A J Bray

The University of Manchester, Manchester, England, United Kingdom

Are you A J Bray?

Claim your profile

Publications (200)369.35 Total impact

  • Source
    Alan J. Bray, Satya N. Majumdar, G. Schehr
    [show abstract] [hide abstract]
    ABSTRACT: In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.
    04/2013;
  • Source
    M. A. Moore, A. J. Bray
    [show abstract] [hide abstract]
    ABSTRACT: 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.
    Physical review. B, Condensed matter 02/2011; 83.
  • Source
    Satya N. Majumdar, Alan J. Bray
    [show abstract] [hide abstract]
    ABSTRACT: We consider three independent Brownian walkers moving on a line. The process terminates when the left-most walker (the `Leader') meets either of the other two walkers. For arbitrary values of the diffusion constants D_1 (the Leader), D_2 and D_3 of the three walkers, we compute the probability distribution P(m|y_2,y_3) of the maximum distance m between the Leader and the current right-most particle (the `Laggard') during the process, where y_2 and y_3 are the initial distances between the leader and the other two walkers. The result has, for large m, the form P(m|y_2,y_3) \sim A(y_2,y_3) m^{-\delta}, where \delta = (2\pi-\theta)/(\pi-\theta) and \theta = cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also determined exactly.
    Journal of Statistical Mechanics Theory and Experiment 06/2010; · 1.87 Impact Factor
  • Source
    Claude Godreche, Alan J. Bray
    [show abstract] [hide abstract]
    ABSTRACT: We study the nonequilibrium properties of directed Ising models with non conserved dynamics, in which each spin is influenced by only a subset of its nearest neighbours. We treat the following models: (i) the one-dimensional chain; (ii) the two-dimensional square lattice; (iii) the two-dimensional triangular lattice; (iv) the three-dimensional cubic lattice. We raise and answer the question: (a) Under what conditions is the stationary state described by the equilibrium Boltzmann-Gibbs distribution? We show that for models (i), (ii), and (iii), in which each spin "sees" only half of its neighbours, there is a unique set of transition rates, namely with exponential dependence in the local field, for which this is the case. For model (iv), we find that any rates satisfying the constraints required for the stationary measure to be Gibbsian should satisfy detailed balance, ruling out the possibility of directed dynamics. We finally show that directed models on lattices of coordination number $z\ge8$ with exponential rates cannot accommodate a Gibbsian stationary state. We conjecture that this property extends to any form of the rates. We are thus led to the conclusion that directed models with Gibbsian stationary states only exist in dimension one and two. We then raise the question: (b) Do directed Ising models, augmented by Glauber dynamics, exhibit a phase transition to a ferromagnetic state? For the models considered above, the answers are open problems, to the exception of the simple cases (i) and (ii). For Cayley trees, where each spin sees only the spins further from the root, we show that there is a phase transition provided the branching ratio, $q$, satisfies $q \ge 3$.
    Journal of Statistical Mechanics Theory and Experiment 11/2009; · 1.87 Impact Factor
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: We study the domain geometry during spinodal decomposition of a 50:50 binary mixture in two dimensions. Extending arguments developed to treat nonconserved coarsening, we obtain approximate analytic results for the distribution of domain areas and perimeters during the dynamics. The main approximation is to regard the interfaces separating domains as moving independently. While this is true in the nonconserved case, it is not in the conserved one. Our results can therefore be considered as a "first-order" approximation for the distributions. In contrast to the celebrated Lifshitz-Slyozov-Wagner distribution of structures of the minority phase in the limit of very small concentration, the distribution of domain areas in the 50:50 case does not have a cutoff. Large structures (areas or perimeters) retain the morphology of a percolative or critical initial condition, for quenches from high temperatures or the critical point, respectively. The corresponding distributions are described by a cA-tau tail, where c and tau are exactly known. With increasing time, small structures tend to have a spherical shape with a smooth surface before evaporating by diffusion. In this regime, the number density of domains with area A scales as A1/2 , as in the Lifshitz-Slyozov-Wagner theory. The threshold between the small and large regimes is determined by the characteristic area A approximately t2/3. Finally, we study the relation between perimeters and areas and the distribution of boundary lengths, finding results that are consistent with the ones summarized above. We test our predictions with Monte Carlo simulations of the two-dimensional Ising model.
    Physical Review E 09/2009; 80(3 Pt 1):031121. · 2.31 Impact Factor
  • [show abstract] [hide abstract]
    ABSTRACT: We obtain the exact distribution of the areas enclosed by domain boundaries (`hulls') during the coarsening dynamics of a two-dimensional nonconserved scalar field. This result represents the first analytical demostration of the dynamical scaling hypothesis for this system. The experimental data for the formation of chiral domains in liquid crystals are in very good agreement with the theory.
    01/2009;
