arXiv subjects

Alan J. Bray

Publications and source records attributed to Alan J. Bray.

At least 19 recordsLinked to original sources

arXiv2013

Persistence and First-Passage Properties in Non-equilibrium Systems

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.

cond-mat.stat-mech↗

arXiv2010

Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers

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^{-δ}, where δ= (2π-θ)/(π-θ) and θ= cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also determined exactly.

cond-mat.stat-mech↗

arXiv2009

Nonequilibrium Stationary States and Phase Transitions in Directed Ising Models

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 $q \ge 3$.

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arXiv2008

Geometry of phase separation

We study the domain geometry during spinodal decomposition of a 50:50 binary mixture in two dimensions. Extending arguments developed to treat non-conserved 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 non-conserved 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 cut-off. 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 $c A^{-τ}$ tail, where $c$ and $τ$ 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 $A^{1/2}$, as in the Lifshitz-Slyozov-Wagner theory. The threshold between the small and large regimes is determined by the characteristic area, ${\rm A} \sim [λ(T) t]^{2/3}$. 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 2d Ising Model.

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arXiv2008

Experimental test of curvature-driven dynamics in the phase ordering of a two dimensional liquid crystal

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 exact result for the distribution of hull-enclosed areas in two-dimensional coarsening in non-conserved scalar order parameter dynamics recently obtained [J.J. Arenzon et al., Phys. Rev. Lett. 98, 061116 (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.

cond-mat.dis-nn↗

arXiv2007

Geometric properties of two-dimensional coarsening with weak disorder

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.

cond-mat.dis-nn↗

arXiv2007

Domain growth morphology in curvature driven two dimensional coarsening

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 $n_h(A,t) = 2c_h/(A + λ_h t)^2$, where $c_h=1/8π\sqrt{3} \approx 0.023$ is a universal constant and $λ_h$ is a material parameter. For a critical initial condition, the same form is obtained, with the same $λ_h$ 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 $n_d(A,t) = 2c_d (λ_d t)^{τ'-2}/(A + λ_d t)^{τ'}$, where $c_d=c_h + {\cal O}(c_h^2)$, $λ_d=λ_h + {\cal O}(c_h)$, and $τ' = 187/91 \approx 2.055$. For critical initial conditions, one replaces $c_d$ by $c_d/2$ (possibly with corrections of ${\cal O}(c_h^2)$) and the exponent is $τ= 379/187 \approx 2.027$. 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.

cond-mat.stat-mech↗

arXiv2007

Survival probability of a diffusing particle constrained by two moving, absorbing boundaries

We calculate the exact asymptotic survival probability, Q, of a one-dimensional 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,λ) = \sum_{n=-\infty}^\infty (-1)^n \cosh(ny) \exp(-n^2λ), where y=cx/D, λ= 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.

cond-mat.stat-mech↗

arXiv2007

Exact results for curvature-driven coarsening in two dimensions

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 $N_h(A,t) = 2c/(A+λt)$, demonstrating the validity of dynamical scaling in this system, where $c=1/8π\sqrt{3}$ is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of $A/λt$. Identical forms are obtained for coarsening from a critical initial state, but with $c$ replaced by $c/2$.

cond-mat.stat-mech↗

arXiv2006

The statistics of critical points of Gaussian fields on large-dimensional spaces

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 to landscape scenarios coming from the anthropic approach to string theory.

cond-mat.dis-nn↗

arXiv2006

Partial Survival and Crossing Statistics for a Diffusing Particle in a Transverse Shear Flow

We consider a non-Gaussian stochastic process where a particle diffuses in the $y$-direction, $dy/dt=η(t)$, subject to a transverse shear flow in the $x$-direction, $dx/dt=f(y)$. Absorption with probability $p$ occurs at each crossing of the line $x=0$. We treat the class of models defined by $f(y) = \pm v_{\pm}(\pm y)^α$ where the upper (lower) sign refers to $y>0$ ($y<0$). We show that the particle survives up to time $t$ with probability $Q(t) \sim t^{-θ(p)}$ and we derive an explicit expression for $θ(p)$ in terms of $α$ and the ratio $v_+/v_-$. From $θ(p)$ we deduce the mean and variance of the density of crossings of the line $x=0$ for this class of non-Gaussian processes.

