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A proof of crystallization in two dimensions
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UNSPECIFIED (2006) A proof of crystallization in two dimensions. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 262 (1). pp. 209236. doi:10.1007/s0022000514587
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Official URL: http://dx.doi.org/10.1007/s0022000514587
Abstract
Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type E({y}) = (1 <= x < x'<= N)Sigma V(vertical bar y(x)  y(x')vertical bar), where y(x) is an element of R2 is the position of particle x and V(r) is an element of R is the pairinteraction energy of two particles which are placed at distance r. Due to the MerminWagner theorem it can't be expected that at finite temperature this system exhibits longrange ordering. We focus on the zero temperature case and show rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the LennardJones potential the ground state energy per particle converges to an explicit constant E*:
(N >infinity)lim (1)/(N) (y:{1...N}> R2)min E({y}) = E*,
where E* is an element of R is the minimum of a simple function on [0, infinity). Furthermore, if suitable Dirichlet or periodic boundary conditions are used, then the minimizers form a triangular lattice. To the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the Euclidean symmetry group consisting of rotations and translations.
Item Type:  Journal Article  

Subjects:  Q Science > QC Physics  
Journal or Publication Title:  COMMUNICATIONS IN MATHEMATICAL PHYSICS  
Publisher:  SPRINGER  
ISSN:  00103616  
Official Date:  February 2006  
Dates: 


Volume:  262  
Number:  1  
Number of Pages:  28  
Page Range:  pp. 209236  
DOI:  10.1007/s0022000514587  
Publication Status:  Published 
Data sourced from Thomson Reuters' Web of Knowledge
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