1,428 research outputs found
Local Complexity of Delone Sets and Crystallinity
This paper characterizes when a Delone set X is an ideal crystal in terms of
restrictions on the number of its local patches of a given size or on the
hetereogeneity of their distribution. Let N(T) count the number of
translation-inequivalent patches of radius T in X and let M(T) be the minimum
radius such that every closed ball of radius M(T) contains the center of a
patch of every one of these kinds. We show that for each of these functions
there is a `gap in the spectrum' of possible growth rates between being bounded
and having linear growth, and that having linear growth is equivalent to X
being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X
then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in
this bound is best possible in all dimensions. For M(T), either M(T) is bounded
or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound
cannot be replaced by any number exceeding 1/2. We also show that every
aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant
c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package
Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra
The paper investigates how correlations can completely specify a uniformly
discrete point process. The setting is that of uniformly discrete point sets in
real space for which the corresponding dynamical hull is ergodic. The first
result is that all of the essential physical information in such a system is
derivable from its -point correlations, . If the system is
pure point diffractive an upper bound on the number of correlations required
can be derived from the cycle structure of a graph formed from the dynamical
and Bragg spectra. In particular, if the diffraction has no extinctions, then
the 2 and 3 point correlations contain all the relevant information.Comment: 16 page
Fluid/solid transition in a hard-core system
We prove that a system of particles in the plane, interacting only with a
certain hard-core constraint, undergoes a fluid/solid phase transition
Statistical mechanics of glass transition in lattice molecule models
Lattice molecule models are proposed in order to study statistical mechanics
of glass transition in finite dimensions. Molecules in the models are
represented by hard Wang tiles and their density is controlled by a chemical
potential. An infinite series of irregular ground states are constructed
theoretically. By defining a glass order parameter as a collection of the
overlap with each ground state, a thermodynamic transition to a glass phase is
found in a stratified Wang tiles model on a cubic lattice.Comment: 18 pages, 8 figure
Dilatancy transition in a granular model
We introduce a model of granular matter and use a stress ensemble to analyze
shearing. Monte Carlo simulation shows the model to exhibit a second order
phase transition, associated with the onset of dilatancy.Comment: Future versions can be obtained from:
http://www.ma.utexas.edu/users/radin/papers/shear2.pd
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