Discrete dynamic models for phase transitions
Johannes Zimmer, Hartmut Schwetlick · The University of Bath Online Publications Store (The University of Bath) · 2007
(joint work with Hartmut Schwetlick) The problem under consideration is easy to formulate. Consider a one-dimensional chain of atoms {qj}j∈Z on a torus (Z: = Z/LZ with L ∈ N) or on the real line (Z: = Z). For each atom, the deformation is given by uk: R → R. The equations of motion are governed by Newton’s law, which, in suitable units, reads (1) ük(t) = V ′(uk+1(t) − uk(t)) − V ′(uk(t) − uk−1(t)) for every k ∈ Z (on the torus, indices are counted modulo L). This is a spatially discretized, one-dimensional version of the well-studied equations of motions of an elastic material (2) utt(x) = Div(σ(Du(x))). Discretized equations as (1) are intrinsically interesting, as they correspond to forward-backward equations. This become apparent in the travelling wave formu-lation (3) uj(t) = u(j − ct) for j ∈ Z, since then Equation (1) transforms into (4) c2ü(x) = V ′(u(x+ 1) − u(x)) − V ′(u(x) − u(x − 1)). This is the Euler-Lagrange function for the action functional φ(u):=