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Alan J. Bray

Publications and source records attributed to Alan J. Bray.

At least 37 records · Page 2Linked to original sources
arXiv2003

Persistence Exponents and the Statistics of Crossings and Occupation Times for Gaussian Stationary Processes

We consider the persistence probability, the occupation-time distribution and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the `partial survival' probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2001)] to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d^n x/dt^n=η(t) with n > 2, incorporating an extrapolation of the series to the limit of continuous sampling. We extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d^n x/dt^n=η(t) processes for n=1 (random walk), n=2 (random acceleration) and larger n, and to diffusion from random initial conditions in 1-3 dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.

cond-mat.stat-mech↗
arXiv2003

Survival Probability of a Ballistic Tracer Particle in the Presence of Diffusing Traps

We calculate the survival probability P_S(t) up to time t of a tracer particle moving along a deterministic trajectory in a continuous d-dimensional space in the presence of diffusing but mutually noninteracting traps. In particular, for a tracer particle moving ballistically with a constant velocity c, we obtain an exact expression for P_S(t), valid for all t, for d<2. For d \geq 2, we obtain the leading asymptotic behavior of P_S(t) for large t. In all cases, P_S(t) decays exponentially for large t, P_S(t) \sim \exp(-θt). We provide an explicit exact expression for the exponent θin dimensions d \leq 2, and for the physically relevant case, d=3, as a function of the system parameters.

cond-mat.stat-mech↗
arXiv2003

Persistence of Manifolds in Nonequilibrium Critical Dynamics

We study the persistence P(t) of the magnetization of a d' dimensional manifold (i.e., the probability that the manifold magnetization does not flip up to time t, starting from a random initial condition) in a d-dimensional spin system at its critical point. We show analytically that there are three distinct late time decay forms for P(t) : exponential, stretched exponential and power law, depending on a single parameter ζ=(D-2+η)/z where D=d-d' and η, z are standard critical exponents. In particular, our theory predicts that the persistence of a line magnetization decays as a power law in the d=2 Ising model at its critical point. For the d=3 critical Ising model, the persistence of the plane magnetization decays as a power law, while that of a line magnetization decays as a stretched exponential. Numerical results are consistent with these analytical predictions.

cond-mat.stat-mech↗
arXiv2002

Formal Solution of a Class of Reaction-Diffusion Models: Reduction to a Single-Particle Problem

We consider the trapping reaction A + B -> B in space dimension d<=2. By formally eliminating the B particles from the problem we derive an effective dynamics for the A particles from which the survival probability of a given A particle and the statistics of its spatial fluctuations can be calculated in a rather general way. The method can be extended to the study of annihilation/coalescence reactions, B + B -> 0 or B, in d=2.

cond-mat.stat-mech↗
arXiv2002

The stiffness exponent of two-dimensional Ising spin glasses for non-periodic boundary conditions using aspect-ratio scaling

We study the scaling behavior of domain-wall energies in two-dimensional Ising spin glasses with Gaussian and bimodal distributions of the interactions and different types of boundary conditions. The domain walls are generated by changing the boundary conditions at T=0 in one direction. The ground states of the original and perturbed system are calculated numerically by applying an efficient matching algorithm. Systems of size LxM with different aspect-ratios 1/8 <= L/M <= 64 are considered. For Gaussian interactions, using the aspect-ratio scaling approach, we find a stiffness exponent theta=-0.287(4), which is independent of the boundary conditions in contrast to earlier results. Furthermore, we find a scaling behavior of the domain-wall energy as predicted by the aspect-ratio approach. Finally, we show that this approach does not work for the bimodal distribution of interactions.

cond-mat.dis-nn↗
arXiv2002

Exact Asymptotics for One-Dimensional Diffusion with Mobile Traps

We consider a diffusing particle, with diffusion constant D', moving in one dimension in an infinite sea of noninteracting mobile traps with diffusion constant D and density rho. We show that the asymptotic behavior of the survival probability, P(t), is given by P(t) ~ exp[-4 rho(Dt/pi)^{1/2}], independent of D'. The result comes from obtaining upper and lower bounds on P(t), and showing that they coincide asymptotically. We also obtain exact results for P(t) to first order in D' for an arbitrary finite number of traps.

