Bourne, D. P., Peletier, M. A. and Theil, Florian (2014) Optimality of the triangular lattice for a particle system with Wasserstein interaction. Communications in Mathematical Physics, Volume 329 (Number 1). pp. 117-140. doi:10.1007/s00220-014-1965-5 ISSN 0010-3616.

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Abstract

We prove strong crystallization results in two dimensions for an energy that arises in the theory of block copolymers. The energy is defined on sets of points and their weights, or equivalently on the set of atomic measures. It consists of two terms; the first term is the sum of the square root of the weights, and the second is the quadratic optimal transport cost between the atomic measure and the Lebesgue measure. We prove that this system admits crystallization in several different ways: (1) the energy is bounded from below by the energy of a triangular lattice (called T); (2) if the energy equals that of T, then the measure is a rotated and translated copy of T; (3) if the energy is close to that of T, then locally the measure is close to a rotated and translated copy of T. These three results require the domain to be a polygon with at most six sides. A fourth result states that the energy of T can be achieved in the limit of large domains, for domains with arbitrary boundaries. The proofs make use of three ingredients. First, the optimal transport cost associates to each point a polygonal cell; the energy can be bounded from below by a sum over all cells of a function that depends only on the cell. Second, this function has a convex lower bound that is sharp at T. Third, Euler's polytope formula limits the average number of sides of the polygonal cells to six, where six is the number corresponding to the triangular lattice.

Item Type:	Journal Article
Divisions:	Faculty of Science, Engineering and Medicine > Science > Physics
Journal or Publication Title:	Communications in Mathematical Physics
Publisher:	Springer
ISSN:	0010-3616
Official Date:	July 2014
Dates:	Date Event July 2014 Published 13 April 2014 Available 18 September 2013 Accepted 16 January 2013 Submitted
Volume:	Volume 329
Number:	Number 1
Page Range:	pp. 117-140
DOI:	10.1007/s00220-014-1965-5
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)

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