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The role of microscopic anisotropy in the macroscopic behavior of a phase boundary. (English) Zbl 0639.58038
A Landau-Ginzburg approach is implemented to study the influence microscopic anisotropy on the interfacial boundary between two phases. The continuum limit for a lattice spin system with anisotropic interactions $$J_ i$$ and spacing $$a_ i$$ leads to the system of equations $\tau (\partial \phi /\partial t)=\sum^{d}_{i=1}\xi^ 2_ i(\partial^ 2\phi /\partial x^ 2_ i)+(\phi -\phi^ 3)+2u,$
$\partial u/\partial t+(\ell /2)(\partial \phi /\partial t)=K\Delta u.$ Here, $$\phi$$ is an order parameter, u is the temperature, $$\tau$$ is a relaxation time, $$\xi_ i$$ are related to $$J_ i$$ and $$a_ i$$, $$\ell$$ is the latent heat of fusion, and K is the diffusivity. Both equilibrium and non-equilibrium conditions are considered. A modified Gibbs-Thompson relation $\Delta su(r,\theta)=-[\sigma (\theta)+\sigma ''(\theta)]\kappa -\tau \nu \sigma (\theta)/\xi^ 2_ A(\theta)+{\mathcal O}(\xi^ 2)$ is obtained, where s is entropy density, $$\kappa$$ is curvature, $$\nu$$ is normal velocity of the interface, and $$\xi_ A(\theta)$$ is a measure of the thickness of the interface.

##### MSC:
 58Z05 Applications of global analysis to the sciences 82B99 Equilibrium statistical mechanics 58J99 Partial differential equations on manifolds; differential operators
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