Inserted: 30 jul 2013
Last Updated: 3 mar 2015
Journal: Archive Rational Mech. Anal.
This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on $\Gamma$-convergence.
We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from $XY$ spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order $\Gamma$-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices.
As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg-Landau framework.
Keywords: minimizing movements, Discrete topological singularities, Dislocations in crystals