cond-mat.stat-mech↗
arXiv2001

Persistence of a Continuous Stochastic Process with Discrete-Time Sampling: Non-Markov Processes

We consider the problem of `discrete-time persistence', which deals with the zero-crossings of a continuous stochastic process, X(T), measured at discrete times, T = n(ΔT). For a Gaussian Stationary Process the persistence (no crossing) probability decays as exp(-θ_D T) = [ρ(a)]^n for large n, where a = \exp[-(ΔT)/2], and the discrete persistence exponent, θ_D, is given by θ_D = \ln(ρ)/2\ln(a). Using the `Independent Interval Approximation', we show how θ_D varies with (ΔT) for small (ΔT) and conclude that experimental measurements of persistence for smooth processes, such as diffusion, are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker. We extend the matrix method developed by us previously [Phys. Rev. E 64, 015151(R) (2001)] to determine ρ(a) for a two-dimensional random walk and the one-dimensional random acceleration problem. We also consider `alternating persistence', which corresponds to a < 0, and calculate ρ(a) for this case.

cond-mat.stat-mech↗
arXiv2001

Series Expansion Calculation of Persistence Exponents

We consider an arbitrary Gaussian Stationary Process X(T) with known correlator C(T), sampled at discrete times T_n = n ΔT. The probability that (n+1) consecutive values of X have the same sign decays as P_n \sim \exp(-θ_D T_n). We calculate the discrete persistence exponent θ_D as a series expansion in the correlator C(ΔT) up to 14th order, and extrapolate to ΔT = 0 using constrained Padé approximants to obtain the continuum persistence exponent θ. For the diffusion equation our results are in exceptionally good agreement with recent numerical estimates.

cond-mat.stat-mech↗
arXiv2001

Persistence of a Continuous Stochastic Process with Discrete-Time Sampling

We introduce the concept of `discrete-time persistence', which deals with zero-crossings of a continuous stochastic process, X(T), measured at discrete times, T = n ΔT. For a Gaussian Markov process with relaxation rate μ, we show that the persistence (no crossing) probability decays as ρ(a)^n for large n, where a = \exp(-μΔT), and we compute ρ(a) to high precision. We also define the concept of `alternating persistence', which corresponds to a<0. For a>1, corresponding to motion in an unstable potential (μ<0), there is a nonzero probability of having no zero-crossings in infinite time, and we show how to calculate it.

cond-mat.stat-mech↗
arXiv2001

Interface Fluctuations, Burgers Equations, and Coarsening under Shear

We consider the interplay of thermal fluctuations and shear on the surface of the domains in various systems coarsening under an imposed shear flow. These include systems with nonconserved and conserved dynamics, and a conserved order parameter advected by a fluid whose velocity field satisfies the Navier-Stokes equation. In each case the equation of motion for the interface height reduces to an anisotropic Burgers equation. The scaling exponents that describe the growth and coarsening of the interface are calculated exactly in any dimension in the case of conserved and nonconserved dynamics. For a fluid-advected conserved order parameter we determine the exponents, but we are unable to build a consistent perturbative expansion to support their validity.

cond-mat.stat-mech↗
arXiv2001

Interface Fluctuations under Shear

Coarsening systems under uniform shear display a long time regime characterized by the presence of highly stretched and thin domains. The question then arises whether thermal fluctuations may actually destroy this layered structure. To address this problem in the case of non-conserved dynamics we study an anisotropic version of the Burgers equation, constructed to describe thermal fluctuations of an interface in the presence of a uniform shear flow. As a result, we find that stretched domains are only marginally stable against thermal fluctuations in $d=2$, whereas they are stable in $d=3$.

cond-mat.stat-mech↗
arXiv2000

Dynamics and delocalisation transition for an interface driven by a uniform shear flow

