237,409 research outputs found
Catastrophic Particle Production under Periodic Perturbation
We develop a formalism to investigate the behavior of quantum field and
quantum ground state when the field is coupled to perturbation that
periodically oscillates. Working in the Schroedinger picture of quantum field
theory, we confirm that the phenomenon of parametric resonance in the classical
theory implies an instability of quantum vacuum, and correspondingly it gives
rise to catastrophic particle production if the oscillation lasts indefinitely;
the produced number of particles exponentially increases without bound as time
proceeds. The density matrix describing the limiting stage of the quantum state
is determined by a small set of parameters. Moreover, the energy spectrum and
the intensity of produced particles are worked out in greatest detail in the
limit of weak coupling or small amplitude perturbation. In the case of strong
coupling or large amplitude perturbation the leading adiabatic formula is
derived. Application to cosmological fate of weakly interacting spinless fields
(WISF) such as the invisible axion, the Polonyi, and the modular fields is
discussed. Although very little effect is expected on the invisible axion, the
Polonyi type field has a chance that it catastrophically decays at an early
epoch without much production of entropy, provided that an intrinsic coupling
is large enough.Comment: 33 page
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
An overview of the proper generalized decomposition with applications in computational rheology
We review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3] and [4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shell-like domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates
Bayesian optimisation for likelihood-free cosmological inference
Many cosmological models have only a finite number of parameters of interest,
but a very expensive data-generating process and an intractable likelihood
function. We address the problem of performing likelihood-free Bayesian
inference from such black-box simulation-based models, under the constraint of
a very limited simulation budget (typically a few thousand). To do so, we adopt
an approach based on the likelihood of an alternative parametric model.
Conventional approaches to approximate Bayesian computation such as
likelihood-free rejection sampling are impractical for the considered problem,
due to the lack of knowledge about how the parameters affect the discrepancy
between observed and simulated data. As a response, we make use of a strategy
previously developed in the machine learning literature (Bayesian optimisation
for likelihood-free inference, BOLFI), which combines Gaussian process
regression of the discrepancy to build a surrogate surface with Bayesian
optimisation to actively acquire training data. We extend the method by
deriving an acquisition function tailored for the purpose of minimising the
expected uncertainty in the approximate posterior density, in the parametric
approach. The resulting algorithm is applied to the problems of summarising
Gaussian signals and inferring cosmological parameters from the Joint
Lightcurve Analysis supernovae data. We show that the number of required
simulations is reduced by several orders of magnitude, and that the proposed
acquisition function produces more accurate posterior approximations, as
compared to common strategies.Comment: 16+9 pages, 12 figures. Matches PRD published version after minor
modification
Quantum System under Periodic Perturbation: Effect of Environment
In many physical situations the behavior of a quantum system is affected by
interaction with a larger environment. We develop, using the method of
influence functional, how to deduce the density matrix of the quantum system
incorporating the effect of environment. After introducing characterization of
the environment by spectral weight, we first devise schemes to approximate the
spectral weight, and then a perturbation method in field theory models, in
order to approximately describe the environment. All of these approximate
models may be classified as extended Ohmic models of dissipation whose
differences are in the high frequency part.
The quantum system we deal with in the present work is a general class of
harmonic oscillators with arbitrary time dependent frequency. The late time
behavior of the system is well described by an approximation that employs a
localized friction in the dissipative part of the correlation function
appearing in the influence functional. The density matrix of the quantum system
is then determined in terms of a single classical solution obtained with the
time dependent frequency. With this one can compute the entropy, the energy
distribution function, and other physical quantities of the system in a closed
form.
Specific application is made to the case of periodically varying frequency.
This dynamical system has a remarkable property when the environmental
interaction is switched off: Effect of the parametric resonance gives rise to
an exponential growth of the populated number in higher excitation levels, or
particle production in field theory models. The effect of the environment is
investigated for this dynamical system and it is demonstrated that there existsComment: 55 pages, LATEX file plus 13 PS figures. A few calculational
mistatkes and corresponding figure 1 in field theory model corrected and some
changes made for publication in Phys. Rev.D (in press
Semiparametric Bayesian inference in multiple equation models
This paper outlines an approach to Bayesian semiparametric regression in multiple equation models which can be used to carry out inference in seemingly unrelated regressions or simultaneous equations models with nonparametric components. The approach treats the points on each nonparametric regression line as unknown parameters and uses a prior on the degree of smoothness of each line to ensure valid posterior inference despite the fact that the number of parameters is greater than the number of observations. We develop an empirical Bayesian approach that allows us to estimate the prior smoothing hyperparameters from the data. An advantage of our semiparametric model is that it is written as a seemingly unrelated regressions model with independent normal-Wishart prior. Since this model is a common one, textbook results for posterior inference, model comparison, prediction and posterior computation are immediately available. We use this model in an application involving a two-equation structural model drawn from the labour and returns to schooling literatures
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
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