The Continuum Limit of the Thomas-Fermi-Dirac-von Weizsacker Model

Abstract

This work studies the atomistic Thomas-Fermi-Dirac-von Weiszacker model on a Bravais lattice, by establishing relation with the continuum elasticity model, thus provides it a solid microscopic foundation at the atomistic level. More specifically, the stored energy density can be derived from the atomistic TFDW functional by homogenization, by assuming a reasonable stability condition, the discrete deformation function we get from the atomistic model converges to the Cauchy-Born solution from solving the continuum model with a quadratic rate due to underlying inversion symmetry of the lattice. In our analysis, we used two-scale ansatz to construct approximate solutions, discrete Fourier analysis in the consistence estimate, and perturbation technique to analyze the Hessian for the stability analysis.