We study random-field xy spin model at T = 0 numerically on lattices of up to 1000 × 1000 × 1000 spins with the accent on the weak random field. Our numerical method is physically equivalent to slow cooling in which the system is gradually losing the energy and relaxing to an energy minimum. The system shows glass properties, the resulting spin states depending strongly on the initial conditions. Random initial condition for the spins leads to the vortex glass (VG) state with short-range spin-spin correlations defined by the average distance between vortex lines. Collinear and some other vortex-free initial conditions result in the vortex-free ferromagnetic (F) states that have a lower energy. The energy difference between the F and VG states correlates with vorticity of the VG state. Correlation functions in the F states agree with the Larkin-Imry-Ma theory at short distances. Hysteresis curves for weak random field are dominated by topologically stable spin walls raptured by vortex loops. We find no relaxation paths from the F, VG, or any other states to the hypothetical vortex-free state with zero magnetization.

We consider an n-component fixed-length order parameter interacting with a weak random field in d=1, 2, 3 dimensions. Relaxation from the initially ordered state and spin-spin correlation functions are studied on lattices containing hundreds of millions of sites. At n ≤ d the presence of topological defects leads to strong metastability and glassy behavior, with the final state depending on the initial condition. At n=d+1, when topological structures are nonsingular, the system possesses a weak metastability. At n>d+1, when topological objects are absent, the final, lowest-energy state is independent of the initial condition. It is characterized by the exponential decay of correlations that agrees quantitatively with the theory based upon the Imry-Ma argument.

The classical Abraham–Lorentz formula for the radiation reaction on a point charge suffers from two notorious defects: runaways and preacceleration. Recently, several authors have advocated as an alternative the Landau–Lifshitz formula, which has neither fault. The latter formula is often presented as an approximation to Abraham–Lorentz, raising the delicate question of how an approximation can be considered more accurate than the original. For a spherical shell of finite size, the equation for the radiation reaction is noncontroversial. We begin there, obtain the Abraham–Lorentz and Landau–Lifshitz expressions as limiting cases, and undertake some numerical studies to determine which is superior.