1,051 research outputs found
Forced Burgers Equation in an Unbounded Domain
The inviscid Burgers equation with random and spatially smooth forcing is
considered in the limit when the size of the system tends to infinity. For the
one-dimensional problem, it is shown both theoretically and numerically that
many of the features of the space-periodic case carry over to infinite domains
as intermediate time asymptotics. In particular, for large time we
introduce the concept of -global shocks replacing the notion of main shock
which was considered earlier in the periodic case (1997, E et al., Phys. Rev.
Lett. 78, 1904). In the case of spatially extended systems these objects are no
anymore global. They can be defined only for a given time scale and their
spatial density behaves as for large . The
probability density function of the age of shocks behaves
asymptotically as . We also suggest a simple statistical model for
the dynamics and interaction of shocks and discuss an analogy with the problem
of distribution of instability islands for a simple first-order stochastic
differential equation.Comment: 9 pages, 10 figures, revtex4, J. Stat. Phys, in pres
Burgers Turbulence
The last decades witnessed a renewal of interest in the Burgers equation.
Much activities focused on extensions of the original one-dimensional
pressureless model introduced in the thirties by the Dutch scientist J.M.
Burgers, and more precisely on the problem of Burgers turbulence, that is the
study of the solutions to the one- or multi-dimensional Burgers equation with
random initial conditions or random forcing. Such work was frequently motivated
by new emerging applications of Burgers model to statistical physics,
cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the
simplest instances of a nonlinear system out of equilibrium. The study of
random Lagrangian systems, of stochastic partial differential equations and
their invariant measures, the theory of dynamical systems, the applications of
field theory to the understanding of dissipative anomalies and of multiscaling
in hydrodynamic turbulence have benefited significantly from progress in
Burgers turbulence. The aim of this review is to give a unified view of
selected work stemming from these rather diverse disciplines.Comment: Review Article, 49 pages, 43 figure
A Bayesian self-clustering analysis of the highest energy cosmic rays detected by the Pierre Auger Observatory
Cosmic rays (CRs) are protons and atomic nuclei that flow into our Solar
system and reach the Earth with energies of up to ~10^21 eV. The sources of
ultra-high energy cosmic rays (UHECRs) with E >~ 10^19 eV remain unknown,
although there are theoretical reasons to think that at least some come from
active galactic nuclei (AGNs). One way to assess the different hypotheses is by
analysing the arrival directions of UHECRs, in particular their
self-clustering. We have developed a fully Bayesian approach to analyzing the
self-clustering of points on the sphere, which we apply to the UHECR arrival
directions. The analysis is based on a multi-step approach that enables the
application of Bayesian model comparison to cases with weak prior information.
We have applied this approach to the 69 highest energy events recorded by the
Pierre Auger Observatory (PAO), which is the largest current UHECR data set. We
do not detect self-clustering, but simulations show that this is consistent
with the AGN-sourced model for a data set of this size. Data sets of several
hundred UHECRs would be sufficient to detect clustering in the AGN model.
Samples of this magnitude are expected to be produced by future experiments,
such as the Japanese Experiment Module Extreme Universe Space Observatory
(JEM-EUSO).Comment: 10 pages, 4 figures; accepted in MNRA
On hyperbolicity of minimizers for 1D random Lagrangian systems
We prove hyperbolicity of global minimizers for random Lagrangian systems in
dimension 1. The proof considerably simplifies a related result in [2]. The
conditions for hyperbolicity are almost optimal: they are essentially the same
as conditions for uniqueness of a global minimizer in [3]
A Bayesian analysis of the 69 highest energy cosmic rays detected by the Pierre Auger Observatory
The origins of ultra-high energy cosmic rays (UHECRs) remain an open
question. Several attempts have been made to cross-correlate the arrival
directions of the UHECRs with catalogs of potential sources, but no definite
conclusion has been reached. We report a Bayesian analysis of the 69 events
from the Pierre Auger Observatory (PAO), that aims to determine the fraction of
the UHECRs that originate from known AGNs in the Veron-Cety & Veron (VCV)
catalog, as well as AGNs detected with the Swift Burst Alert Telescope
(Swift-BAT), galaxies from the 2MASS Redshift Survey (2MRS), and an additional
volume-limited sample of 17 nearby AGNs. The study makes use of a multi-level
Bayesian model of UHECR injection, propagation and detection. We find that for
reasonable ranges of prior parameters, the Bayes factors disfavour a purely
isotropic model. For fiducial values of the model parameters, we report 68%
credible intervals for the fraction of source originating UHECRs of
0.09+0.05-0.04, 0.25+0.09-0.08, 0.24+0.12-0.10, and 0.08+0.04-0.03 for the VCV,
Swift-BAT and 2MRS catalogs, and the sample of 17 AGNs, respectively
Mathematical and computational modelling of post-transcriptional gene relation by micro-RNA
Mathematical models and computational simulations have proved valuable in many areas of cell biology, including gene regulatory networks. When properly calibrated against experimental data, kinetic models can be used to describe how the concentrations of key species evolve over time. A reliable model allows âwhat ifâ scenarios to be investigated quantitatively in silico, and also provides a means to compare competing hypotheses about the underlying biological mechanisms at work. Moreover, models at different scales of resolution can be merged into a bigger picture âsystemsâ level description. In the case where gene regulation is post-transcriptionally affected by microRNAs, biological understanding and experimental techniques have only recently matured to the extent that we can postulate and test kinetic models. In this chapter, we summarize some recent work that takes the first steps towards realistic modelling, focusing on the contributions of the authors. Using a deterministic ordinary differential equation framework, we derive models from first principles and test them for consistency with recent experimental data, including microarray and mass spectrometry measurements. We first consider typical mis-expression experiments, where the microRNA level is instantaneously boosted or depleted and thereafter remains at a fixed level. We then move on to a more general setting where the microRNA is simply treated as another species in the reaction network, with microRNA-mRNA binding forming the basis for the post-transcriptional repression. We include some speculative comments about the potential for kinetic modelling to contribute to the more widespread sequence and network based approaches in the qualitative investigation of microRNA based gene regulation. We also consider what new combinations of experimental data will be needed in order to make sense of the increased systems-level complexity introduced by microRNAs
Chemical master versus chemical langevin for first-order reaction networks
Markov jump processes are widely used to model interacting species in circumstances where discreteness and stochasticity are relevant. Such models have been particularly successful in computational cell biology, and in this case, the interactions are typically rst-order. The Chemical Langevin Equation is a stochastic dierential equation that can be regarded as an approximation to the underlying jump process. In particular, the Chemical Langevin Equation allows simulations to be performed more eectively. In this work, we obtain expressions for the rst and second moments of the Chemical Langevin Equation for a generic rst-order reaction network. Moreover, we show that these moments exactly match those of the under-lying jump process. Hence, in terms of means, variances and correlations, the Chemical Langevin Equation is an excellent proxy for the Chemical Master Equation. Our work assumes that a unique solution exists for the Chemical Langevin Equation. We also show that the moment matching re- sult extends to the case where a gene regulation model of Raser and O'Shea (Science, 2004) is replaced by a hybrid model that mixes elements of the Master and Langevin equations. We nish with numerical experiments on a dimerization model that involves second order reactions, showing that the two regimes continue to give similar results
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