 97381 Jacek Miekisz
 An Ultimate Frustration in ClassicalLattice Gas Models
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Jun 27, 97

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Abstract. We compare tiling systems with squarelike tiles
and classical latticegas models with translationinvariant,
finiterange interactions between particles.
For a given tiling, there is a natural construction
of a corresponding latticegas model. With onetoone
correspondence between particles and tiles,
we simply assign a positive energy to pairs of nearestneighbor
particles which do not match as tiles; otherwise the energy
of interaction is zero. Such models of interacting particles
are called nonfrustrated  all interactions can attain their
minima simultaneously. Groundstate configurations of these
models correspond to tilings; they have the minimal energy
density equal to zero. There are frustrated latticegas models;
antiferromagnetic Ising model on the triangular lattice
is a standard example. However, in all such models known so far,
one could always find a nonfrustrated interaction having
the same groundstate configurations.
Here we constructed an uncountable family of classical
latticegas models with unique groundstate measures which
are not uniquely ergodic measures of any tiling system,
or more generally, of any system of finite type. Therefore,
we have shown that the family of structures which are unique
ground states of some translationinvariant, finiterange
interactions is larger than the family of tilings which
form single isomorphism classes. Such groundstate measures
cannot be groundstate measures of any translationinvariant,
finiterange, nonfrustrated potential.
Our groundstate configurations are twodimensional
analogs of onedimensional, most homogeneous groundstate
configurations of infiniterange, convex, repulsive interactions
in models with devil's staircases.
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