3,895 research outputs found
Detecting independence of random vectors: generalized distance covariance and Gaussian covariance
Distance covariance is a quantity to measure the dependence of two random
vectors. We show that the original concept introduced and developed by
Sz\'{e}kely, Rizzo and Bakirov can be embedded into a more general framework
based on symmetric L\'{e}vy measures and the corresponding real-valued
continuous negative definite functions. The L\'{e}vy measures replace the
weight functions used in the original definition of distance covariance. All
essential properties of distance covariance are preserved in this new
framework. From a practical point of view this allows less restrictive moment
conditions on the underlying random variables and one can use other distance
functions than Euclidean distance, e.g. Minkowski distance. Most importantly,
it serves as the basic building block for distance multivariance, a quantity to
measure and estimate dependence of multiple random vectors, which is introduced
in a follow-up paper [Distance Multivariance: New dependence measures for
random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the
present article.Comment: Published at https://doi.org/10.15559/18-VMSTA116 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
Strong Effect of Weak Charging in Suspensions of Anisotropic Colloids
Suspensions of hard colloidal particles frequently serve as model systems in
studies on fundamental aspects of phase transitions. But often colloidal
particles that are considered as ``hard'' are in fact weakly charged. If the
colloids are spherical, weak charging has a only a weak effect on the
structural properties of the suspension, which can be easily corrected for.
However, this does not hold for anisotropic particles.
We introduce a model for the interaction potential between charged ellipsoids
of revolution (spheroids) based on the Derjaguin approximation of
Debye--H\"uckel Theory and present a computer simulation study on aspects of
the system's structural properties and phase behaviour. In line with previous
experimental observations, we find that even a weak surface charge has a strong
impact on the correlation functions. A likewise strong impact is seen on the
phase behaviour, in particular, we find stable cubatic order in suspensions of
oblate ellipsoids
Solid phase properties and crystallization in simple model systems
We review theoretical and simulational approaches to the description of
equilibrium bulk crystal and interface properties as well as to the
nonequilibrium processes of homogeneous and heterogeneous crystal nucleation
for the simple model systems of hard spheres and Lennard-Jones particles. For
the equilibrium properties of bulk and interfaces, density functional theories
employing fundamental measure functionals prove to be a precise and versatile
tool, as exemplified with a closer analysis of the hard sphere crystalliquid
interface. A detailed understanding of the dynamic process of nucleation in
these model systems nevertheless still relies on simulational approaches. We
review bulk nucleation and nucleation at structured walls and examine in closer
detail the influence of walls with variable strength on nucleation in the
Lennard-Jones fluid. We find that a planar crystalline substrate induces the
growth of a crystalline film for a large range of lattice spacings and
interaction potentials. Only a strongly incommensurate substrate and a very
weakly attractive substrate potential lead to crystal growth with a non-zero
contact angle
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