We study the effect of a uniform shear flow on an interface separating the two broken-symmetry ordered phases of a two-dimensional system with nonconserved scalar order parameter. The interface, initially flat and perpendicular to the flow, is distorted by the shear flow. We show that there is a critical shear rate, γ_c, proportional to 1/L^2, (where L is the system width perpendicular to the flow) below which the interface can sustain the shear. In this regime the countermotion of the interface under its curvature balances the shear flow, and the stretched interface stabilizes into a time-independent shape whose form we determine analytically. For γ> γ_c, the interface acquires a non-zero velocity, whose profile is shown to reach a time-independent limit which we determine exactly. The analytical results are checked by numerical integration of the equations of motion.

cond-mat.stat-mech↗
arXiv2000

Spatial Persistence of Fluctuating Interfaces

We show that the probability, P_0(l), that the height of a fluctuating (d+1)-dimensional interface in its steady state stays above its initial value up to a distance l, along any linear cut in the d-dimensional space, decays as P_0(l) \sim l^(-θ). Here θis a `spatial' persistence exponent, and takes different values, θ_s or θ_0, depending on how the point from which l is measured is specified. While θ_s is related to fractional Brownian motion, and can be determined exactly, θ_0 is non-trivial even for Gaussian interfaces.

cond-mat.stat-mech↗
arXiv2000

Ohta-Jasnow-Kawasaki Approximation for Nonconserved Coarsening under Shear

We analytically study coarsening dynamics in a system with nonconserved scalar order parameter, when a uniform time-independent shear flow is present. We use an anisotropic version of the Ohta-Jasnow-Kawasaki approximation to calculate the growth exponents in two and three dimensions: for d=3 the exponents we find are the same as expected on the basis of simple scaling arguments, that is 3/2 in the flow direction and 1/2 in all the other directions, while for d=2 we find an unusual behavior, in that the domains experience an unlimited narrowing for very large times and a nontrivial dynamical scaling appears. In addition, we consider the case where an oscillatory shear is applied to a two-dimensional system, finding in this case a standard t^1/2 growth, modulated by periodic oscillations. We support our two-dimensional results by means of numerical simulations and we propose to test our predictions by experiments on twisted nematic liquid crystals.

cond-mat.stat-mech↗
arXiv2000

Novel glassy behavior in a ferromagnetic p-spin model

Recent work has suggested the existence of glassy behavior in a ferromagnetic model with a four-spin interaction. Motivated by these findings, we have studied the dynamics of this model using Monte Carlo simulations with particular attention being paid to two-time quantities. We find that the system shares many features in common with glass forming liquids. In particular, the model exhibits: (i) a very long-lived metastable state, (ii) autocorrelation functions that show stretched exponential relaxation, (iii) a non-equilibrium timescale that appears to diverge at a well defined temperature, and (iv) low temperature aging behaviour characteristic of glasses.

cond-mat.stat-mech↗
arXiv2000

Coarsening Dynamics of a Nonconserved Field Advected by a Uniform Shear Flow

We consider the ordering kinetics of a nonconserved scalar field advected by a uniform shear flow. Using the Ohta-Jasnow-Kawasaki approximation, modified to allow for shear-induced anisotropy, we calculate the asymptotic time dependence of the characteristic length scales, L_parallel and L_perp, that describe the growth of order parallel and perpendicular to the mean domain orientation. In space dimension d=3 we find, up to constants, L_parallel = gamma t^{3/2}, L_perp = t^{1/2}, where gamma is the shear rate, while for d = 2 we find L_parallel = gamma^{1/2} t (ln t)^{1/4}, L_perp = gamma^{-1/2}(ln t)^{-1/4} . Our predictions for d=2 can be tested by experiments on twisted nematic liquid crystals.

cond-mat.stat-mech↗
arXiv1998

Survival-Time Distribution for Inelastic Collapse

In a recent publication [PRL {\bf 81}, 1142 (1998)] it was argued that a randomly forced particle which collides inelastically with a boundary can undergo inelastic collapse and come to rest in a finite time. Here we discuss the survival probability for the inelastic collapse transition. It is found that the collapse-time distribution behaves asymptotically as a power-law in time, and that the exponent governing this decay is non-universal. An approximate calculation of the collapse-time exponent confirms this behaviour and shows how inelastic collapse can be viewed as a generalised persistence phenomenon.

cond-mat.stat-mech↗