THESIS
2017
viii, 36 pages : illustrations (some color) ; 30 cm
Abstract
A lot of systems display domain patterns at equilibrium. Some commonly observed
patterns are bubbles and stripes in two-dimensional systems; or droplets, tubes and sheets
in three-dimensional systems. In this thesis, the 2D patterns are studied by an energetic
approach, using the Ising model with generalized interactions as the model. Several concepts
in statistical physics, such as Legendre transformation and the correlation function,
are proved to be helpful in the understanding of the behavior of the model. One important
implication is that a power-law interaction may be approximated as a simple short-range
interaction plus a chemical potential. These concepts also help in developing a simple
and efficient method in prediction of the ground state.
Another focus of this thesi...[
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A lot of systems display domain patterns at equilibrium. Some commonly observed
patterns are bubbles and stripes in two-dimensional systems; or droplets, tubes and sheets
in three-dimensional systems. In this thesis, the 2D patterns are studied by an energetic
approach, using the Ising model with generalized interactions as the model. Several concepts
in statistical physics, such as Legendre transformation and the correlation function,
are proved to be helpful in the understanding of the behavior of the model. One important
implication is that a power-law interaction may be approximated as a simple short-range
interaction plus a chemical potential. These concepts also help in developing a simple
and efficient method in prediction of the ground state.
Another focus of this thesis is to investigate the assumptions made in the method.
Our method first assumes some structures like bubbles and stripes, but complex patterns
are sometimes possible. We identify that the preference of particle separation is the
product of two factors, namely, the particle density and an interaction-dependent term.
Complex patterns can be understood as the result of competitions between two factors of
the preference and the qualitative features of such interactions are discussed. Finally, a
proof of ground state periodicity is given using Fourier series.
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