Publications (229)624.27 Total impact
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ABSTRACT: In this review we discuss the persistence and the related firstpassage properties in extended manybody 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 nonMarkovian. Several exact and approximate methods have been developed to compute the persistence of such nonMarkov 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.  [Show abstract] [Hide abstract]
ABSTRACT: We show that the AlmeidaThouless line in Ising spin glasses vanishes when their dimension d > 6 as h_{AT}^2/T_c^2 = C(d6)^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 onedimensional Ising spin glass with longrange interactions. It is shown that replica symmetry breaking also stops as d > 6.  [Show abstract] [Hide abstract]
ABSTRACT: We consider three independent Brownian walkers moving on a line. The process terminates when the leftmost 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(my_2,y_3) of the maximum distance m between the Leader and the current rightmost 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(my_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.  [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 onedimensional chain; (ii) the twodimensional square lattice; (iii) the twodimensional triangular lattice; (iv) the threedimensional cubic lattice. We raise and answer the question: (a) Under what conditions is the stationary state described by the equilibrium BoltzmannGibbs 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 $q \ge 3$. 
Article: Geometry of phase separation
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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 "firstorder" approximation for the distributions. In contrast to the celebrated LifshitzSlyozovWagner 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 cAtau 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 LifshitzSlyozovWagner 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 twodimensional Ising model.  [Show abstract] [Hide abstract]
ABSTRACT: We obtain the exact distribution of the areas enclosed by domain boundaries (`hulls') during the coarsening dynamics of a twodimensional 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.  [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 curvaturedriven process. We test this prediction using the recently obtained exact result for the distribution of hullenclosed areas in twodimensional 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 bentcore liquid crystals belongs to the AllenCahn 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.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the statistics of the areas enclosed by domain boundaries (‘hulls’) during the curvaturedriven coarsening dynamics of a twodimensional 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 ‘meanfield’ 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. 
Article: Phase Transition in a Random Minima Model: Mean Field Theory and Exact Solution on the Bethe
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ABSTRACT: We consider the number and distribution of minima in random landscapes defined on nonEuclidean lattices. Using an ensemble where random landscapes are reweighted by a fugacity factor $z$ for each minimum they contain, we construct first a `twobox' 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.  [Show abstract] [Hide abstract]
ABSTRACT: The domain morphology of weakly disordered ferromagnets, quenched from the hightemperature phase to the lowtemperature phase, is studied using numerical simulations. We find that the geometrical properties of the coarsening domain structure, e.g. the distributions of hullenclosed 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 "superuniversality" property previously noted for the pair correlation function.  [Show abstract] [Hide abstract]
ABSTRACT: We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvaturedriven twodimensional 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.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a Brownian particle, with diffusion constant D, moving inside an expanding ddimensional spherewhose 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 r0 from the centre. The asymptotic (t >infinity) probability, Q, obtained by integrating over all final positions, that the particle survives, starting from the centre of the sphere, is given by Q = [4/Gamma(nu+1)lambda(nu+1)]Sigma(n) b(n) exp[(alpha(n)(nu))(2)/lambda], where lambda = cL(0)/D, b(n) = (alpha(n)(nu))(2 nu)/[J(nu+1)(alpha(n)(nu))](2), nu = (d  2)/2 and alpha(n)(nu) is the nth positive zero of the Bessel function J(nu)(z). The cases d = 1 and d = 3 are especially simple, and may be solved elegantly using backward FokkerPlanck methods.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the statistics of the areas enclosed by domain boundaries ("hulls") during the curvaturedriven coarsening dynamics of a twodimensional 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.  [Show abstract] [Hide abstract]
ABSTRACT: We calculate the average number of critical points of a Gaussian field on a highdimensional 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.  [Show abstract] [Hide abstract]
ABSTRACT: We calculate the exact asymptotic survival probability, Q, of a onedimensional Brownian particle, initially located located at the point x in (L,L), in the presence of two moving absorbing boundaries located at \pm(L+ct). The result is Q(y,\lambda) = \sum_{n=\infty}^\infty (1)^n \cosh(ny) \exp(n^2\lambda), 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.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a nonGaussian stochastic process where a particle diffuses in the ydirection, dy/dt = eta(t), subject to a transverse shear flow in the xdirection, dx/dt = f(y). Absorption with probability p occurs at each crossing of the line x = 0. We treat the class of models defined by f(y) = ±v±(±y)alpha where the upper (lower) sign refers to y > 0 (y < 0). We show that the particle survives up to time t with probability Q(t) ~ ttheta(p) and we derive an explicit expression for theta(p) in terms of alpha and the ratio v+/v. From theta(p) we deduce the mean and variance of the density of crossings of the line x = 0 for this class of nonGaussian processes.  [Show abstract] [Hide abstract]
ABSTRACT: The ordered phase of shortrange spin glasses is described in terms of the scaling behaviour associated with a zerotemperature 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 longdistance behaviour of connected correlation functions, G(r) ∝ r−y, and the singular response to a weak magnetic field, msing ∝ hd/(d2y), The decay of the connected correlation functions at large distances implies that the purestate overlap distribution function P(q) is trivial, in contrast to the SherringtonKirkpatrick model. The dynamics of the system are also discussed, as is the extension to vector spin models. 
Article: Mechanism for the failure of the Edwards hypothesis in the SherringtonKirkpatrick spin glass
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ABSTRACT: The dynamics of the SherringtonKirkpatrick 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.  [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 OhtaJasnowKawasaki (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. 
Article: Free energy landscapes, dynamics and the edge of chaos in meanfield models of spin glasses
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ABSTRACT: Metastable states in Ising spinglass models are studied by finding iterative solutions of meanfield equations for the local magnetizations. Two different equations are studied: the TAP equations which are exact for the SK model, and the simpler `naivemeanfield' (NMF) equations. The freeenergy 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 indexone (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 (ff_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 ff_0 is of order 1/N, the barriers scale as N^{1/3}, but between states for which ff_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 timeconsuming computing methods and suggest that this method may be of general utility. Comment: 13 pages
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9k  Citations  
624.27  Total Impact Points  
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Institutions

19772013

The University of Manchester
 • School of Physics and Astronomy
 • Theoretical Physics Division
Manchester, England, United Kingdom


2003

Paul Sabatier University  Toulouse III
 Laboratoire de Physique Théorique  UMR 5152  LPT
Tolosa de Llenguadoc, MidiPyrénées, France


2001

University of Maryland, College Park
Maryland, United States 
University of Kent
Cantorbery, England, United Kingdom


2000

Imperial College London
 Department of Mathematics
London, ENG, United Kingdom


1998

Jawaharlal Nehru University
 School of Physical Sciences
New Delhi, NCT, India
