December  2014, 34(12): 5085-5097. doi: 10.3934/dcds.2014.34.5085

Mathematical theory of solids: From quantum mechanics to continuum models

1. 

Department of Mathematics, Princeton University, Princeton, N.J. 08544

2. 

Departments of Mathematics, Physics, and Chemistry, Duke University, Durham NC, United States

Received  March 2014 Revised  June 2014 Published  June 2014

N/A
Citation: Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085
References:
[1]

M. Arroyo and T. Belytschko, Nonlinear mechanical response and rippling of thick multiwalled Carbon nanotubes,, Phys. Rev. Lett., 91 (2003).   Google Scholar

[2]

M. Arroyo and T. Belytschko, Finite element methods for the nonlinear mechanics of crystalline sheets and nanotubes,, Int. J. Numer. Methods Eng., 59 (2004), 419.  doi: 10.1002/nme.944.  Google Scholar

[3]

X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics,, Arch. Ration. Mech. Anal., 164 (2002), 341.  doi: 10.1007/s00205-002-0218-5.  Google Scholar

[4]

M. Born, Thermodynamics of crystals and melting,, J. Chem. Phys., 7 (1939), 591.   Google Scholar

[5]

A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: The one-dimensional case,, Arch. Ration. Mech. Anal., 146 (1999), 23.  doi: 10.1007/s002050050135.  Google Scholar

[6]

E. Cancès and M. Lewin, The dielectric permittivity of crystals in the reduced Hartree-Fock approximation,, Arch. Ration. Mech. Anal., 197 (2010), 139.  doi: 10.1007/s00205-009-0275-0.  Google Scholar

[7]

E. Cancès and G. Stoltz, A mathematical formulation of the random phase approximation for crystals,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 887.  doi: 10.1016/j.anihpc.2012.05.004.  Google Scholar

[8]

A.-L. Cauchy, Sur l'equilibre et le mouvement d'un système de points materiels sollicités par forces d'attraction ou de répulsion mutuelle,, Ex. de Math., 3 (1828), 277.   Google Scholar

[9]

A.-L. Cauchy, De la pression on tension dans un système de points matériels,, Ex. de Math., 3 (1828), 253.   Google Scholar

[10]

S. Conti, G. Dolzmann, B. Kirchheim and S. Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to $SO(n)$,, J. Eur. Math. Soc., 8 (2006), 515.  doi: 10.4171/JEMS/65.  Google Scholar

[11]

W. E and J. Lu, The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born rule,, Arch. Ration. Mech. Anal., 199 (2011), 407.  doi: 10.1007/s00205-010-0339-1.  Google Scholar

[12]

W. E and J. Lu, Stability and the continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsacker model,, J. Math. Phys., 53 (2012).   Google Scholar

[13]

W. E and J. Lu, The Kohn-Sham equation for deformed crystals,, Mem. Amer. Math. Soc., 221 (2013).  doi: 10.1090/S0065-9266-2012-00659-9.  Google Scholar

[14]

W. E, J. Lu and X. Yang, Effective Maxwell equations from time-dependent density functional theory,, Acta Math. Sin. (Eng. Ser.), 27 (2011), 339.  doi: 10.1007/s10114-011-0555-0.  Google Scholar

[15]

W. E and P. B. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems,, Arch. Ration. Mech. Anal., 183 (2007), 241.  doi: 10.1007/s00205-006-0031-7.  Google Scholar

[16]

G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice,, J. Nonlinear Sci., 12 (2002), 445.  doi: 10.1007/s00332-002-0495-z.  Google Scholar

[17]

P. Hohenberg and W. Kohn, Inhomogeneous electron gas,, Phys. Rev. (2), 136 (1964).  doi: 10.1103/PhysRev.136.B864.  Google Scholar

[18]

W. Kohn and L. Sham, Self-consistent equations including exchange and correlation effects,, Phys. Rev., 140 (1965).  doi: 10.1103/PhysRev.140.A1133.  Google Scholar

[19]

F. A. Lindemann, Uber die Berechnung Molecular Eigenfrequnzen,, Physik. Z., 11 (1910), 609.   Google Scholar

[20]

M. P. Marder, Condensed Matter Physics,, Second edition, (2010).   Google Scholar

[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis,, Second edition, (1980).   Google Scholar

[22]

E. B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids,, Philos. Mag. A, 73 (1996), 1529.   Google Scholar

[23]

J. Z. Yang and W. E, Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models,, Phys. Rev. B, 74 (2006).   Google Scholar

show all references

References:
[1]

M. Arroyo and T. Belytschko, Nonlinear mechanical response and rippling of thick multiwalled Carbon nanotubes,, Phys. Rev. Lett., 91 (2003).   Google Scholar

[2]

M. Arroyo and T. Belytschko, Finite element methods for the nonlinear mechanics of crystalline sheets and nanotubes,, Int. J. Numer. Methods Eng., 59 (2004), 419.  doi: 10.1002/nme.944.  Google Scholar

[3]

X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics,, Arch. Ration. Mech. Anal., 164 (2002), 341.  doi: 10.1007/s00205-002-0218-5.  Google Scholar

[4]

M. Born, Thermodynamics of crystals and melting,, J. Chem. Phys., 7 (1939), 591.   Google Scholar

[5]

A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: The one-dimensional case,, Arch. Ration. Mech. Anal., 146 (1999), 23.  doi: 10.1007/s002050050135.  Google Scholar

[6]

E. Cancès and M. Lewin, The dielectric permittivity of crystals in the reduced Hartree-Fock approximation,, Arch. Ration. Mech. Anal., 197 (2010), 139.  doi: 10.1007/s00205-009-0275-0.  Google Scholar

[7]

E. Cancès and G. Stoltz, A mathematical formulation of the random phase approximation for crystals,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 887.  doi: 10.1016/j.anihpc.2012.05.004.  Google Scholar

[8]

A.-L. Cauchy, Sur l'equilibre et le mouvement d'un système de points materiels sollicités par forces d'attraction ou de répulsion mutuelle,, Ex. de Math., 3 (1828), 277.   Google Scholar

[9]

A.-L. Cauchy, De la pression on tension dans un système de points matériels,, Ex. de Math., 3 (1828), 253.   Google Scholar

[10]

S. Conti, G. Dolzmann, B. Kirchheim and S. Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to $SO(n)$,, J. Eur. Math. Soc., 8 (2006), 515.  doi: 10.4171/JEMS/65.  Google Scholar

[11]

W. E and J. Lu, The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born rule,, Arch. Ration. Mech. Anal., 199 (2011), 407.  doi: 10.1007/s00205-010-0339-1.  Google Scholar

[12]

W. E and J. Lu, Stability and the continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsacker model,, J. Math. Phys., 53 (2012).   Google Scholar

[13]

W. E and J. Lu, The Kohn-Sham equation for deformed crystals,, Mem. Amer. Math. Soc., 221 (2013).  doi: 10.1090/S0065-9266-2012-00659-9.  Google Scholar

[14]

W. E, J. Lu and X. Yang, Effective Maxwell equations from time-dependent density functional theory,, Acta Math. Sin. (Eng. Ser.), 27 (2011), 339.  doi: 10.1007/s10114-011-0555-0.  Google Scholar

[15]

W. E and P. B. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems,, Arch. Ration. Mech. Anal., 183 (2007), 241.  doi: 10.1007/s00205-006-0031-7.  Google Scholar

[16]

G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice,, J. Nonlinear Sci., 12 (2002), 445.  doi: 10.1007/s00332-002-0495-z.  Google Scholar

[17]

P. Hohenberg and W. Kohn, Inhomogeneous electron gas,, Phys. Rev. (2), 136 (1964).  doi: 10.1103/PhysRev.136.B864.  Google Scholar

[18]

W. Kohn and L. Sham, Self-consistent equations including exchange and correlation effects,, Phys. Rev., 140 (1965).  doi: 10.1103/PhysRev.140.A1133.  Google Scholar

[19]

F. A. Lindemann, Uber die Berechnung Molecular Eigenfrequnzen,, Physik. Z., 11 (1910), 609.   Google Scholar

[20]

M. P. Marder, Condensed Matter Physics,, Second edition, (2010).   Google Scholar

[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis,, Second edition, (1980).   Google Scholar

[22]

E. B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids,, Philos. Mag. A, 73 (1996), 1529.   Google Scholar

[23]

J. Z. Yang and W. E, Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models,, Phys. Rev. B, 74 (2006).   Google Scholar

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