539 research outputs found
Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
The goal of this paper is to quantitatively describe some statistical
properties of higher-dimensional determinantal point processes with a primary
focus on the nearest-neighbor distribution functions. Toward this end, we
express these functions as determinants of matrices and then
extrapolate to . This formulation allows for a quick and accurate
numerical evaluation of these quantities for point processes in Euclidean
spaces of dimension . We also implement an algorithm due to Hough \emph{et.
al.} \cite{hough2006dpa} for generating configurations of determinantal point
processes in arbitrary Euclidean spaces, and we utilize this algorithm in
conjunction with the aforementioned numerical results to characterize the
statistical properties of what we call the Fermi-sphere point process for to 4. This homogeneous, isotropic determinantal point process, discussed
also in a companion paper \cite{ToScZa08}, is the high-dimensional
generalization of the distribution of eigenvalues on the unit circle of a
random matrix from the circular unitary ensemble (CUE). In addition to the
nearest-neighbor probability distribution, we are able to calculate Voronoi
cells and nearest-neighbor extrema statistics for the Fermi-sphere point
process and discuss these as the dimension is varied. The results in this
paper accompany and complement analytical properties of higher-dimensional
determinantal point processes developed in \cite{ToScZa08}.Comment: 42 pages, 17 figure
Controlling the Short-Range Order and Packing Densities of Many-Particle Systems
Questions surrounding the spatial disposition of particles in various
condensed-matter systems continue to pose many theoretical challenges. This
paper explores the geometric availability of amorphous many-particle
configurations that conform to a given pair correlation function g(r). Such a
study is required to observe the basic constraints of non-negativity for g(r)
as well as for its structure factor S(k). The hard sphere case receives special
attention, to help identify what qualitative features play significant roles in
determining upper limits to maximum amorphous packing densities. For that
purpose, a five-parameter test family of g's has been considered, which
incorporates the known features of core exclusion, contact pairs, and damped
oscillatory short-range order beyond contact. Numerical optimization over this
five-parameter set produces a maximum-packing value for the fraction of covered
volume, and about 5.8 for the mean contact number, both of which are within the
range of previous experimental and simulational packing results. However, the
corresponding maximum-density g(r) and S(k) display some unexpected
characteristics. A byproduct of our investigation is a lower bound on the
maximum density for random sphere packings in dimensions, which is sharper
than a well-known lower bound for regular lattice packings for d >= 3.Comment: Appeared in Journal of Physical Chemistry B, vol. 106, 8354 (2002).
Note Errata for the journal article concerning typographical errors in Eq.
(11) can be found at http://cherrypit.princeton.edu/papers.html However, the
current draft on Cond-Mat (posted on August 8, 2002) is correct
Modeling Heterogeneous Materials via Two-Point Correlation Functions: II. Algorithmic Details and Applications
In the first part of this series of two papers, we proposed a theoretical
formalism that enables one to model and categorize heterogeneous materials
(media) via two-point correlation functions S2 and introduced an efficient
heterogeneous-medium (re)construction algorithm called the "lattice-point"
algorithm. Here we discuss the algorithmic details of the lattice-point
procedure and an algorithm modification using surface optimization to further
speed up the (re)construction process. The importance of the error tolerance,
which indicates to what accuracy the media are (re)constructed, is also
emphasized and discussed. We apply the algorithm to generate three-dimensional
digitized realizations of a Fontainebleau sandstone and a boron
carbide/aluminum composite from the two- dimensional tomographic images of
their slices through the materials. To ascertain whether the information
contained in S2 is sufficient to capture the salient structural features, we
compute the two-point cluster functions of the media, which are superior
signatures of the micro-structure because they incorporate the connectedness
information. We also study the reconstruction of a binary laser-speckle pattern
in two dimensions, in which the algorithm fails to reproduce the pattern
accurately. We conclude that in general reconstructions using S2 only work well
for heterogeneous materials with single-scale structures. However, two-point
information via S2 is not sufficient to accurately model multi-scale media.