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: We study electric field driven deracemization in an achiral liquid crystal through the formation and coarsening of chiral domains. It is proposed that deracemization in this system is a curvature-driven process. We test this prediction using the recently obtained exact result for the distribution of hull-enclosed areas in two-dimensional coarsening with nonconserved scalar order parameter dynamics [J. J. Arenzon et al., Phys. Rev. Lett. 98, 145701 (2007)]. The experimental data are in very good agreement with the theory. We thus demonstrate that deracemization in such bent-core liquid crystals belongs to the Allen-Cahn universality class, and that the exact formula, which gives us the statistics of domain sizes during coarsening, can also be used as a strict test for this dynamic universality class.
    Physical Review Letters 12/2008; 101(19):197801. · 7.94 Impact Factor
  • [show abstract] [hide abstract]
    ABSTRACT: We consider the statistics of the areas enclosed by domain boundaries (‘hulls’) during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area, n h (A, t)dA, with enclosed area in the range (A,A + dA), is described, for large time t, by the scaling form n h (A, t) = 2c h /(A + λ h t)2, demonstrating the validity of dynamical scaling in this system. Here $ c_h = {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8}\pi \sqrt 3 $ c_h = {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8}\pi \sqrt 3 is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and λ h is a material parameter. The distribution of domain areas, n d (A, t), is apparently very similar to that of hull areas up to very large values of A/λ h t. Identical forms are obtained for coarsening from a critical initial state, but with c h replaced by c h /2. The similarity of the two distributions (of areas enclosed by hulls, and of domain areas) is accounted for by the smallness of c h . By applying a ‘mean-field’ type of approximation we obtain the form n d (A, t) ≃ 2c d [λ d (t+t 0)] τ−2/[A+λ d (t+t 0)] τ , where t 0 is a microscopic timescale and τ = 187/91 ≃ 2.055, for a disordered initial state, and a similar result for a critical initial state but with c d → c d /2 and τ → τ c = 379/187 ≃ 2.027. We also find that c d = c h + O(c h 2) and λ d = λ h (1 + O(c h )). These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.
    Physics of Condensed Matter 07/2008; 64(3):403-407. · 1.28 Impact Factor
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: We consider the number and distribution of minima in random landscapes defined on non-Euclidean lattices. Using an ensemble where random landscapes are reweighted by a fugacity factor $z$ for each minimum they contain, we construct first a `two-box' mean field theory. This exhibits an ordering phase transition at $z\c=2$ above which one box contains an extensive number of minima. The onset of order is governed by an unusual order parameter exponent $\beta=1$, motivating us to study the same model on the Bethe lattice. Here we find from an exact solution that for any connectivity $\mu+1>2$ there is an ordering transition with a conventional mean field order parameter exponent $\beta=1/2$, but with the region where this behaviour is observable shrinking in size as $1/\mu$ in the mean field limit of large $\mu$. We show that the behaviour in the transition region can also be understood directly within a mean field approach, by making the assignment of minima `soft'. Finally we demonstrate, in the simplest mean field case, how the analysis can be generalized to include both maxima and minima. In this case an additional first order phase transition appears, to a landscape in which essentially all sites are either minima or maxima.
    Journal of Statistical Mechanics Theory and Experiment 07/2008; · 1.87 Impact Factor
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: The domain morphology of weakly disordered ferromagnets, quenched from the high-temperature phase to the low-temperature phase, is studied using numerical simulations. We find that the geometrical properties of the coarsening domain structure, e.g. the distributions of hull-enclosed areas and domain perimeter lengths, are described by a scaling phenomenology in which the growing domain scale R(t) is the only relevant parameter. Furthermore, the scaling functions have forms identical to those of the corresponding pure system, extending the "super-universality" property previously noted for the pair correlation function.
    EPL (Europhysics Letters) 03/2008; 82(1):10001. · 2.26 Impact Factor
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, n_{h}(A,t)dA , with enclosed area in the interval (A,A+dA) , is described, for a disordered initial condition, by the scaling function n_{h}(A,t)=2c_{h}(A+lambda_{h}t);{2} , where c_{h}=18pi sqrt[3] approximately 0.023 is a universal constant and lambda_{h} is a material parameter. For a critical initial condition, the same form is obtained, with the same lambda_{h} but with c_{h} replaced by c_{h}2 . For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form n_{d}(A,t)=2c_{d}(lambda_{d}t);{tau'-2}(A+lambda_{d}t);{tau'} , where c_{d} and lambda_{d} are numerically very close to c_{h} and lambda_{h} , respectively, and tau'=18791 approximately 2.055 . For critical initial conditions, one replaces c_{d} by c_{d}2 and the exponent is tau=379187 approximately 2.027 . These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas.
    Physical Review E 01/2008; 76(6 Pt 1):061116. · 2.31 Impact Factor
  • Source
    A. J. Bray, B. Derrida, C. Godréche
    [show abstract] [hide abstract]
