1,121 research outputs found
Cell shape analysis of random tessellations based on Minkowski tensors
To which degree are shape indices of individual cells of a tessellation
characteristic for the stochastic process that generates them? Within the
context of stochastic geometry and the physics of disordered materials, this
corresponds to the question of relationships between different stochastic
models. In the context of image analysis of synthetic and biological materials,
this question is central to the problem of inferring information about
formation processes from spatial measurements of resulting random structures.
We address this question by a theory-based simulation study of shape indices
derived from Minkowski tensors for a variety of tessellation models. We focus
on the relationship between two indices: an isoperimetric ratio of the
empirical averages of cell volume and area and the cell elongation quantified
by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for
these quantities, as well as for distributions thereof and for correlations of
cell shape and volume, are presented for Voronoi mosaics of the Poisson point
process, determinantal and permanental point processes, and Gibbs hard-core and
random sequential absorption processes as well as for Laguerre tessellations of
polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data
are complemented by mechanically stable crystalline sphere and disordered
ellipsoid packings and area-minimising foam models. We find that shape indices
of individual cells are not sufficient to unambiguously identify the generating
process even amongst this limited set of processes. However, we identify
significant differences of the shape indices between many of these tessellation
models. Given a realization of a tessellation, these shape indices can narrow
the choice of possible generating processes, providing a powerful tool which
can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their
Applications in Stochastic Geometry and Imaging" in Lecture Notes in
Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense
Entanglement in fermion systems and quantum metrology
Entanglement in fermion many-body systems is studied using a generalized
definition of separability based on partitions of the set of observables,
rather than on particle tensor products. In this way, the characterizing
properties of non-separable fermion states can be explicitly analyzed, allowing
a precise description of the geometric structure of the corresponding state
space. These results have direct applications in fermion quantum metrology:
sub-shot noise accuracy in parameter estimation can be obtained without the
need of a preliminary state entangling operation.Comment: 26 pages, LaTe
An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit
We construct an extensive adiabatic invariant for a Klein-Gordon chain in the
thermodynamic limit. In particular, given a fixed and sufficiently small value
of the coupling constant , the evolution of the adiabatic invariant is
controlled up to times scaling as for any large enough
value of the inverse temperature . The time scale becomes a stretched
exponential if the coupling constant is allowed to vanish jointly with the
specific energy. The adiabatic invariance is exhibited by showing that the
variance along the dynamics, i.e. calculated with respect to time averages, is
much smaller than the corresponding variance over the whole phase space, i.e.
calculated with the Gibbs measure, for a set of initial data of large measure.
All the perturbative constructions and the subsequent estimates are consistent
with the extensive nature of the system.Comment: 60 pages. Minor corrections with respect to the first version. To
appear in Annales Henri Poincar\'
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