Moreover, we construct realizations of hypothetical materials with desired
structural characteristics obtained by manipulating their two-point correlation
functions.Comment: 35 pages, 19 figure
Inverse Statistical Mechanics: Probing the Limitations of Isotropic Pair Potentials to Produce Ground-State Structural Extremes
Inverse statistical-mechanical methods have recently been employed to design
optimized short-ranged radial (isotropic) pair potentials that robustly produce
novel targeted classical ground-state many-particle configurations. The target
structures considered in those studies were low-coordinated crystals with a
high degree of symmetry. In this paper, we further test the fundamental
limitations of radial pair potentials by targeting crystal structures with
appreciably less symmetry, including those in which the particles have
different local structural environments. These challenging target
configurations demanded that we modify previous inverse optimization
techniques. Using this modified optimization technique, we have designed
short-ranged radial pair potentials that stabilize the two-dimensional kagome
crystal, the rectangular kagome crystal, and rectangular lattices, as well as
the three-dimensional structure of CaF crystal inhabited by a single
particle species. We verify our results by cooling liquid configurations to
absolute zero temperature via simulated annealing and ensuring that such states
have stable phonon spectra. Except for the rectangular kagome structure, all of
the target structures can be stabilized with monotonic repulsive potentials.
Our work demonstrates that single-component systems with short-ranged radial
pair potentials can counterintuitively self-assemble into crystal ground states
with low symmetry and different local structural environments. Finally, we
present general principles that offer guidance in determining whether certain
target structures can be achieved as ground states by radial pair potentials
Classical many-particle systems with unique disordered ground states
Classical ground states (global energy-minimizing configurations) of
many-particle systems are typically unique crystalline structures, implying
zero enumeration entropy of distinct patterns (aside from trivial symmetry
operations). By contrast, the few previously known disordered classical ground
states of many-particle systems are all high-entropy (highly degenerate)
states. Here we show computationally that our recently-proposed "perfect-glass"
many-particle model [Sci. Rep., 6, 36963 (2016)] possesses disordered classical
ground states with a zero entropy: a highly counterintuitive situation. For all
of the system sizes, parameters, and space dimensions that we have numerically
investigated, the disordered ground states are unique such that they can always
be superposed onto each other or their mirror image. At low energies, the
density of states obtained from simulations matches those calculated from the
harmonic approximation near a single ground state, further confirming
ground-state uniqueness. Our discovery provides singular examples in which
entropy and disorder are at odds with one another. The zero-entropy ground
states provide a unique perspective on the celebrated Kauzmann-entropy crisis
in which the extrapolated entropy of a supercooled liquid drops below that of
the crystal. We expect that our disordered unique patterns to be of value in
fields beyond glass physics, including applications in cryptography as
pseudo-random functions with tunable computational complexity
Transport, Geometrical and Topological Properties of Stealthy Disordered Hyperuniform Two-Phase Systems
Disordered hyperuniform many-particle systems have attracted considerable
recent attention. One important class of such systems is the classical ground
states of "stealthy potentials." The degree of order of such ground states
depends on a tuning parameter. Previous studies have shown that these
ground-state point configurations can be counterintuitively disordered,
infinitely degenerate, and endowed with novel physical properties (e.g.,
negative thermal expansion behavior). In this paper, we focus on the disordered
regime in which there is no long-range order, and control the degree of
short-range order. We map these stealthy disordered hyperuniform point
configurations to two-phase media by circumscribing each point with a possibly
overlapping sphere of a common radius : the "particle" and "void" phases are
taken to be the space interior and exterior to the spheres, respectively. We
study certain transport properties of these systems, including the effective
diffusion coefficient of point particles diffusing in the void phase as well as
static and time-dependent characteristics associated with diffusion-controlled
reactions. Besides these effective transport properties, we also investigate
several related structural properties, including pore-size functions, quantizer
error, an order metric, and percolation threshold. We show that these
transport, geometrical and topological properties of our two-phase media
derived from decorated stealthy ground states are distinctly different from
those of equilibrium hard-sphere systems and spatially uncorrelated overlapping
spheres
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