7,499 research outputs found
A Simple Derivation of the Refined Sphere Packing Bound Under Certain Symmetry Hypotheses
A judicious application of the Berry-Esseen theorem via suitable Augustin
information measures is demonstrated to be sufficient for deriving the sphere
packing bound with a prefactor that is
for all codes on certain
families of channels -- including the Gaussian channels and the non-stationary
Renyi symmetric channels -- and for the constant composition codes on
stationary memoryless channels. The resulting non-asymptotic bounds have
definite approximation error terms. As a preliminary result that might be of
interest on its own, the trade-off between type I and type II error
probabilities in the hypothesis testing problem with (possibly non-stationary)
independent samples is determined up to some multiplicative constants, assuming
that the probabilities of both types of error are decaying exponentially with
the number of samples, using the Berry-Esseen theorem.Comment: 20 page
Packing-Limited Growth
We consider growing spheres seeded by random injection in time and space.
Growth stops when two spheres meet leading eventually to a jammed state. We
study the statistics of growth limited by packing theoretically in d dimensions
and via simulation in d=2, 3, and 4. We show how a broad class of such models
exhibit distributions of sphere radii with a universal exponent. We construct a
scaling theory that relates the fractal structure of these models to the decay
of their pore space, a theory that we confirm via numerical simulations. The
scaling theory also predicts an upper bound for the universal exponent and is
in exact agreement with numerical results for d=4.Comment: 6 pages, 5 figures, 4 tables, revtex4 to appear in Phys. Rev. E, May
200
Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology
We present a model of (modified) gravity on spacetimes with fractal structure
based on packing of spheres, which are (Euclidean) variants of the Packed Swiss
Cheese Cosmology models. As the action functional for gravity we consider the
spectral action of noncommutative geometry, and we compute its expansion on a
space obtained as an Apollonian packing of 3-dimensional spheres inside a
4-dimensional ball. Using information from the zeta function of the Dirac
operator of the spectral triple, we compute the leading terms in the asymptotic
expansion of the spectral action. They consist of a zeta regularization of a
divergent sum which involves the leading terms of the spectral actions of the
individual spheres in the packing. This accounts for the contribution of the
points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There
is an additional term coming from the residue at the additional point in the
real dimension spectrum that corresponds to the packing constant, as well as a
series of fluctuations coming from log-periodic oscillations, created by the
points of the dimension spectrum that are off the real line. These terms detect
the fractality of the residue set of the sphere packing. We show that the
presence of fractality influences the shape of the slow-roll potential for
inflation, obtained from the spectral action. We also discuss the effect of
truncating the fractal structure at a certain scale related to the energy scale
in the spectral action.Comment: 38 pages LaTe
Tagged-particle dynamics in a hard-sphere system: mode-coupling theory analysis
The predictions of the mode-coupling theory of the glass transition (MCT) for
the tagged-particle density-correlation functions and the mean-squared
displacement curves are compared quantitatively and in detail to results from
Newtonian- and Brownian-dynamics simulations of a polydisperse
quasi-hard-sphere system close to the glass transition. After correcting for a
17% error in the dynamical length scale and for a smaller error in the
transition density, good agreement is found over a wide range of wave numbers
and up to five orders of magnitude in time. Deviations are found at the highest
densities studied, and for small wave vectors and the mean-squared
displacement. Possible error sources not related to MCT are discussed in
detail, thereby identifying more clearly the issues arising from the MCT
approximation itself. The range of applicability of MCT for the different types
of short-time dynamics is established through asymptotic analyses of the
relaxation curves, examining the wave-number and density-dependent
characteristic parameters. Approximations made in the description of the
equilibrium static structure are shown to have a remarkable effect on the
predicted numerical value for the glass-transition density. Effects of small
polydispersity are also investigated, and shown to be negligible.Comment: 20 pages, 23 figure
Packing Hyperspheres in High-Dimensional Euclidean Spaces
We present the first study of disordered jammed hard-sphere packings in
four-, five- and six-dimensional Euclidean spaces. Using a collision-driven
packing generation algorithm, we obtain the first estimates for the packing
fractions of the maximally random jammed (MRJ) states for space dimensions
, 5 and 6 to be , 0.31 and 0.20, respectively. To
a good approximation, the MRJ density obeys the scaling form , where and , which appears to be
consistent with high-dimensional asymptotic limit, albeit with different
coefficients. Calculations of the pair correlation function and
structure factor for these states show that short-range ordering
appreciably decreases with increasing dimension, consistent with a recently
proposed ``decorrelation principle,'' which, among othe things, states that
unconstrained correlations diminish as the dimension increases and vanish
entirely in the limit . As in three dimensions (where ), the packings show no signs of crystallization, are isostatic,
and have a power-law divergence in at contact with power-law
