Harnack Inequalities and Discrete–Continuous Error Estimates for a Chain of Atoms with Two-Body Interactions

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We consider deformations in ℝ3 of an infinite linear chain of atoms where each atom interacts with all others through a two-body potential. We compute the effect of an external force applied to the chain. At equilibrium, the positions of the particles satisfy an Euler–Lagrange equation. For large classes of potentials, we prove that every solution is well approximated by the solution of a continuous model when applied forces and displacements of the atoms are small. We establish an error estimate between the discrete and the continuous solution based on a Harnack lemma of independent interest. Finally we apply our results to some Lennard-Jones potentials.

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