134,640 research outputs found
Meso-scale turbulence in living fluids
Turbulence is ubiquitous, from oceanic currents to small-scale biological and
quantum systems. Self-sustained turbulent motion in microbial suspensions
presents an intriguing example of collective dynamical behavior amongst the
simplest forms of life, and is important for fluid mixing and molecular
transport on the microscale. The mathematical characterization of turbulence
phenomena in active non-equilibrium fluids proves even more difficult than for
conventional liquids or gases. It is not known which features of turbulent
phases in living matter are universal or system-specific, or which
generalizations of the Navier-Stokes equations are able to describe them
adequately. Here, we combine experiments, particle simulations, and continuum
theory to identify the statistical properties of self-sustained meso-scale
turbulence in active systems. To study how dimensionality and boundary
conditions affect collective bacterial dynamics, we measured energy spectra and
structure functions in dense Bacillus subtilis suspensions in quasi-2D and 3D
geometries. Our experimental results for the bacterial flow statistics agree
well with predictions from a minimal model for self-propelled rods, suggesting
that at high concentrations the collective motion of the bacteria is dominated
by short-range interactions. To provide a basis for future theoretical studies,
we propose a minimal continuum model for incompressible bacterial flow. A
detailed numerical analysis of the 2D case shows that this theory can reproduce
many of the experimentally observed features of self-sustained active
turbulence.Comment: accepted PNAS version, 6 pages, click doi for Supplementary
Informatio
Probabilistic Infinite Secret Sharing
The study of probabilistic secret sharing schemes using arbitrary probability
spaces and possibly infinite number of participants lets us investigate
abstract properties of such schemes. It highlights important properties,
explains why certain definitions work better than others, connects this topic
to other branches of mathematics, and might yield new design paradigms.
A probabilistic secret sharing scheme is a joint probability distribution of
the shares and the secret together with a collection of secret recovery
functions for qualified subsets. The scheme is measurable if the recovery
functions are measurable. Depending on how much information an unqualified
subset might have, we define four scheme types: perfect, almost perfect, ramp,
and almost ramp. Our main results characterize the access structures which can
be realized by schemes of these types.
We show that every access structure can be realized by a non-measurable
perfect probabilistic scheme. The construction is based on a paradoxical pair
of independent random variables which determine each other.
For measurable schemes we have the following complete characterization. An
access structure can be realized by a (measurable) perfect, or almost perfect
scheme if and only if the access structure, as a subset of the Sierpi\'nski
space , is open, if and only if it can be realized by a span
program. The access structure can be realized by a (measurable) ramp or almost
ramp scheme if and only if the access structure is a set
(intersection of countably many open sets) in the Sierpi\'nski topology, if and
only if it can be realized by a Hilbert-space program
Projective Techniques and Functional Integration
A general framework for integration over certain infinite dimensional spaces
is first developed using projective limits of a projective family of compact
Hausdorff spaces. The procedure is then applied to gauge theories to carry out
integration over the non-linear, infinite dimensional spaces of connections
modulo gauge transformations. This method of evaluating functional integrals
can be used either in the Euclidean path integral approach or the Lorentzian
canonical approach. A number of measures discussed are diffeomorphism invariant
and therefore of interest to (the connection dynamics version of) quantum
general relativity. The account is pedagogical; in particular prior knowledge
of projective techniques is not assumed. (For the special JMP issue on
Functional Integration, edited by C. DeWitt-Morette.)Comment: 36 pages, latex, no figures, Preprint CGPG/94/10-
Noncommutative Lattices and Their Continuum Limits
We consider finite approximations of a topological space by
noncommutative lattices of points. These lattices are structure spaces of
noncommutative -algebras which in turn approximate the algebra \cc(M) of
continuous functions on . We show how to recover the space and the
algebra \cc(M) from a projective system of noncommutative lattices and an
inductive system of noncommutative -algebras, respectively.Comment: 22 pages, 8 Figures included in the LaTeX Source New version, minor
modifications (typos corrected) and a correction in the list of author
Non-local energetics of random heterogeneous lattices
In this paper, we study the mechanics of statistically non-uniform two-phase
elastic discrete structures. In particular, following the methodology proposed
in (Luciano and Willis, Journal of the Mechanics and Physics of Solids 53,
1505-1522, 2005), energetic bounds and estimates of the Hashin-Shtrikman-Willis
type are developed for discrete systems with a heterogeneity distribution
quantified by second-order spatial statistics. As illustrated by three
numerical case studies, the resulting expressions for the ensemble average of
the potential energy are fully explicit, computationally feasible and free of
adjustable parameters. Moreover, the comparison with reference Monte-Carlo
simulations confirms a notable improvement in accuracy with respect to
approaches based solely on the first-order statistics.Comment: 32 pages, 8 figure
Continuum limit of amorphous elastic bodies: A finite-size study of low frequency harmonic vibrations
The approach of the elastic continuum limit in small amorphous bodies formed
by weakly polydisperse Lennard-Jones beads is investigated in a systematic
finite-size study. We show that classical continuum elasticity breaks down when
the wavelength of the sollicitation is smaller than a characteristic length of
approximately 30 molecular sizes. Due to this surprisingly large effect
ensembles containing up to N=40,000 particles have been required in two
dimensions to yield a convincing match with the classical continuum predictions
for the eigenfrequency spectrum of disk-shaped aggregates and periodic bulk
systems. The existence of an effective length scale \xi is confirmed by the
analysis of the (non-gaussian) noisy part of the low frequency vibrational
eigenmodes. Moreover, we relate it to the {\em non-affine} part of the
displacement fields under imposed elongation and shear. Similar correlations
(vortices) are indeed observed on distances up to \xi~30 particle sizes.Comment: 28 pages, 13 figures, 3 table
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