Dynamics of kinks in the Ginzburg-Landau equation: Approach to a metastable shape and collapse of embedded pairs of kinks

We consider initial data for the real Ginzburg-Landau equation having two widely separated zeros. We require these initial conditions to be locally close to a stationary solution (the ``kink'' solution) except for a perturbation supported in a small interval between the two kinks. We show that such a perturbation vanishes on a time scale much shorter than the time scale for the motion of the kinks. The consequences of this bound, in the context of earlier studies of the dynamics of kinks in the Ginzburg-Landau equation, [ER], are as follows: we consider initial conditions $v_0$ whose restriction to a bounded interval $I$ have several zeros, not too regularly spaced, and other zeros of $v_0$ are very far from $I$. We show that all these zeros eventually disappear by colliding with each other. This relaxation process is very slow: it takes a time of order exponential of the length of $I$

On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at infinity. Even in the special case of Euclidean space, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.Comment: 18 page

Zero area singularities in general relativity and inverse mean curvature flow

First we restate the definition of a Zero Area Singularity, recently introduced by H. Bray. We then consider several definitions of mass for these singularities. We use the Inverse Mean Curvature Flow to prove some new results about the mass of a singularity, the ADM mass of the manifold, and the capacity of the singularity.Comment: 13 page

Non-equilibrium Phase-Ordering with a Global Conservation Law

In all dimensions, infinite-range Kawasaki spin exchange in a quenched Ising model leads to an asymptotic length-scale $L \sim (\rho t)^{1/2} \sim t^{1/3}$ at $T=0$ because the kinetic coefficient is renormalized by the broken-bond density, $\rho \sim L^{-1}$. For $T>0$, activated kinetics recovers the standard asymptotic growth-law, $L \sim t^{1/2}$. However, at all temperatures, infinite-range energy-transport is allowed by the spin-exchange dynamics. A better implementation of global conservation, the microcanonical Creutz algorithm, is well behaved and exhibits the standard non-conserved growth law, $L \sim t^{1/2}$, at all temperatures.Comment: 2 pages and 2 figures, uses epsf.st

Corrections to Scaling in Phase-Ordering Kinetics

The leading correction to scaling associated with departures of the initial condition from the scaling morphology is determined for some soluble models of phase-ordering kinetics. The result for the pair correlation function has the form C(r,t) = f_0(r/L) + L^{-\omega} f_1(r/L) + ..., where L is a characteristic length scale extracted from the energy. The correction-to-scaling exponent \omega has the value \omega=4 for the d=1 Glauber model, the n-vector model with n=\infty, and the approximate theory of Ohta, Jasnow and Kawasaki. For the approximate Mazenko theory, however, \omega has a non-trivial value: omega = 3.8836... for d=2, and \omega = 3.9030... for d=3. The correction-to-scaling functions f_1(x) are also calculated.Comment: REVTEX, 7 pages, two figures, needs epsf.sty and multicol.st

Corrections to Scaling in the Phase-Ordering Dynamics of a Vector Order Parameter

Corrections to scaling, associated with deviations of the order parameter from the scaling morphology in the initial state, are studied for systems with O(n) symmetry at zero temperature in phase-ordering kinetics. Including corrections to scaling, the equal-time pair correlation function has the form C(r,t) = f_0(r/L) + L^{-omega} f_1(r/L) + ..., where L is the coarsening length scale. The correction-to-scaling exponent, omega, and the correction-to-scaling function, f_1(x), are calculated for both nonconserved and conserved order parameter systems using the approximate Gaussian closure theory of Mazenko. In general, omega is a non-trivial exponent which depends on both the dimensionality, d, of the system and the number of components, n, of the order parameter. Corrections to scaling are also calculated for the nonconserved 1-d XY model, where an exact solution is possible.Comment: REVTeX, 20 pages, 2 figure

Drowsy Cheetah Hunting Antelopes: A Diffusing Predator Seeking Fleeing Prey

We consider a system of three random walkers (a `cheetah' surrounded by two `antelopes') diffusing in one dimension. The cheetah and the antelopes diffuse, but the antelopes experience in addition a deterministic relative drift velocity, away from the cheetah, proportional to their distance from the cheetah, such that they tend to move away from the cheetah with increasing time. Using the backward Fokker-Planck equation we calculate, as a function of their initial separations, the probability that the cheetah has caught neither antelope after infinite time.Comment: 5 page

Spin-resolved electron-impact ionization of lithium

Electron-impact ionization of lithium is studied using the convergent close-coupling (CCC) method at 25.4 and 54.4 eV. Particular attention is paid to the spin-dependence of the ionization cross sections. Convergence is found to be more rapid for the spin asymmetries, which are in good agreement with experiment, than for the underlying cross sections. Comparison with the recent measured and DS3C-calculated data of Streun et al (1999) is most intriguing. Excellent agreement is found with the measured and calculated spin asymmetries, yet the discrepancy between the CCC and DS3C cross sections is very large

Phase Ordering Dynamics of the O(n) Model - Exact Predictions and Numerical Results

We consider the pair correlation functions of both the order parameter field and its square for phase ordering in the $O(n)$ model with nonconserved order parameter, in spatial dimension $2\le d\le 3$ and spin dimension $1\le n\le d$. We calculate, in the scaling limit, the exact short-distance singularities of these correlation functions and compare these predictions to numerical simulations. Our results suggest that the scaling hypothesis does not hold for the $d=2$ $O(2)$ model. Figures (23) are available on request - email [email protected]: 23 pages, Plain LaTeX, M/C.TH.93/2

Defect energy of infinite-component vector spin glasses

We compute numerically the zero temperature defect energy, Delta E, of the vector spin glass in the limit of an infinite number of spin components m, for a range of dimensions 2 <= d <= 5. Fitting to Delta E ~ L^theta, where L is the system size, we obtain: theta = -1.54 (d=2), theta = -1.04 (d=3), theta = -0.67 (d=4) and theta = -0.37 (d=5). These results show that the lower critical dimension, d_l (the dimension where theta changes sign), is significantly higher for m=infinity than for finite m (where 2 < d_l < 3).Comment: 4 pages, 5 figure