361,274 research outputs found
Local Lagrangian Approximations for the Evolution of the Density Distribution Function in Large-Scale Structure
We examine local Lagrangian approximations for the gravitational evolution of
the density distribution function. In these approximations, the final density
at a Lagrangian point q at a time t is taken to be a function only of t and of
the initial density at the same Lagrangian point. A general expression is given
for the evolved density distribution function for such approximations, and we
show that the vertex generating function for a local Lagrangian mapping applied
to an initially Gaussian density field bears a simple relation to the mapping
itself. Using this result, we design a local Lagrangian mapping which
reproduces nearly exactly the hierarchical amplitudes given by perturbation
theory for gravitational evolution. When extended to smoothed density fields
and applied to Gaussian initial conditions, this mapping produces a final
density distribution function in excellent agreement with full numerical
simulations of gravitational clustering. We also examine the application of
these local Lagrangian approximations to non-Gaussian initial conditions.Comment: LaTeX, 22 pages, and 11 postscript figure
Modelling diffusion in crystals under high internal stress gradients
Diffusion of vacancies and impurities in metals is important in many processes occurring in structural materials. This diffusion often takes place in the presence of spatially rapidly varying stresses. Diffusion under stress is frequently modelled by local approximations to the vacancy formation and diffusion activation enthalpies which are linear in the stress, in order to account for its dependence on the local stress state and its gradient. Here, more accurate local approximations to the vacancy formation and diffusion activation enthalpies, and the simulation methods needed to implement them, are introduced. The accuracy of both these approximations and the linear approximations are assessed via comparison to full atomistic studies for the problem of vacancies around a Lomer dislocation in Aluminium. Results show that the local and linear approximations for the vacancy formation enthalpy and diffusion activation enthalpy are accurate to within 0.05 eV outside a radius of about 13 Å (local) and 17 Å (linear) from the centre of the dislocation core or, more generally, for a strain gradient of roughly up to 6 × 10^6 m^-1 and 3 × 10^6 m^-1, respectively. These results provide a basis for the development of multiscale models of diffusion under highly non-uniform stress
On Local Approximations to the Nonlinear Evolution of Large-Scale Structure
We present a comparative analysis of several methods, known as local
Lagrangian approximations, which are aimed to the description of the nonlinear
evolution of large-scale structure. We have investigated various aspects of
these approximations, such as the evolution of a homogeneous ellipsoid,
collapse time as a function of initial conditions, and asymptotic behavior. As
one of the common features of the local approximations, we found that the
calculated collapse time decreases asymptotically with the inverse of the
initial shear. Using these approximations, we have computed the cosmological
mass function, finding reasonable agreement with N-body simulations and the
Press-Schechter formula.Comment: revised version with color figures, minor changes, accepted for
publication in the Astrophysical Journal, 30 pages, 13 figure
Malliavin calculus for difference approximations of multidimensional diffusions: truncated local limit theorem
For a difference approximations of multidimensional diffusion, the truncated
local limit theorem is proved. Under very mild conditions on the distribution
of the difference terms, this theorem provides that the transition
probabilities of these approximations, after truncation of some asymptotically
negligible terms, possess a densities that converge uniformly to the transition
probability density for the limiting diffusion and satisfy a uniform
diffusion-type estimates. The proof is based on the new version of the
Malliavin calculus for the product of finite family of measures, that may
contain non-trivial singular components. An applications for uniform estimates
for mixing and convergence rates for difference approximations to SDE's and for
convergence of difference approximations for local times of multidimensional
diffusions are given.Comment: 34 page
Finite Approximations of Physical Models over Local Fields
We show that the Schr\"odinger operator associated with a physical system
over a local field can be approximated in a very strong sense by finite
Schr\"odinger operators. Some striking numerical results are included at the
end of the article
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