7,822 research outputs found
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 model with nonconserved order
parameter, in spatial dimension and spin dimension .
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 model. Figures (23) are available on request - email
[email protected]: 23 pages, Plain LaTeX, M/C.TH.93/2
A counter-example to a recent version of the Penrose conjecture
By considering suitable axially symmetric slices on the Kruskal spacetime, we
construct counterexamples to a recent version of the Penrose inequality in
terms of so-called generalized apparent horizons.Comment: 12 pages. Appendix added with technical details. To appear in
Classical and Quantum Gravit
Phase Ordering Kinetics of One-Dimensional Non-Conserved Scalar Systems
We consider the phase-ordering kinetics of one-dimensional scalar systems.
For attractive long-range () interactions with ,
``Energy-Scaling'' arguments predict a growth-law of the average domain size for all . Numerical results for ,
, and demonstrate both scaling and the predicted growth laws. For
purely short-range interactions, an approach of Nagai and Kawasaki is
asymptotically exact. For this case, the equal-time correlations scale, but the
time-derivative correlations break scaling. The short-range solution also
applies to systems with long-range interactions when , and in that limit the amplitude of the growth law is exactly
calculated.Comment: 19 pages, RevTex 3.0, 8 FIGURES UPON REQUEST, 1549
Why temperature chaos in spin glasses is hard to observe
The overlap length of a three-dimensional Ising spin glass on a cubic lattice
with Gaussian interactions has been estimated numerically by transfer matrix
methods and within a Migdal-Kadanoff renormalization group scheme. We find that
the overlap length is large, explaining why it has been difficult to observe
spin glass chaos in numerical simulations and experiment.Comment: 4 pages, 6 figure
Growth Laws for Phase Ordering
We determine the characteristic length scale, , in phase ordering
kinetics for both scalar and vector fields, with either short- or long-range
interactions, and with or without conservation laws. We obtain
consistently by comparing the global rate of energy change to the energy
dissipation from the local evolution of the order parameter. We derive growth
laws for O(n) models, and our results can be applied to other systems with
similar defect structures.Comment: 12 pages, LaTeX, second tr
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
The Energy-Scaling Approach to Phase-Ordering Growth Laws
We present a simple, unified approach to determining the growth law for the
characteristic length scale, , in the phase ordering kinetics of a system
quenched from a disordered phase to within an ordered phase. This approach,
based on a scaling assumption for pair correlations, determines
self-consistently for purely dissipative dynamics by computing the
time-dependence of the energy in two ways. We derive growth laws for conserved
and non-conserved models, including two-dimensional XY models and
systems with textures. We demonstrate that the growth laws for other systems,
such as liquid-crystals and Potts models, are determined by the type of
topological defect in the order parameter field that dominates the energy. We
also obtain generalized Porod laws for systems with topological textures.Comment: LATeX 18 pages (REVTeX macros), one postscript figure appended,
REVISED --- rearranged and clarified, new paragraph on naive dimensional
analysis at end of section I
Mean-field theory for a spin-glass model of neural networks: TAP free energy and paramagnetic to spin-glass transition
An approach is proposed to the Hopfield model where the mean-field treatment
is made for a given set of stored patterns (sample) and then the statistical
average over samples is taken. This corresponds to the approach made by
Thouless, Anderson and Palmer (TAP) to the infinite-range model of spin
glasses. Taking into account the fact that in the Hopfield model there exist
correlations between different elements of the interaction matrix, we obtain
its TAP free energy explicitly, which consists of a series of terms exhibiting
the cluster effect. Nature of the spin-glass transition in the model is also
examined and compared with those given by the replica method as well as the
cavity method.Comment: 12 pages, LaTex, 1 PostScript figur
Perturbative Corrections to the Ohta-Jasnow-Kawasaki Theory of Phase-Ordering Dynamics
A perturbation expansion is considered about the Ohta-Jasnow-Kawasaki theory
of phase-ordering dynamics; the non-linear terms neglected in the OJK
calculation are reinstated and treated as a perturbation to the linearised
equation. The first order correction term to the pair correlation function is
calculated in the large-d limit and found to be of order 1/(d^2).Comment: Revtex, 27 pages including 2 figures, submitted to Phys. Rev. E,
references adde
Self Consistent Screening Approximation For Critical Dynamics
We generalise Bray's self-consistent screening approximation to describe the
critical dynamics of the theory. In order to obtain the dynamical
exponent , we have to make an ansatz for the form of the scaling functions,
which fortunately can be much constrained by general arguments. Numerical
values of for , and are obtained using two different
ans\"atze, and differ by a very small amount. In particular, the value of obtained for the 3-d Ising model agrees well with recent
Monte-Carlo simulations.Comment: 21 pages, LaTeX file + 4 (EPS) figure
- …