    ABSTRACT: A non-trivial exponent β characterising non-equilibrium coarsening processes is calculated in a soluble model. For a spin model, the exponent describes how the fraction p0 of spins which have never flipped (or, equivalently, the fraction of space which has never been traversed by a domain wall) depends on the characteristic domain scale L: p0 ~ Lβ-1. For the one-dimensional time-dependent Ginzburg-Landau equation at zero temperature we show that the critical exponent β is the zero of a transcendental equation, and find β = 0.824 924 12....
    EPL (Europhysics Letters) 07/2007; 27(3):175. · 2.26 Impact Factor
  • Source
    Alan J. Bray, Richard Smith
    [show abstract] [hide abstract]
    ABSTRACT: We consider a Brownian particle, with diffusion constant D, moving inside an expanding d-dimensional sphere whose surface is an absorbing boundary for the particle. The sphere has initial radius L_0 and expands at a constant rate c. We calculate the joint probability density, p(r,t|r_0), that the particle survives until time t, and is at a distance r from the centre of the sphere, given that it started at a distance r_0 from the centre. Comment: 5 pages
    Journal of Physics A Mathematical and Theoretical 05/2007; · 1.77 Impact Factor
  • Source
    Alan J Bray, David S Dean
    [show abstract] [hide abstract]
    ABSTRACT: We calculate the average number of critical points of a Gaussian field on a high-dimensional space as a function of their energy and their index. Our results give a complete picture of the organization of critical points and are of relevance to glassy and disordered systems and landscape scenarios coming from the anthropic approach to string theory.
    Physical Review Letters 05/2007; 98(15):150201. · 7.94 Impact Factor
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: We consider the statistics of the areas enclosed by domain boundaries ("hulls") during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than A has, for large time t, the scaling form Nh(A,t)=2c/(A+lambdat), demonstrating the validity of dynamical scaling in this system, where c=1/8pisquare root 3 is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of A/lambdat. Identical forms are obtained for coarsening from a critical initial state, but with c replaced by c/2.
    Physical Review Letters 05/2007; 98(14):145701. · 7.94 Impact Factor
  • Source
    Alan J. Bray, Richard Smith
    [show abstract] [hide abstract]
    ABSTRACT: We calculate the exact asymptotic survival probability, Q, of a one-dimensional Brownian particle, initially located at the point x ∈ (-L, L), in the presence of two moving, absorbing boundaries located at ±(L + ct). The result is Q(y, lambda) = ∑∞n=-∞(-1)ncosh(ny)exp(-n2lambda), where y = cx/D, lambda = cL/D and D is the diffusion constant of the particle. The results may be extended to the case where the absorbing boundaries have different speeds. As an application, we compute the asymptotic survival probability for the trapping reaction A + B --> B, for evanescent traps with a long decay time.
    Journal of Physics A Mathematical and Theoretical 01/2007; 40(10). · 1.77 Impact Factor
  • A J Bray, M A Moore
    [show abstract] [hide abstract]
    ABSTRACT: The ordered phase of short-range spin glasses is described in terms of the scaling behaviour associated with a zero-temperature fixed point. The main ingredient of the theory is the exponent y which describes the growth with length scale L of the characteristic coupling at zero temperature, J(L) − JLY. The exponent y is estimated numerically for dimensions d=2,3. For Ising spin glasses we find y − -0.3 for d=2 and y − 0.2 for d=3, implying scaling to weak (strong) coupling for d=2(3), i.e. the “lower critical dimension” dℓ satisfies 2<dℓ<3. For d<dℓ y determines the divergence of the correlation length for T→O, while for d>dℓ it determines the large scale properties of the ordered phase, such as the long-distance behaviour of connected correlation functions, G(r) ∝ r−y, and the singular response to a weak magnetic field, msing ∝ hd/(d-2y), The decay of the connected correlation functions at large distances implies that the pure-state overlap distribution function P(q) is trivial, in contrast to the Sherrington-Kirkpatrick model. The dynamics of the system are also discussed, as is the extension to vector spin models.
    07/2006: pages 121-153;
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: The dynamics of the Sherrington-Kirkpatrick 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 modeling 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.
    Physical review. B, Condensed matter 07/2006; 74(2).
  • Source
    N P Rapapa, A J Bray
    [show abstract] [hide abstract]
    ABSTRACT: 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 nontrivial 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.
    Physical Review E 05/2006; 73(4 Pt 2):046123. · 2.31 Impact Factor
  • Source
    [show abstract] [hide abstract]
    ABSTRACT: 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. Comment: 13 pages
    Physical review. B, Condensed matter 02/2006;

Publication Stats

2k Citations
369.35 Total Impact Points

Institutions

  • 1990–2011
    • The University of Manchester
      • • School of Physics and Astronomy
      • • Theoretical Physics Division
      Manchester, England, United Kingdom
  • 2002–2003
    • Paul Sabatier University - Toulouse III
      • Laboratoire de Physique Théorique - UMR 5152 - LPT
      Tolosa de Llenguadoc, Midi-Pyrénées, France
  • 2000
    • Imperial College London
      • Department of Mathematics
      London, ENG, United Kingdom
  • 1995–1998
    • Jawaharlal Nehru University
      • School of Physical Sciences
      New Delhi, NCT, India