Survival Probability of a Gaussian Non-Markovian Process: Application to the T=0 Dynamics of the Ising Model

Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks

Persistence, defined as the probability that a signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. Often, persistence decays algebraically with time with non trivial exponents. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing transiently after an imposed initial perturbation. Here, we introduce a theoretical framework that enables the non-perturbative determination of persistence exponents of Gaussian non-Markovian processes with non stationary dynamics relaxing to a steady state after an initial perturbation. Two situations are analyzed: either the system is subjected to a temperature quench at initial time, or its past trajectory is assumed to have been observed and thus known. Our theory covers the case of spatial dimension higher than one, opening the way to characterize non-trivial reaction kinetics for complex systems with non-equilibrium initial conditions.

Zero-temperature coarsening in the two-dimensional long-range Ising model

We investigate the nonequilibrium dynamics following a quench to zero temperature of the nonconserved Ising model with power-law decaying long-range interactions ∝1/r^{d+σ} in d=2 spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent α, the persistence exponent θ, and the fractal dimension d_{f}. It is found that the growth exponent α≈3/4 is independent of σ and different from α=1/2, as expected for nearest-neighbor models. In the large σ regime of the tunable interactions only the fractal dimension d_{f} of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponents θ this is a direct consequence of the different growth exponents α as can be understood from the relation d-d_{f}=θ/α; they just differ by the ratio of the growth exponents ≈3/2. This relation has been proposed for annihilation processes and later numerically tested for the d=2 nearest-neighbor Ising model. We confirm this relation for all σ studied, reinforcing its general validity.

Anomalous persistence exponents for normal yet aging diffusion

The persistence exponent θ, which characterizes the long-time decay of the survival probability of stochastic processes in the presence of an absorbing target, plays a key role in quantifying the dynamics of fluctuating systems. So far, anomalous values of the persistence exponent (θ≠1/2) were obtained, but only for anomalous processes (i.e., with Hurst exponent H≠1/2). Here we exhibit examples of ageing processes which, even if they display asymptotically a normal diffusive scaling (H=1/2), are characterized by anomalous persistent exponents that we determine analytically. Based on this analysis, we propose the following general criterion: The persistence exponent of asymptotically diffusive processes is anomalous if the increments display ageing and depend on the observation time T at all timescales.

Statistics of occupation times and connection to local properties of nonhomogeneous random walks

We consider the statistics of occupation times, the number of visits at the origin, and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single exponent, that is connected to a local property of the probability density function of the process, viz., the probability of occupying the origin at time t, P(t). We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: When P(t) shows the same long-time behavior, each observable follows indeed the same distribution.

Persistence in Brownian motion of an ellipsoidal particle in two dimensions

We investigate the persistence probability p(t) of the position of a Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability that a stochastic variable has not changed its sign in the given time interval. We explicitly consider two cases-diffusion of a free particle and that of a harmonically trapped particle. The latter is particularly relevant in experiments that use trapping and tracking techniques to measure the displacements. We provide analytical expressions of p(t) for both the scenarios and show that in the absence of the shape asymmetry, the results reduce to the case of an isotropic particle. The analytical expressions of p(t) are further validated against numerical simulation of the underlying overdamped dynamics. We also illustrate that p(t) can be a measure to determine the shape asymmetry of a colloid and the translational and rotational diffusivities can be estimated from the measured persistence probability. The advantage of this method is that it does not require the tracking of the orientation of the particle.

Universal first-passage statistics in aging media

It has been known for a long time that the kinetics of diffusion-limited reactions can be quantified by the time needed for a diffusing molecule to reach a target: the first-passage time (FPT). So far the general determination of the mean first-passage time to a target in confinement has left aside aging media, such as glassy materials, cellular media, or cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant diffusion processes, which model a broad class of transport processes of molecules in aging media, and demonstrate that all the moments of the FPT obey universal scalings with the confining volume with nontrivial exponents. Our analysis shows that a nonlinear scaling with the volume of the mean FPT, which quantities the mean reaction time, is the hallmark of aging and provides a general tool to quantify its impact on reaction kinetics in confinement.

