408 research outputs found
Exponentially convergent data assimilation algorithm for Navier-Stokes equations
The paper presents a new state estimation algorithm for a bilinear equation
representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS)
equations on a torus in R2. This state equation is subject to uncertain but
bounded noise in the input (Kolmogorov forcing) and initial conditions, and its
output is incomplete and contains bounded noise. The algorithm designs a
time-dependent gain such that the estimation error converges to zero
exponentially. The sufficient condition for the existence of the gain are
formulated in the form of algebraic Riccati equations. To demonstrate the
results we apply the proposed algorithm to the reconstruction a chaotic fluid
flow from incomplete and noisy data
Preheating after N-flation
We study preheating in N-flation, assuming the Mar\v{c}enko-Pastur mass
distribution, equal energy initial conditions at the beginning of inflation and
equal axion-matter couplings, where matter is taken to be a single, massless
bosonic field. By numerical analysis we find that preheating via parametric
resonance is suppressed, indicating that the old theory of perturbative
preheating is applicable. While the tensor-to-scalar ratio, the non-Gaussianity
parameters and the scalar spectral index computed for N-flation are similar to
those in single field inflation (at least within an observationally viable
parameter region), our results suggest that the physics of preheating can
differ significantly from the single field case.Comment: 14 pages, 14 figures, references added, fixed typo
Models of Passive and Reactive Tracer Motion: an Application of Ito Calculus
By means of Ito calculus it is possible to find, in a straight-forward way,
the analytical solution to some equations related to the passive tracer
transport problem in a velocity field that obeys the multidimensional Burgers
equation and to a simple model of reactive tracer motion.Comment: revised version 7 pages, Latex, to appear as a letter to J. of
Physics
On the mass transport by a Burgers velocity field
The mass transport by a Burgers velocity field is investigated in the
framework of the theory of stochastic processes. Much attention is devoted to
the limit of vanishing viscosity (inviscid limit) describing the "adhesion
model" for the early stage of the evolution of the Universe. In particular the
mathematical foundations for the ansatz currently used in the literature to
compute the mass distribution in the inviscid limit are provided.Comment: 14 pages, Latex, revised version submitted to Physica
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
Small Noise Asymptotics for Invariant Densities for a Class of Diffusions: A Control Theoretic View (with Erratum)
The uniqueness argument in the proof of Theorem 5, p. 483, of "Small noise
asymptotics for invariant densities for a class of diffusions: a control
theoretic view, J. Math. Anal. and Appl. (2009) " is flawed. We give here a
corrected proof.Comment: 23 pages; Journal of Mathematical Analysis and Applications, 200
Equation-Free Dynamic Renormalization: Self-Similarity in Multidimensional Particle System Dynamics
We present an equation-free dynamic renormalization approach to the
computational study of coarse-grained, self-similar dynamic behavior in
multidimensional particle systems. The approach is aimed at problems for which
evolution equations for coarse-scale observables (e.g. particle density) are
not explicitly available. Our illustrative example involves Brownian particles
in a 2D Couette flow; marginal and conditional Inverse Cumulative Distribution
Functions (ICDFs) constitute the macroscopic observables of the evolving
particle distributions.Comment: 7 pages, 5 figure
The quantum theory of measurement within dynamical reduction models
We analyze in mathematical detail, within the framework of the QMUPL model of
spontaneous wave function collapse, the von Neumann measurement scheme for the
measurement of a 1/2 spin particle. We prove that, according to the equation of
the model: i) throughout the whole measurement process, the pointer of the
measuring device is always perfectly well localized in space; ii) the
probabilities for the possible outcomes are distributed in agreement with the
Born probability rule; iii) at the end of the measurement the state of the
microscopic system has collapsed to the eigenstate corresponding to the
measured eigenvalue. This analysis shows rigorously how dynamical reduction
models provide a consistent solution to the measurement problem of quantum
mechanics.Comment: 24 pages, RevTeX. Minor changes mad
Cosmological Inflation and the Quantum Measurement Problem
According to cosmological inflation, the inhomogeneities in our universe are
of quantum mechanical origin. This scenario is phenomenologically very
appealing as it solves the puzzles of the standard hot big bang model and
naturally explains why the spectrum of cosmological perturbations is almost
scale invariant. It is also an ideal playground to discuss deep questions among
which is the quantum measurement problem in a cosmological context. Although
the large squeezing of the quantum state of the perturbations and the
phenomenon of decoherence explain many aspects of the quantum to classical
transition, it remains to understand how a specific outcome can be produced in
the early universe, in the absence of any observer. The Continuous Spontaneous
Localization (CSL) approach to quantum mechanics attempts to solve the quantum
measurement question in a general context. In this framework, the wavefunction
collapse is caused by adding new non linear and stochastic terms to the
Schroedinger equation. In this paper, we apply this theory to inflation, which
amounts to solving the CSL parametric oscillator case. We choose the
wavefunction collapse to occur on an eigenstate of the Mukhanov-Sasaki variable
and discuss the corresponding modified Schroedinger equation. Then, we compute
the power spectrum of the perturbations and show that it acquires a universal
shape with two branches, one which remains scale invariant and one with nS=4, a
spectral index in obvious contradiction with the Cosmic Microwave Background
(CMB) anisotropy observations. The requirement that the non-scale invariant
part be outside the observational window puts stringent constraints on the
parameter controlling the deviations from ordinary quantum mechanics...
(Abridged).Comment: References added, minor corrections, conclusions unchange
The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps
Consider jump-type stochastic differential equations with the drift,
diffusion and jump terms. Logarithmic derivatives of densities for the solution
process are studied, and the Bismut-Elworthy-Li type formulae can be obtained
under the uniformly elliptic condition on the coefficients of the diffusion and
jump terms. Our approach is based upon the Kolmogorov backward equation by
making full use of the Markovian property of the process.Comment: 29 pages, to appear in Journal of Theoretical Probabilit
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