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$Q$-Tensor Continuum Energies as Limits of Head-to-Tail Symmetric Spin Systems

2015; Society for Industrial and Applied Mathematics; Volume: 47; Issue: 4 Linguagem: Inglês

10.1137/130941341

1095-7154

Andrea Braides, Marco Cicalese, Francesco Solombrino,

Quantum chaos and dynamical systems

We consider a class of spin-type discrete systems and analyze their continuum limit as the lattice spacing goes to zero. Under standard coerciveness and growth assumptions together with an additional head-to-tail symmetry condition, we observe that this limit can be conveniently written as a functional in the space of $Q$-tensors. We further characterize the limit energy density in several cases (both in two and three dimensions). In the planar case we also develop a second-order theory and we derive gradient or concentration-type models according to the chosen scaling.