On Convergence to Equilibrium Distribution,I.¶The Klein–Gordon Equation with Mixing

Consider the Klein–Gordon equation (KGE) in ℝ ⁿ , n≥ 2, with constant or variable coefficients. We study the distribution μ _t of the random solution at time t∈ℝ. We assume that the initial probability measure μ₀ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also assume that μ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The main result is the convergence of μ _t to a Gaussian probability measure as t→∞ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's “room-corridor” argument. The case of variable coefficients is treated by using an “averaged” version ofthe scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay. Received: 4 January 2001 / Accepted: 2 July 2001     

Effect of high-energy electron irradiation on forsterite laser crystals

The effect of 21-MeV electron irradiation on the optical absorption characteristics of Czochralski-grown forsterite (Mg₂SiO₄) single crystals (both undoped and chromium-doped) has been investigated. The irradiation is found to induce additional optical absorption (AOA) in the crystals in the range of 225–1200 nm due to the formation of color centers based on intrinsic host point defects and the change in the oxidation state of chromium ions. The AOA spectra have been decomposed into elementary bands. The influence of the chromium concentration in crystals, the oxygen content in the growth atmosphere, and additional doping with lithium on the behavior of these bands has been analyzed. A possible structure of the color centers responsible for the AOA is discussed. It is shown that the electron irradiation somewhat decreases the intensity of the characteristic absorption bands of tri- and tetravalent chromium ions and gives rise to a new absorption band in Mg₂SiO₄:Cr and Mg₂SiO₄:Cr,Li crystals heavily doped with chromium.     

Laser-triggered ion acceleration at moderate intensity and pulse duration

Two-dimensional particle-in-cell (PIC) simulations of laser-triggered ion acceleration in overdense plasma at moderate intensity (≃1.4×10¹⁸ W/cm²) and pulse duration (≃0.5 ps) are presented. We focus on the comparison of the efficiency of ion acceleration for normal and oblique incidence of the laser light, for backward and forward directions of ion emission, and for large and small focal spots. We discuss the correlation between the properties of accelerated ions and hot electrons, and identify the tendency of the ion spectra in the forward direction to those typical for the isothermal and adiabatic regimes of plasma expansion.     

On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

The paper considers the wave equation, with constant or variable coefficients in ⁿ , with odd n3. We study the asymptotics of the distribution _t of the random solution at time t as t . It is assumed that the initial measure ₀ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of _t to a Gaussian measure as t , which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's room-corridor argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.     

Deriving hydrodynamic equations for lattice systems

We study the dynamics of lattice systems in ℤ^d, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ₀^ɛ, ɛ > 0}. The measures μ₀^ɛ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order o(ɛ⁻¹ ) and inhomogeneous under shifts of the order ɛ⁻¹ . Moreover, correlations of the measures μ₀^ɛ decrease uniformly in ɛ at large distances. For all τ ∈ ℝ \ 0, r ∈ ℝ^d, and κ > 0, we consider distributions of a random solution at the instants t = τ/ɛ^κ at points close to [r/ɛ] ∈ ℤ^d. Our main goal is to study the asymptotic behavior of these distributions as ɛ → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type.     

Interaction of impurity and intrinsic defects in forsterite and its role in the formation of spectral and luminescence properties

This paper reports on the results of investigations of the coefficients of chromium distribution between a crystal and a melt of forsterite, the absorption and luminescence spectra, and the electron paramagnetic resonance spectra of chromium centers in Mg₂SiO₄: Cr, Mg₂SiO₄: Cr: Sc, and Mg₂SiO₄: Cr: Li crystals. It has been established that the concentration dependences of these properties vary upon changing over from the range of trace concentrations of chromium, scandium, or lithium impurities in the melt to the range of higher concentrations of these impurities. The observed phenomenon is explained by the interaction of impurities with intrinsic defects of the crystal, which is called as the microimpurity trapping effect. According to the performed estimations, the concentration of predominant intrinsic defects (magnesium Frenkel defects) in the forsterite crystals grown from the melt is equal to (7.5 ± 0.3) × 10⁻⁶ atomic fractions. The energy of the formation of magnesium Frenkel defects can be estimated as 4.2 ± 0.2 eV.     

Laser-triggered quasi-monoenergetic ion beams at a moderate intensity and pulse duration

Plasma produced by short laser pulses from thin homogeneous foils with light and heavy ions is capable of generating quasi-monoenergetic light ions. This happens for the tail of light ions near the front of heavy ions. It was found that this effect is well pronounced for a moderate laser intensity (～10¹⁸ W/cm²) and pulse duration (～1 ps) by using a 2D particle-in-cell simulation of the laser interaction with thin CD₂ foils. Quasi-monoenergetic deuterons form a jet from the rear side of the foil with the energy ～1 MeV. The conversion efficiency to these quasi-monoenergetic ions is 10^?3.