arXiv subjects
Publications and source records attributed to A. J. Bray.
We show that the Almeida-Thouless line in Ising spin glasses vanishes when their dimension d -> 6 as h_{AT}^2/T_c^2 = C(d-6)^4(1- T/T_c)^{d/2 - 1}, where C is a constant of order unity. An equivalent result which could be checked by simulations is given for the one-dimensional Ising spin glass with long-range interactions. It is shown that replica symmetry breaking also stops as d -> 6.
The dynamics of the SK model at T=0 starting from random spin configurations is considered. The metastable states reached by such dynamics are atypical of such states as a whole, in that the probability density of site energies, $p(λ)$, is small at $λ=0$. Since virtually all metastable states have a much larger $p(0)$, this behavior demonstrates a qualitative failure of the Edwards hypothesis. We look for its origins by modelling the changes in the site energies during the dynamics as a Markov process. We show how the small $p(0)$ arises from features of the Markov process that have a clear physical basis in the spin-glass, and hence explain the failure of the Edwards hypothesis.
Metastable states in Ising spin-glass models are studied by finding iterative solutions of mean-field equations for the local magnetizations. Two different equations are studied: the TAP equations which are exact for the SK model, and the simpler `naive-mean-field' (NMF) equations. The free-energy landscapes that emerge are very different. For the TAP equations, the numerical studies confirm the analytical results of Aspelmeier et al., which predict that TAP states consist of close pairs of minima and index-one (one unstable direction) saddle points, while for the NMF equations saddle points with large indices are found. For TAP the barrier height between a minimum and its nearby saddle point scales as (f-f_0)^{-1/3} where f is the free energy per spin of the solution and f_0 is the equilibrium free energy per spin. This means that for `pure states', for which f-f_0 is of order 1/N, the barriers scale as N^{1/3}, but between states for which f-f_0 is of order one the barriers are finite and also small so such metastable states will be of limited physical significance. For the NMF equations there are saddles of index K and we can demonstrate that their complexity Sigma_K scales as a function of K/N. We have also employed an iterative scheme with a free parameter that can be adjusted to bring the system of equations close to the `edge of chaos'. Both for the TAP and NME equations it is possible with this approach to find metastable states whose free energy per spin is close to f_0. As N increases, it becomes harder and harder to find solutions near the edge of chaos, but nevertheless the results which can be obtained are competitive with those achieved by more time-consuming computing methods and suggest that this method may be of general utility.
We analytically study the effect of a uniform shear flow on the persistence properties of coarsening systems. The study is carried out within the anisotropic Ohta-Jasnow-Kawasaki (OJK) approximation for a system with nonconserved scalar order parameter. We find that the persistence exponent theta has a non-trivial value: theta = 0.5034... in space dimension d=3, and theta = 0.2406... for d=2, the latter being exactly twice the value found for the unsheared system in d=1. We also find that the autocorrelation exponent lambda is affected by shear in d=3 but not in d=2.
We consider the low-temperature coarsening dynamics of a one-dimensional Ising ferromagnet with conserved Kawasaki-like dynamics in the domain representation. Domains diffuse with size-dependent diffusion constant, $D(l) \propto l^γ$ with $γ= -1$. We generalize this model to arbitrary $γ$, and derive an expression for the domain density, $N(t) \sim t^{-ϕ}$ with $ϕ=1/(2-γ)$, using a scaling argument. We also investigate numerically the persistence exponent $θ$ characterizing the power-law decay of the number, $N_p(t)$, of persistent (unflipped) spins at time $t$, and find $N_{p}(t)\sim t^{-θ}$ where $θ$ depends on $γ$. We show how the results for $ϕ$ and $θ$ are related to similar calculations in diffusion-limited cluster-cluster aggregation (DLCA) where clusters with size-dependent diffusion constant diffuse through an immobile `empty' phase and aggregate irreversibly on impact. Simulations show that, while $ϕ$ is the same in both models, $θ$ is different except for $γ=0$. We also investigate models that interpolate between symmetric domain diffusion and DLCA.
