WIAS Preprint No. 2589, (2019)

Surface energy and boundary layers for a chain of atoms at low temperature



Authors

  • Jansen, Sabine
  • König, Wolfgang
  • Schmidt, Bernd
  • Theil, Florian

2010 Mathematics Subject Classification

  • 82B21 74B20 74G65 60F10

Keywords

  • Atomistic models of elasticity, surface energy and boundary layers, semi-classical limit of transfer operators, uniform decay of correlations, path large deviations for, stationary processes

DOI

10.20347/WIAS.PREPRINT.2589

Abstract

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature goes to zero. Our main results are: (1) As the temperature goes to zero and at fixed positive pressure, the Gibbs measures  for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in the inverse temperature.

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