700 research outputs found
A Multiscale Guide to Brownian Motion
We revise the Levy's construction of Brownian motion as a simple though still
rigorous approach to operate with various Gaussian processes. A Brownian path
is explicitly constructed as a linear combination of wavelet-based "geometrical
features" at multiple length scales with random weights. Such a wavelet
representation gives a closed formula mapping of the unit interval onto the
functional space of Brownian paths. This formula elucidates many classical
results about Brownian motion (e.g., non-differentiability of its path),
providing intuitive feeling for non-mathematicians. The illustrative character
of the wavelet representation, along with the simple structure of the
underlying probability space, is different from the usual presentation of most
classical textbooks. Similar concepts are discussed for fractional Brownian
motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional
Gaussian fields. Wavelet representations and dyadic decompositions form the
basis of many highly efficient numerical methods to simulate Gaussian processes
and fields, including Brownian motion and other diffusive processes in
confining domains
Unsupervised landmark analysis for jump detection in molecular dynamics simulations
Molecular dynamics is a versatile and powerful method to study diffusion in
solid-state ionic conductors, requiring minimal prior knowledge of equilibrium
or transition states of the system's free energy surface. However, the analysis
of trajectories for relevant but rare events, such as a jump of the diffusing
mobile ion, is still rather cumbersome, requiring prior knowledge of the
diffusive process in order to get meaningful results. In this work, we present
a novel approach to detect the relevant events in a diffusive system without
assuming prior information regarding the underlying process. We start from a
projection of the atomic coordinates into a landmark basis to identify the
dominant features in a mobile ion's environment. Subsequent clustering in
landmark space enables a discretization of any trajectory into a sequence of
distinct states. As a final step, the use of the smooth overlap of atomic
positions descriptor allows distinguishing between different environments in a
straightforward way. We apply this algorithm to ten Li-ionic systems and
conduct in-depth analyses of cubic LiLaZrO, tetragonal
LiGePS, and the -eucryptite LiAlSiO. We
compare our results to existing methods, underscoring strong points,
weaknesses, and insights into the diffusive behavior of the ionic conduction in
the materials investigated
IST Austria Thesis
In this Thesis, I study composite quantum impurities with variational techniques, both inspired by machine learning as well as fully analytic. I supplement this with exploration of other applications of machine learning, in particular artificial neural networks, in many-body physics. In Chapters 3 and 4, I study quasiparticle systems with variational approach. I derive a Hamiltonian describing the angulon quasiparticle in the presence of a magnetic field. I apply analytic variational treatment to this Hamiltonian. Then, I introduce a variational approach for non-additive systems, based on artificial neural networks. I exemplify this approach on the example of the polaron quasiparticle (Fröhlich Hamiltonian). In Chapter 5, I continue using artificial neural networks, albeit in a different setting. I apply artificial neural networks to detect phases from snapshots of two types physical systems. Namely, I study Monte Carlo snapshots of multilayer classical spin models as well as molecular dynamics maps of colloidal systems. The main type of networks that I use here are convolutional neural networks, known for their applicability to image data
Nucleation of a sodium droplet on C60
We investigate theoretically the progressive coating of C60 by several sodium
atoms. Density functional calculations using a nonlocal functional are
performed for NaC60 and Na2C60 in various configurations. These data are used
to construct an empirical atomistic model in order to treat larger sizes in a
statistical and dynamical context. Fluctuating charges are incorporated to
account for charge transfer between sodium and carbon atoms. By performing
systematic global optimization in the size range 1<=n<=30, we find that Na_nC60
is homogeneously coated at small sizes, and that a growing droplet is formed
above n=>8. The separate effects of single ionization and thermalization are
also considered, as well as the changes due to a strong external electric
field. The present results are discussed in the light of various experimental
data.Comment: 17 pages, 10 figure
Spectral modeling of type II supernovae. I. Dilution factors
We present substantial extensions to the Monte Carlo radiative transfer code
TARDIS to perform spectral synthesis for type II supernovae. By incorporating a
non-LTE ionization and excitation treatment for hydrogen, a full account of
free-free and bound-free processes, a self-consistent determination of the
thermal state and by improving the handling of relativistic effects, the
improved code version includes the necessary physics to perform spectral
synthesis for type II supernovae to high precision as required for the reliable
inference of supernova properties. We demonstrate the capabilities of the
extended version of TARDIS by calculating synthetic spectra for the
prototypical type II supernova SN1999em and by deriving a new and independent
set of dilution factors for the expanding photosphere method. We have
investigated in detail the dependence of the dilution factors on photospheric
properties and, for the first time, on changes in metallicity. We also compare
our results with two previously published sets of dilution factors by Eastman
et al. (1996) and by Dessart & Hillier (2005), and discuss the potential
sources of the discrepancies between studies.Comment: 16 pages, 12 figures, 2 tables, accepted for publication in A&
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