arXiv subjects

A. J. Bray

Publications and source records attributed to A. J. Bray.

50 records · Page 3Linked to original sources

arXiv1996

Structure-Factor Tail for the Ordering Kinetics of Nonconserved Fields without Topological Defects

Using a cell dynamic system (CDS) simulation scheme, we investigate the phase-ordering dynamics of non-conserved O(n) models without topological defects, i.e. for $n > d+1$ where $d$ is the spatial dimensionality. In particular, we consider zero-temperature quenches for $d=2$, $n=4,5$, and for $d=1$, $n=3,4,5$. We find, in agreement with previous simulations using fixed-length spins, that dynamical scaling is obtained, with characteristic length $L(t) = t^{1/2}$. We show that the asymptotic behaviour of the structure-factor scaling function $g(q)$ is well fitted by the stretched exponential form $g(q)\sim \exp(-bq^δ)$, with an exponent $δ$ that appears to depend on both n and d. An analytical treatment of an approximate large-N equation for the pair correlation function yields $g(q) \sim q^{-(d-1)/2}\exp(-bq)$, with $b \sim (\ln n)^{1/2}$ for large $n$, in agreement with recent simulations of the same equation.

cond-mat↗

arXiv1995

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.

cond-mat↗

arXiv1995

Lifshitz-Slyozov Scaling For Late-Stage Coarsening With An Order-Parameter-Dependent Mobility

The coarsening dynamics of the Cahn-Hilliard equation with order-parameter dependent mobility, $λ(ϕ) \propto (1-ϕ^2)^α$, is addressed at zero temperature in the Lifshitz-Slyozov limit where the minority phase occupies a vanishingly small volume fraction. Despite the absence of bulk diffusion for $α>0$, the mean domain size is found to grow as $ \propto t^{1/(3+α)}$, due to subdiffusive transport of the order parameter through the majority phase. The domain-size distribution is determined explicitly for the physically relevant case $α= 1$.

cond-mat↗

arXiv1995

Ordering Kinetics of Conserved XY Models

The zero-temperature ordering kinetics of conserved XY models in spatial dimensions $d=2$ and 3 is studied using cell dynamical simulations. The growth of the characteristic length scale $L(t)$ is fully consistent with recent theoretical predictions: $L(t) \sim t^{1/4}$ for $d=2$ and $L(t) \sim (t\ln t)^{1/4}$ for $d=3$. A gaussian closure approximation describes the form of the structure factor rather well for $d=3$.

cond-mat↗

arXiv1995

Theory of Phase Ordering Kinetics

The theory of phase ordering dynamics -- the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase -- is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time-dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, e.g. vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings, monopoles) provides a unifying framework for discussing coarsening in these different systems.

cond-mat↗

arXiv1994

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, $L(t)$, 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 $L(t)$ 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 $O(n)$ 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.

cond-mat↗

arXiv1994

Phase Ordering of 2D XY Systems Below T_{KT}

We consider quenches in non-conserved two-dimensional XY systems between any two temperatures below the Kosterlitz-Thouless transition. The evolving systems are defect free at coarse-grained scales, and can be exactly treated. Correlations scale with a characteristic length $L(t) \propto t^{1/2}$ at late times. The autocorrelation decay exponent, $\barλ = (η_i+η_f)/2$, depends on both the initial and the final state of the quench through the respective decay exponents of equilibrium correlations, $C_{EQ}(r) \sim r^{-η}$. We also discuss time-dependent quenches.

cond-mat↗

arXiv1994

Exact Exponent $λ$ of the Autocorrelation Function for a Soluble Model of Coarsening

The exponent $λ$ that describes the decay of the autocorrelation function $A(t)$ in a phase ordering system, $A(t) \sim L^{-(d-λ)}$, where $d$ is the dimension and $L$ the characteristic length scale at time $t$, is calculated exactly for the time-dependent Ginzburg-Landau equation in $d=1$. We find $λ= 0.399\,383\,5\ldots$. We also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as $M(t) \sim L^μ$ and prove that for the $1d$ Ginzburg-Landau equation, $μ=λ$, exemplifying a general result.

cond-mat↗

arXiv1994

Unwinding Scaling Violations in Phase Ordering

The one-dimensional $O(2)$ model is the simplest example of a system with topological textures. The model exhibits anomalous ordering dynamics due to the appearance of two characteristic length scales: the phase coherence length, $L \sim t^{1/z}$, and the phase winding length, $L_{w} \sim L^χ$. We derive the scaling law $z=2+μχ$, where $μ=0$ ($μ=2$) for nonconserved (conserved) dynamics and $χ=1/2$ for uncorrelated initial orientations. From hard-spin equations of motion, we consider the evolution of the topological defect density and recover a simple scaling description. (please email ar@v2.ph.man.ac.uk for a hard copy by mail)

cond-mat↗

arXiv1994

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 ($r^{-(1+σ)}$) interactions with $σ>0$, ``Energy-Scaling'' arguments predict a growth-law of the average domain size $L \sim t^{1/(1+σ)}$ for all $σ>0$. Numerical results for $σ=0.5$, $1.0$, and $1.5$ 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 $σ\rightarrow \infty$, and in that limit the amplitude of the growth law is exactly calculated.

cond-mat↗

arXiv1993

Gaussian Approach for Phase Ordering in Nonconserved Scalar Systems with Long-Range Interactions

We have applied the gaussian auxiliary field method introduced by Mazenko to the ordering dynamics of a non-conserved scalar system with attractive long-range interactions. This study provides a test-bed for the approach and shows some of the difficulties encountered in constructing a closed theory for the pair correlation function. The equation obtained for the equal-time two-point correlation function is studied in the limiting cases of small and large values of the scaling variable. A Porod regime at short distance and an asymptotic power-law decay at large distance are obtained. The theory, is not, however, consistent with the expected growth-law, and attempts to retrieve the correct growth lead to inconsistencies. These results indicate a failure of the gaussian assumption (at least in the form in which we use it) for this system.

cond-mat↗

arXiv1993

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 bray@v2.ph.man.ac.uk

cond-mat↗

arXiv1993

Generalizations of the KPZ equation (minor changes to enable easy latexing)

We generalize the KPZ equation to an O(3) $N=2j+1$ component model. In the limit $N \to \infty$ we show that the mode coupling equations become exact. Solving these approximately we find that the dynamic exponent $z$ increases from $3/2$ for $d=1$ to $2$ at the dimension $d\approx3.6$. For $d=1$ it can be shown analytically that $z=3/2$ for all $j$. The case $j=2$ for $d=2$ is investigated by numerical integration of the KPZ equation.

cond-mat↗

arXiv1993

Growth Laws for Phase Ordering

We determine the characteristic length scale, $L(t)$, 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 $L(t)$ 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.

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