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

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Abstract

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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References

  1. Bardenhagen, S., Triantafyllidis, N.: Derivation of higher order gradient continuum theories in 2,3-D nonlinear elasticity from periodic lattice models. J. Mech. Phys. Solids 42, 111–139 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berezhnyy, M., Berlyand, L.: Continuum limit for three-dimensional mass-spring networks and discrete Korn’s inequality. J. Mech. Phys. Solids 54(3), 635–669 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blanc, X., Monneau, R.: Screening of an applied electric field inside a metallic layer described by the Thomas-Fermi-von Weizsäcker model. Adv. Differ. Equ. 7, 847–876 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Blanc, X., Le Bris, C., Legoll, F.: Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. Math. Model. Numer. Anal. 39(4), 797–826 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blanc, X., Le Bris, C., Legoll, F.: Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics: the convex case. Acta Math. Appl. Sin. Engl. Ser. 23, 209–216 (2007)

    Article  MathSciNet  Google Scholar 

  6. Blanc, X., Le Bris, C., Lions, P.-L.: From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164, 341–381 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Braides, A., Gelli, M.S.: Limits of discrete systems with long-range interactions. J. Convex Anal. 9, 363–399 (2002). Special issue on optimization (Montpellier, 2000)

    MATH  MathSciNet  Google Scholar 

  8. Braides, A., Dal Maso, G., Garroni, A.: Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Ration. Mech. Anal. 146(1), 23–58 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Braides, A., Gelli, M.S., Sigalotti, M.: The passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case. Tr. Ordena Lenina Mat. Inst. Im. V.A. Steklova 236, 408–427 (2002)

    MathSciNet  Google Scholar 

  10. Braides, A., Lew, A.J., Ortiz, M.: Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180(2), 151–182 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. E, W., Huang, Z.: Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett. 87, 135501 (2001)

    Article  Google Scholar 

  12. E, W., Huang, Z.: A dynamic atomistic-continuum method for the simulation of crystalline materials. J. Comput. Phys. 182, 234–261 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. E, W., Ming, P.: Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. E, W., Ming, P.: Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Acta Math. Appl. Sin. Engl. Ser. 23(4), 529–550 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Friesecke, G., James, R.D.: A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids 48, 1519–1540 (2000). The J.R. Willis 60th anniversary volume

    Article  MATH  MathSciNet  Google Scholar 

  16. Friesecke, G., Theil, F.: Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12, 445–478 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gardner, C.S., Radin, C.: The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys. 20, 719–724 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  18. Heitmann, R.C., Radin, C.: The ground state for sticky disks. J. Stat. Phys. 22, 281–287 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  19. Mielke, A.: Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner-Husimi transforms. Arch. Ration. Mech. Anal. 181, 401–448 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Monneau, R.: Uniform elliptic estimate for an infinite plate in linear elasticity. Commun. Partial Differ. Equ. 29, 989–1016 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Monneau, R.: A new 3d-2d interior error estimate independent on the geometry of a linear elastic plate. Asymptot. Anal. 49, 331–344 (2006)

    MATH  MathSciNet  Google Scholar 

  22. Radin, C.: The ground state for soft disks. J. Stat. Phys. 26, 365–373 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  23. Radin, C.: Classical ground states in one dimension. J. Stat. Phys. 35, 109–117 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  24. Radin, C.: Low temperature and the origin of crystalline symmetry. Int. J. Mod. Phys. B 1, 1157–1191 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  25. Radin, C., Schulman, L.S.: Periodicity of classical ground states. Phys. Rev. Lett. 51, 621–622 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  26. Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method. J. Mech. Phys. Solids 47, 611–642 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tadmor, E.B., Ortiz, M., Phillips, R.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73, 1529–1563 (1996)

    Article  Google Scholar 

  28. Tadmor, E.B., Phillips, R., Ortiz, M.: Mixed atomistic and continuum models of deformation in solids. Langmuir 12, 4529–4534 (1996)

    Article  Google Scholar 

  29. Tadmor, E.B., Smith, G.S., Bernstein, N., Kaxiras, E.: Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59, 235–245 (1999)

    Article  Google Scholar 

  30. Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Triantafyllidis, N., Bardenhagen, S.: On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models. J. Elast. 33, 259–293 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  32. Truskinovsky, L.: Fracture as a phase transition. In: Batra, R.C., Beatty, M.F. (eds.) Contemporary Research in the Mechanics and Mathematics of Materials, pp. 322–332. CIMNE, Barcelona (1996)

    Google Scholar 

  33. Ventevogel, W.J.: On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Physica A: Stat. Theor. Phys. 92, 343–361 (1978)

    Article  ADS  Google Scholar 

  34. Ventevogel, W.J., Nijboer, B.R.A.: On the configuration of systems of interacting particles with minimum potential energy per particle. Physica A 98, 274–288 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  35. Ventevogel, W.J., Nijboer, B.R.A.: On the configuration of systems of interacting particles with minimum potential energy per particle. Physica A 99, 569–580 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  36. Zanzotto, G.: On the material symmetry group of elastic crystals and the Born rule. Arch. Ration. Mech. Anal. 121, 1–36 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Fisica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

    R. Benguria

  2. Ceremade, UMR CNRS 7534, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris Cédex 16, France

    J. Dolbeault

  3. CERMICS, Ecole nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455, Marne-la-Vallée Cedex 2, France

    R. Monneau

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  1. R. Benguria
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  2. J. Dolbeault
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  3. R. Monneau
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Correspondence to J. Dolbeault.

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Benguria, R., Dolbeault, J. & Monneau, R. Harnack Inequalities and Discrete—Continuous Error Estimates for a Chain of Atoms with Two—Body Interactions. J Stat Phys 134, 27–51 (2009). https://doi.org/10.1007/s10955-008-9662-4

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  • DOI: https://doi.org/10.1007/s10955-008-9662-4

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