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Long range order in atomistic models for solids
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Giuliani, Alessandro and Theil, Florian (2021) Long range order in atomistic models for solids. Journal of the European Mathematical Society, 24 (10). pp. 3505-3555. doi:10.4171/jems/1169 ISSN 1435-9855.
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Official URL: https://doi.org/10.4171/JEMS/1169
Abstract
The emergence of long range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in physics. To address this challenge we study a discrete microscopic model for an elastic crystal with dislocations in three dimensions, previously introduced by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read– Shockley law for grain boundaries, which we rigorously derive in a simple, explicit geometry. We analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza–Ortiz Hamiltonian plus a contribution from the dislocation cores. Our main result is that the model exhibits long range positional order at low temperatures. The proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QD Chemistry | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| SWORD Depositor: | Library Publications Router | ||||||
| Library of Congress Subject Headings (LCSH): | Dislocations in crystals, Dislocations in crystals -- Mathematical models , Gibbs' equation, Crystals, Grain boundaries | ||||||
| Journal or Publication Title: | Journal of the European Mathematical Society | ||||||
| Publisher: | European Mathematical Society - EMS - Publishing House GmbH | ||||||
| ISSN: | 1435-9855 | ||||||
| Official Date: | 19 November 2021 | ||||||
| Dates: |
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| Volume: | 24 | ||||||
| Number: | 10 | ||||||
| Page Range: | pp. 3505-3555 | ||||||
| DOI: | 10.4171/jems/1169 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 1 February 2022 | ||||||
| Date of first compliant Open Access: | 1 February 2022 | ||||||
| RIOXX Funder/Project Grant: |
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