We develop a rigorous error analysis for coarse-graining of defect-formation free energy. For a one-dimensional constrained atomistic system, we establish the thermodynamic limit of the defect-formation free energy and obtain explicitly the rate of convergence. We then construct a sequence of coarse-grained energies with the same rate but significantly reduced computational cost. We illustrate our analytical results through explicit computations for the case of harmonic potentials and through numerical simulations.
Accepted:
DOI: 10.1051/m2an/2017052
Keywords: Defect formation free energy, finite temperature, material defects, Cauchy–Born rule
Dobson, Matthew 1; Duong, Manh Hong 2; Ortner, Christoph 2
@article{M2AN_2018__52_4_1315_0,
author = {Dobson, Matthew and Duong, Manh Hong and Ortner, Christoph},
title = {On assessing the accuracy of defect free energy computations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1315--1352},
year = {2018},
publisher = {EDP Sciences},
volume = {52},
number = {4},
doi = {10.1051/m2an/2017052},
mrnumber = {3875288},
zbl = {1455.74023},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2017052/}
}
TY - JOUR AU - Dobson, Matthew AU - Duong, Manh Hong AU - Ortner, Christoph TI - On assessing the accuracy of defect free energy computations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1315 EP - 1352 VL - 52 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2017052/ DO - 10.1051/m2an/2017052 LA - en ID - M2AN_2018__52_4_1315_0 ER -
%0 Journal Article %A Dobson, Matthew %A Duong, Manh Hong %A Ortner, Christoph %T On assessing the accuracy of defect free energy computations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1315-1352 %V 52 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2017052/ %R 10.1051/m2an/2017052 %G en %F M2AN_2018__52_4_1315_0
Dobson, Matthew; Duong, Manh Hong; Ortner, Christoph. On assessing the accuracy of defect free energy computations. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 52 (2018) no. 4, pp. 1315-1352. doi: 10.1051/m2an/2017052
[1] , , and , Finite-temperature coarse-graining of one-dimensional models: Mathematical analysis and computational approaches. J. Nonlin. Sci. 20 (2010) 241–275. | MR | Zbl | DOI
[2] and , A numerical strategy for coarse-graining two-dimensional atomistic models at finite temperature: The membrane case. Comput. Mat. Sci. 66 (2013) 84–95. | DOI
[3] , Uniform Poincaré inequalities for unbounded conservative spin systems: the non-interacting case. Stochastic Process. Appl. 106 (2003) 223–244. | MR | Zbl | DOI
[4] and . Materials Science and Engineering: An Introduction. Wiley (2010).
[5] , Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab. 3 (1975) 146–158.
[6] and and , On assessing the accuracy of defect free energy computations. To published in: ESAIM: M2AN DOI: (2018). | MR | Zbl | Numdam | DOI
[7] and , Conditional limit for exponential families and finite versions of de Finetti’s theorem. J. Theor. Probab. 1 (1988) 381–410. | MR | Zbl | DOI
[8] , , and , Finite-temperature quasicontinuum: Molecular dynamics without all the atoms. Phys. Rev. Lett. 95 (2005) 060202.
[9] and , Refinements of the Gibbs conditioning principle. Probability Theory and Related Fields 104 (1996) 1–14. | MR | Zbl | DOI
[10] and . Large deviation techniques and applications. Springer, New York. (1998). | MR | Zbl | DOI
[11] , , and , A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 302–351. | MR | Zbl | Numdam | DOI
[12] , , , and , Dynamics of point defect formation, clustering and pit initiation on the pyrite surface. Electr. Acta 127 (2014) 416–426. | DOI
[13] , and . Free Energy Computations. Imperial College Press (2012). | MR | Zbl
[14] , LSI for Kawasaki dynamics with weak interaction. Commun. Math. Phys. 307 (2011) 817–860. | MR | Zbl | DOI
[15] , , , , and , Finite-temperature extension of the quasicontinuum method using langevin dynamics: entropy losses and analysis of errors. Modell. Simul. Mater. Sci. Eng. 18 (2010) 015003.
[16] . An Introduction to Mineral Sciences. Cambridge University Press. Cambridge Books Online (1992) | DOI
[17] and , Fundamentals of defect ionization and transport. In Charged Semiconductor Defects, Engineering Materials and Processes. Springer London (2009). | DOI
[18] and , Accuracy of computation of crystalline defects at finite temperature. Preprint arXiv:1409.5739 (2014).
[19] , , , and , Finite-Temperature Quasi-Continuum. ASME. Appl. Mech. Rev. 65 (2013) 010803–010803-27. | DOI
[20] , and , Free energy of defect formation: Thermodynamics of anion frenkel pairs in indium oxide. Phys. Rev. B 83 (2011) 224105. | DOI
Cited by Sources:
