14,460 research outputs found

### 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

### 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

### 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 number of metastable states in spin glasses

In this letter, we show that the formulae of Bray and Moore for the average
logarithm of the number of metastable states in spin glasses can be obtained by
calculating the partition function with $m$ coupled replicas with the symmetry
among these explicitly broken according to a generalization of the `two-group'
ansatz. This equivalence allows us to find solutions of the BM equations where
the lower `band-edge' free energy equals the standard static free energy. We
present these results for the Sherrington-Kirkpatrick model, but we expect them
to apply to all mean-field spin glasses.Comment: 6 pages, LaTeX, no figures. Postscript directly available
http://chimera.roma1.infn.it/index_papers_complex.htm

### Velocity Distribution of Topological Defects in Phase-Ordering Systems

The distribution of interface (domain-wall) velocities ${\bf v}$ in a
phase-ordering system is considered. Heuristic scaling arguments based on the
disappearance of small domains lead to a power-law tail, $P_v(v) \sim v^{-p}$
for large v, in the distribution of $v \equiv |{\bf v}|$. The exponent p is
given by $p = 2+d/(z-1)$, where d is the space dimension and 1/z is the growth
exponent, i.e. z=2 for nonconserved (model A) dynamics and z=3 for the
conserved case (model B). The nonconserved result is exemplified by an
approximate calculation of the full distribution using a gaussian closure
scheme. The heuristic arguments are readily generalized to conserved case
(model B). The nonconserved result is exemplified by an approximate calculation
of the full distribution using a gaussian closure scheme. The heuristic
arguments are readily generalized to systems described by a vector order
parameter.Comment: 5 pages, Revtex, no figures, minor revisions and updates, to appear
in Physical Review E (May 1, 1997

### 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

### Phase Ordering Kinetics with External Fields and Biased Initial Conditions

The late-time phase-ordering kinetics of the O(n) model for a non-conserved
order parameter are considered for the case where the O(n) symmetry is broken
by the initial conditions or by an external field. An approximate theoretical
approach, based on a `gaussian closure' scheme, is developed, and results are
obtained for the time-dependence of the mean order parameter, the pair
correlation function, the autocorrelation function, and the density of
topological defects [e.g. domain walls ($n=1$), or vortices ($n=2$)]. The
results are in qualitative agreement with experiments on nematic films and
related numerical simulations on the two-dimensional XY model with biased
initial conditions.Comment: 35 pages, latex, no figure

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