Explicit mass renormalization and consistent derivation of radiative response of classical electron

The radiative response of the classical electron is commonly described by the Lorentz–Abraham–Dirac (LAD) equation. Dirac’s derivation of this equation is based on energy and momentum conservation laws and on regularization of the field singularities and infinite energies of the point charge by subtraction of certain quantities: “We... shall try to get over difficulties associated with the infinite energy of the process by a process of direct omission or subtraction of unwanted terms”. To substantiate Dirac’s approach and clarify the mass renormalization, we introduce the point charge as a limit of extended charges contracting to a point; the fulfillment of conservation laws follows from the relativistic covariant Lagrangian formulation of the problem. We derive the relativistic point charge dynamics described by the LAD equation from the extended charge dynamics in a localization limit by a method which can be viewed as a refinement of Dirac’s approach in the spirit of Ehrenfest theorem. The model exhibits the mass renormalization as the cancellation of Coulomb energy with the Poincaré cohesive energy. The value of the renormalized mass is not postulated as an arbitrary constant, but is explicitly calculated. The analysis demonstrates that the local energy–momentum conservation laws yield dynamics of a point charge which involves three constants: mass, charge and radiative response coefficient θ$\theta$. The value of θ$\theta$ depends on the composition of the adjacent potential which generates Poincaré forces. The classical value of the radiative response coefficient is singled out by the global requirement that the adjacent potential does not affect the radiated energy balance and affects only the local energy balance involved in the renormalization.     

Polar nanoregions in Pb(Mg<Subscript>1/3</Subscript>Nb<Subscript>2/3</Subscript>)O<Subscript>3</Subscript> (PMN): insights from a supercell approach

We report construction of a model of polar nanoregions in the PMN relaxor ferroelectric based on first-principles lattice dynamics for chemically ordered supercells [S.A. Prosandeev et al., Phys. Rev. B 70, 134110 (2004)], combined with invariance under permutations and dipole-dipole interaction as a source supporting randomly oriented residual polarization. Representative analytical estimates of polar nanore-gion — supercell mapping reproduce both nonzero local and zero macroscopic polarization of the structure, as well as the temperature change of the supercell anisotropy at cooling and field cooling.     

Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data

Potential Analysis - We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for...     

Method of Best Successive Approximations for Nonlinear Operators

We present a new approach to the approximation of nonlinear operators in probability spaces. The approach is based on a combination of the specific iterative procedure and the best approximation problem solution with a quadratic approximant. We show that the combination of these new techniques allow us to build a computationally efficient and flexible method. The algorithm of the method and its application to the optimal filtering of stochastic signals are given.