetic relations

Martensitic materials pose good test problems for the analysis of the passage from atomistic to continuum. The reason is that on the microscopic level, they can be well approximated by a chain of atoms governed by Newton’s equations; thus, the system is microscopically Hamiltonian. However, martensitic materials can undergo phase transitions, and moving phase boundaries can generate dissipation, so that the system is macroscopically dissipative. The simplest possible situation for which this phenomenon can be understood is that of a single phase boundary propagating in a one-dimensional chain of atoms {qj}j∈Z on the real line. Neighbouring atoms are linked by a spring with elastic potential V , and it is convenient to assume that only nearest neighbours interact. The longitudinal elongation of atom k is given by uk : R → R. The argument of the elastic potential V is the discrete strain, that is, the difference of the deformations, uk+1(t) − uk(t). The springs are bistable, with the two stable states representing two stable phases. The equations of motion are assumed to be governed by Newton’s law, so that in suitable units