Many materials can be modeled either as discrete systems or as
continua, depending on the scale. At intermediate scales it is
necessary to understand the transition from discrete to continuous
models and variational methods have proved successful in this task,
especially for systems, both stochastic and deterministic, that
depend on lattice energies. This is the first systematic and
unified presentation of research in the area over the last 20
years. The authors begin with a very general and flexible
compactness and representation result, complemented by a thorough
exploration of problems for ferromagnetic energies with
applications ranging from optimal design to quasicrystals and
percolation. This leads to a treatment of frustrated systems, and
infinite-dimensional systems with diffuse interfaces. Each topic is
presented with examples, proofs and applications. Written by
leading experts, it is suitable as a graduate course text as well
as being an invaluable reference for researchers.
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