exponent . Across dimensions, the cumulative number of neighbors
equals the kissing number of the conjectured densest packing close to where
has its first minimum. We obtain estimates for the freezing and
melting desnities for the equilibrium hard-sphere fluid-solid transition,
and , respectively, for , and
and , respectively, for .Comment: 28 pages, 9 figures. To appear in Physical Review
Asymptotic laws for tagged-particle motion in glassy systems
Within the mode-coupling theory for structural relaxation in simple systems
the asymptotic laws and their leading-asymptotic correction formulas are
derived for the motion of a tagged particle near a glass-transition
singularity. These analytic results are compared with numerical ones of the
equations of motion evaluated for a tagged hard sphere moving in a hard-sphere
system. It is found that the long-time part of the two-step relaxation process
for the mean-squared displacement can be characterized by the -relaxation-scaling law and von Schweidler's power-law decay while the
critical-decay regime is dominated by the corrections to the leading power-law
behavior. For parameters of interest for the interpretations of experimental
data, the corrections to the leading asymptotic laws for the non-Gaussian
parameter are found to be so large that the leading asymptotic results are
altered qualitatively by the corrections. Results for the non-Gaussian
parameter are shown to follow qualitatively the findings reported in the
molecular-dynamics-simulations work by Kob and Andersen [Phys. Rev. E 51, 4626
(1995)]
Finite Dimensional Infinite Constellations
In the setting of a Gaussian channel without power constraints, proposed by
Poltyrev, the codewords are points in an n-dimensional Euclidean space (an
infinite constellation) and the tradeoff between their density and the error
probability is considered. The capacity in this setting is the highest
achievable normalized log density (NLD) with vanishing error probability. This
capacity as well as error exponent bounds for this setting are known. In this
work we consider the optimal performance achievable in the fixed blocklength
(dimension) regime. We provide two new achievability bounds, and extend the
validity of the sphere bound to finite dimensional infinite constellations. We
also provide asymptotic analysis of the bounds: When the NLD is fixed, we
provide asymptotic expansions for the bounds that are significantly tighter
than the previously known error exponent results. When the error probability is
fixed, we show that as n grows, the gap to capacity is inversely proportional
(up to the first order) to the square-root of n where the proportion constant
is given by the inverse Q-function of the allowed error probability, times the
square root of 1/2. In an analogy to similar result in channel coding, the
dispersion of infinite constellations is 1/2nat^2 per channel use. All our
achievability results use lattices and therefore hold for the maximal error
probability as well. Connections to the error exponent of the power constrained
Gaussian channel and to the volume-to-noise ratio as a figure of merit are
discussed. In addition, we demonstrate the tightness of the results numerically
and compare to state-of-the-art coding schemes.Comment: 54 pages, 13 figures. Submitted to IEEE Transactions on Information
Theor
Atomic Transport in Dense, Multi-Component Metallic Liquids
Pd43Ni10Cu27P0 has been investigated in its equilibrium liquid state with
incoherent, inelastic neutron scattering. As compared to simple liquids, liquid
PdNiCuP is characterized by a dense packing with a packing fraction above 0.5.
The intermediate scattering function exhibits a fast relaxation process that
precedes structural relaxation. Structural relaxation obeys a time-temperature
superposition that extends over a temperature range of 540K. The mode-coupling
theory of the liquid to glass transition (MCT) gives a consistent description
of the dynamics which governs the mass transport in liquid PdNiCuP alloys. MCT
scaling laws extrapolate to a critical temperature Tc at about 20% below the
liquidus temperature. Diffusivities derived from the mean relaxation times
compare well with Co diffusivities from recent tracer diffusion measurements
and diffsuivities calculated from viscosity via the Stokes-Einstein relation.
In contrast to simple metallic liquids, the atomic transport in dense, liquid
PdNiCuP is characterized by a drastical slowing down of dynamics on cooling, a
q^{-2} dependence of the mean relaxation times at intermediate q and a
vanishing isotope effect as a result of a highly collective transport
mechanism. At temperatures as high as 2Tc diffusion in liquid PdNiCuP is as
fast as in simple liquids at the melting point. However, the difference in the
underlying atomic transport mechanism indicates that the diffusion mechanism in
liquids is not controlled by the value of the diffusivity but rather by that of
the packing fraction
Glass transition of hard spheres in high dimensions
We have investigated analytically and numerically the liquid-glass transition
of hard spheres for dimensions in the framework of mode-coupling
theory. The numerical results for the critical collective and self
nonergodicity parameters and exhibit
non-Gaussian -dependence even up to . and
differ for , but become identical on a scale
, which is proven analytically. The critical packing fraction
is above the corresponding Kauzmann packing
fraction derived by a small cage expansion. Its quadratic
pre-exponential factor is different from the linear one found earlier. The
numerical values for the exponent parameter and therefore the critical
exponents and depend on , even for the largest values of .Comment: 11 pages, 8 figures, Phys. Rev. E (in print
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