We consider the trapping reaction, $A+B\to B$, where $A$ and $B$ particles have a diffusive dynamics characterized by diffusion constants $D_A$ and $D_B$. The interaction with $B$ particles can be formally incorporated in an effective dynamics for one $A$ particle as was recently shown by Bray {\it et al}. [Phys. Rev. E {\bf 67}, 060102 (2003)]. We use this method to compute, in space dimension $d=1$, the asymptotic behaviour of the spatial fluctuation, $ ^{1/2}$, for a surviving $A$ particle in the perturbative regime, $D_A/D_B\ll 1$, for the case of an initially uniform distribution of $B$ particles. We show that, for $t\gg 1$, $ ^{1/2} \propto t^ϕ$ with $ϕ=1/4$. By contrast, the fluctuations of paths constrained to return to their starting point at time $t$ grow with the larger exponent 1/3. Numerical tests are consistent with these predictions.
We compute the complexity (logarithm of the number of TAP states) associated with minima and index-one saddle points of the TAP free energy. Higher-index saddles have smaller complexities. The two leading complexities are equal, consistent with the Morse theorem on the total number of turning points, and have the value given in [A. J. Bray and M. A. Moore, J. Phys. C 13, L469 (1980)]. In the thermodynamic limit, TAP states of all free energies become marginally stable.
This paper has been replaced by cond-mat/0309113
Finite-size scaling (FSS) is a standard technique for measuring scaling exponents in spin glasses. Here we present a critique of this approach, emphasizing the need for all length scales to be large compared to microscopic scales. In particular we show that the replacement, in FSS analyses, of the correlation length by its asymptotic scaling form can lead to apparently good scaling collapses with the wrong values of the scaling exponents.
The problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d. We consider the case where the traps are initially randomly distributed in space, with uniform density rho, and derive upper and lower bounds for the probability Q(t) (averaged over all particle and trap trajectories) that the particle survives up to time t. We show that, for 1<=d<2, the bounds converge asymptotically to give $Q(t) \sim exp(-λ_d t^{d/2})$ where $λ_d = (2/πd) sin(πd/2) (4πD)^{d/2} ρ$ and D is the diffusion constant of the traps, and that $Q(t) \sim exp(- 4πρD t/ln t)$ for d=2. For d>2 bounds can still be derived, but they no longer converge for large t. For 1<=d<=2, these asymptotic form are independent of the diffusion constant of the particle. The results are compared with simulation results obtained using a new algorithm [V. Mehra and P. Grassberger, Phys. Rev. E v65 050101 (2002)] which is described in detail. Deviations from the predicted asymptotic forms are found to be large even for very small values of Q(t), indicating slowly decaying corrections whose form is consistent with the bounds. We also present results in d=1 for the case where the trap densities on either side of the particle are different. For this case we can still obtain exact bounds but they no longer converge.
We consider the survival probability of a particle in the presence of a finite number of diffusing traps in one dimension. Since the general solution for this quantity is not known when the number of traps is greater than two, we devise a perturbation series expansion in the diffusion constant of the particle. We calculate the persistence exponent associated with the particle's survival probability to second order and find that it is characterised by the asymmetry in the number of traps initially positioned on each side of the particle.
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.
The dynamics of the one-dimensional q-state Potts model, in the zero temperature limit, can be formulated through the motion of random walkers which either annihilate (A + A -> 0) or coalesce (A + A -> A) with a q-dependent probability. We consider all of the walkers in this model to be mutually infectious. Whenever two walkers meet, they experience mutual contamination. Walkers which avoid an encounter with another random walker up to time t remain uninfected. The fraction of uninfected walkers is investigated numerically and found to decay algebraically, U(t) \sim t^{-ϕ(q)}, with a nontrivial exponent ϕ(q). Our study is extended to include the coupled diffusion-limited reaction A+A -> B, B+B -> A in one dimension with equal initial densities of A and B particles. We find that the density of walkers decays in this model as ρ(t) \sim t^{-1/2}. The fraction of sites unvisited by either an A or a B particle is found to obey a power law, P(t) \sim t^{-θ} with θ\simeq 1.33. We discuss these exponents within the context of the q-state Potts model and present numerical evidence that the fraction of walkers which remain uninfected decays as U(t) \sim t^{-ϕ}, where ϕ\simeq 1.13 when infection occurs between like particles only, and ϕ\simeq 1.93 when we also include cross-species contamination.