cond-mat.stat-mech↗

arXiv2005

Survival probabilities in the double trapping reaction A +B -> B, B + C -> C

We consider the double trapping reaction A + B -> B, B + C -> C in one dimension. The survival probability of a given A particle is calculated under various conditions on the diffusion constants of the reactants, and on the ratio of initial B and C particle densities. The results are of more general form than those obtained in previous work on the problem.

cond-mat.stat-mech↗

arXiv2005

Survival of a Diffusing Particle in a Transverse Flow Field

We consider a particle diffusing in the y-direction, dy/dt=η(t) where η(t) is Gaussian white noise, and subject to a transverse flow field in the x-direction, dx/dt=f(y), where x \ge 0 and x=0 is an absorbing boundary. We discuss the time-dependence of the survival probability of the particle for a class of functions f(y) that are positive in some regions of space and negative in others.

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arXiv2004

Drowsy Cheetah Hunting Antelopes: A Diffusing Predator Seeking Fleeing Prey

We consider a system of three random walkers (a `cheetah' surrounded by two `antelopes') diffusing in one dimension. The cheetah and the antelopes diffuse, but the antelopes experience in addition a deterministic relative drift velocity, away from the cheetah, proportional to their distance from the cheetah, such that they tend to move away from the cheetah with increasing time. Using the backward Fokker-Planck equation we calculate, as a function of their initial separations, the probability that the cheetah has caught neither antelope after infinite time.

cond-mat.stat-mech↗

arXiv2004

Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process

We introduce a general method, based on a mapping onto quantum mechanics, for investigating the large-T limit of the distribution P(r,T) of the nonlinear functional r[V] = (1/T)\int_0^T dT' V[X(T')], where V(X) is an arbitrary function of the stationary Gaussian Markov process X(T). For T tending to infinity at fixed r we find that P(r,T) behaves as exp[-theta(r) T], where theta(r) is a large deviation function. We present explicit results for a number of special cases, including the case V(X) = X θ(X) which is related to the cooling and the heating degree days relevant to weather derivatives.

cond-mat.stat-mech↗

arXiv2004

Approach to Asymptotic Behaviour in the Dynamics of the Trapping Reaction

We consider the trapping reaction A + B -> B in space dimension d=1, where the A and B particles have diffusion constants D_A, D_B respectively. We calculate the probability, Q(t), that a given A particle has not yet reacted at time t. Exploiting a recent formulation in which the B particles are eliminated from the problem we find, for t -> \infty, $Q(t) \sim \exp[-(4/\sqrtπ)(ρ^2 D_Bt)^{1/2} - (C ρ^2 D_A t)^{1/3} + ...]$, where $ρ$ is the density of B particles and $C \propto D_A/D_B$ for $D_A/D_B << 1$.

cond-mat.stat-mech↗

arXiv2004

Survival of a Diffusing Particle in a Transverse Shear Flow: A First-Passage Problem with Continuously Varying Persistence Exponent

We consider a particle diffusing in the y-direction, dy/dt=η(t), subject to a transverse shear flow in the x-direction, dx/dt=f(y), where x \ge 0 and x=0 is an absorbing boundary. We treat the class of models defined by f(y) = \pm v_{\pm}(\pm y)^αwhere the upper (lower) sign refers to y>0 (y<0). We show that the particle survives with probability Q(t) \sim t^{-θ} with θ= 1/4, independent of α, if v_{+}=v_{-}. If v_{+} \ne v_{-}, however, we show that θdepends on both αand the ratio v_{+}/v_{-}, and we determine this dependence.

cond-mat.stat-mech↗

arXiv2004

Vicious Walkers in a Potential

We consider N vicious walkers moving in one dimension in a one-body potential v(x). Using the backward Fokker-Planck equation we derive exact results for the asymptotic form of the survival probability Q(x,t) of vicious walkers initially located at (x_1,...,x_N) = x, when v(x) is an arbitrary attractive potential. Explicit results are given for a square-well potential with absorbing or reflecting boundary conditions at the walls, and for a harmonic potential with an absorbing or reflecting boundary at the origin and the walkers starting on the positive half line. By mapping the problem of N vicious walkers in zero potential onto the harmonic potential problem, we rederive the results of Fisher [J. Stat. Phys. 34, 667 (1984)] and Krattenthaler et al. [J. Phys. A 33}, 8835 (2000)] respectively for vicious walkers on an infinite line and on a semi-infinite line with an absorbing wall at the origin. This mapping also gives a new result for vicious walkers on a semi-infinite line with a reflecting boundary at the origin: Q(x,t) \sim t^{-N(N-1)/2}.

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