1,842 research outputs found
Modelling quasicrystals at positive temperature
We consider a two-dimensional lattice model of equilibrium statistical
mechanics, using nearest neighbor interactions based on the matching conditions
for an aperiodic set of 16 Wang tiles. This model has uncountably many ground
state configurations, all of which are nonperiodic. The question addressed in
this paper is whether nonperiodicity persists at low but positive temperature.
We present arguments, mostly numerical, that this is indeed the case. In
particular, we define an appropriate order parameter, prove that it is
identically zero at high temperatures, and show by Monte Carlo simulation that
it is nonzero at low temperatures
Two-point correlation properties of stochastic "cloud processes''
We study how the two-point density correlation properties of a point particle
distribution are modified when each particle is divided, by a stochastic
process, into an equal number of identical "daughter" particles. We consider
generically that there may be non-trivial correlations in the displacement
fields describing the positions of the different daughters of the same "mother"
particle, and then treat separately the cases in which there are, or are not,
correlations also between the displacements of daughters belonging to different
mothers. For both cases exact formulae are derived relating the structure
factor (power spectrum) of the daughter distribution to that of the mother.
These results can be considered as a generalization of the analogous equations
obtained in ref. [1] (cond-mat/0409594) for the case of stochastic displacement
fields applied to particle distributions. An application of the present results
is that they give explicit algorithms for generating, starting from regular
lattice arrays, stochastic particle distributions with an arbitrarily high
degree of large-scale uniformity.Comment: 14 pages, 3 figure
A cluster expansion approach to exponential random graph models
The exponential family of random graphs is among the most widely-studied
network models. We show that any exponential random graph model may
alternatively be viewed as a lattice gas model with a finite Banach space norm.
The system may then be treated by cluster expansion methods from statistical
mechanics. In particular, we derive a convergent power series expansion for the
limiting free energy in the case of small parameters. Since the free energy is
the generating function for the expectations of other random variables, this
characterizes the structure and behavior of the limiting network in this
parameter region.Comment: 15 pages, 1 figur
The structure of the hard sphere solid
We show that near densest-packing the perturbations of the HCP structure
yield higher entropy than perturbations of any other densest packing. The
difference between the various structures shows up in the correlations between
motions of nearest neighbors. In the HCP structure random motion of each sphere
impinges slightly less on the motion of its nearest neighbors than in the other
structures.Comment: For related papers see:
http://www.ma.utexas.edu/users/radin/papers.htm
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
Crystalline ground states for classical particles
Pair interactions whose Fourier transform is nonnegative and vanishes above a
wave number K_0 are shown to give rise to periodic and aperiodic infinite
volume ground state configurations (GSCs) in any dimension d. A typical three
dimensional example is an interaction of asymptotic form cos(K_0 r)/r^4. The
result is obtained for densities rho >= rho_d where rho_1=K_0/2pi,
rho_2=(sqrt{3}/8)(K_0/pi)^2 and rho_3=(1/8sqrt{2})(K_0/pi)^3. At rho_d there is
a unique periodic GSC which is the uniform chain, the triangular lattice and
the bcc lattice for d=1,2,3, respectively. For rho>rho_d the GSC is nonunique
and the degeneracy is continuous: Any periodic configuration of density rho
with all reciprocal lattice vectors not smaller than K_0, and any union of such
configurations, is a GSC. The fcc lattice is a GSC only for rho>=(1/6
sqrt{3})(K_0/pi)^3.Comment: final versio
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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