We consider a system of unbiased diffusing walkers (A0 <-> 0A) in one dimension with random initial conditions. We investigate numerically the relation between the fraction of walkers, U(t), which have never encountered another walker up to time t, calling such walkers ``uninfected'' and the fraction of sites, P(t), which have never been visited by a diffusing particle. We extend our study to include the A + B -> 0 diffusion-limited reaction in one-dimension, with equal initial densities of A and B particles distributed homogeneously at t=0. We find U(t) \simeq [P(t)]^γ, with γ\simeq 1.39, in both models, though there is evidence that a smaller value of γis required for t -> infinity.
We introduce the technique of aspect-ratio scaling to study the scale-dependence of interfacial energies in Ising spin glasses, and we show how one can use it to determine the stiffness exponent $θ$ in a clean way, with results that are independent of the domain-wall-forcing boundary conditions imposed on the system. In space dimension d=2 we obtain $θ= -0.282(3)$ for a Gaussian distribution of exchange interactions.
The persistence properties of a set of random walkers obeying the A+B -> 0 reaction, with equal initial density of particles and homogeneous initial conditions, is studied using two definitions of persistence. The probability, P(t), that an annihilation process has not occurred at a given site has the asymptotic form $P(t) -> const + t^{-θ}$, where $θ$ is the persistence exponent (``type I persistence''). We argue that, for a density of particles $ρ>> 1$, this non-trivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, where $θ\approx 0.1207$. In the case of an initially low density, $ρ_0 << 1$, we find $θ\approx 1/4$ asymptotically. The probability that a site remains unvisited by any random walker (``type II persistence'') is also investigated and found to decay with a stretched exponential form, $P(t) \sim \exp(-const ρ_0^{1/2}t^{1/4})$, provided $ρ_0 << 1$. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.
The distribution, n(k,t), of the interval sizes, k, between clusters of persistent sites in the dynamical evolution of the one-dimensional q-state Potts model is studied using a combination of numerical simulations, scaling arguments, and exact analysis. It is shown to have the scaling form n(k,t) = t^{-2z} f(k/t^z), with z= max(1/2,theta), where theta(q) is the persistence exponent which characterizes the fraction of sites which have not changed their state up to time t. When theta > 1/2, the scaling length, t^theta, for the interval-size distribution is larger than the coarsening length scale, t^{1/2}, that characterizes spatial correlations of the Potts variables.
The Langevin equation for a particle (`random walker') moving in d-dimensional space under an attractive central force, and driven by a Gaussian white noise, is considered for the case of a power-law force, F(r) = - Ar^{-sigma}. The `persistence probability', P_0(t), that the particle has not visited the origin up to time t, is calculated. For sigma > 1, the force is asymptotically irrelevant (with respect to the noise), and the asymptotics of P_0(t) are those of a free random walker. For sigma < 1, the noise is (dangerously) irrelevant and the asymptotics of P_0(t) can be extracted from a weak noise limit within a path-integral formalism. For the case sigma=1, corresponding to a logarithmic potential, the noise is exactly marginal. In this case, P_0(t) decays as a power-law, P_0(t) \sim t^{-theta}, with an exponent theta that depends continuously on the ratio of the strength of the potential to the strength of the noise. This case, with d=2, is relevant to the annihilation dynamics of a vortex-antivortex pair in the two-dimensional XY model. Although the noise is multiplicative in the latter case, the relevant Langevin equation can be transformed to the standard form discussed in the first part of the paper. The mean annihilation time for a pair initially separated by r is given by t(r) \sim r^2 ln(r/a) where a is a microscopic cut-off (the vortex core size). Implications for the nonequilibrium critical dynamics of the system are discussed and compared to numerical simulation results.