Persistence of non-Markovian Gaussian stationary processes in discrete time

The persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval. Although persistence is a simple concept to understand, it is in general hard to calculate. Here we consider zero mean Gaussian stationary processes in discrete time n. Few results are known for the persistence P_{0}(n) in discrete time, except the large time behavior which is characterized by the nontrivial constant θ through P_{0}(n)∼θ^{n}. Using a modified version of the independent interval approximation (IIA) that we developed before, we are able to calculate P_{0}(n) analytically in z-transform space in terms of the autocorrelation function A(n). If A(n)→0 as n→∞, we extract θ numerically, while if A(n)=0, for finite n>N, we find θ exactly (within the IIA). We apply our results to three special cases: the nearest-neighbor-correlated "first order moving average process", where A(n)=0 for n>1, the double exponential-correlated "second order autoregressive process", where A(n)=c_{1}λ_{1}^{n}+c_{2}λ_{2}^{n}, and power-law-correlated variables, where A(n)∼n^{-μ}. Apart from the power-law case when μ<5, we find excellent agreement with simulations.

Long-time predictability in disordered spin systems following a deep quench

We study the problem of predictability, or "nature vs nurture," in several disordered Ising spin systems evolving at zero temperature from a random initial state: How much does the final state depend on the information contained in the initial state, and how much depends on the detailed history of the system? Our numerical studies of the "dynamical order parameter" in Edwards-Anderson Ising spin glasses and random ferromagnets indicate that the influence of the initial state decays as dimension increases. Similarly, this same order parameter for the Sherrington-Kirkpatrick infinite-range spin glass indicates that this information decays as the number of spins increases. Based on these results, we conjecture that the influence of the initial state on the final state decays to zero in finite-dimensional random-bond spin systems as dimension goes to infinity, regardless of the presence of frustration. We also study the rate at which spins "freeze out" to a final state as a function of dimensionality and number of spins; here the results indicate that the number of "active" spins at long times increases with dimension (for short-range systems) or number of spins (for infinite-range systems). We provide theoretical arguments to support these conjectures, and also study analytically several mean-field models: the random energy model, the uniform Curie-Weiss ferromagnet, and the disordered Curie-Weiss ferromagnet. We find that for these models, the information contained in the initial state does not decay in the thermodynamic limit-in fact, it fully determines the final state. Unlike in short-range models, the presence of frustration in mean-field models dramatically alters the dynamical behavior with respect to the issue of predictability.

Interplay of interfacial noise and curvature-driven dynamics in two dimensions

We explore the effect of interplay of interfacial noise and curvature-driven dynamics in a binary spin system. An appropriate model is the generalized two-dimensional voter model proposed earlier [M. J. de Oliveira, J. F. F. Mendes, and M. A. Santos, J. Phys. A: Math. Gen. 26, 2317 (1993)JPHAC50305-447010.1088/0305-4470/26/10/006], where the flipping probability of a spin depends on the state of its neighbors and is given in terms of two parameters, x and y. x=0.5andy=1 correspond to the conventional voter model which is purely interfacial noise driven, while x=1 and y=1 correspond to the Ising model, where coarsening is fully curvature driven. The coarsening phenomena for 0.5<x<1 keeping y=1 is studied in detail. The dynamical behavior of the relevant quantities show characteristic differences from both x=0.5 and 1. The most remarkable result is the existence of two time scales for x≥x_{c} where x_{c}≈0.7. On the other hand, we have studied the exit probability which shows Ising-like behavior with a universal exponent for any value of x>0.5; the effect of x appears in altering the value of the parameter occurring in the scaling function only.

How long do particles spend in vortical regions in turbulent flows?

We obtain the probability distribution functions (PDFs) of the time that a Lagrangian tracer or a heavy inertial particle spends in vortical or strain-dominated regions of a turbulent flow, by carrying out direct numerical simulations of such particles advected by statistically steady, homogeneous, and isotropic turbulence in the forced, three-dimensional, incompressible Navier-Stokes equation. We use the two invariants, Q and R, of the velocity-gradient tensor to distinguish between vortical and strain-dominated regions of the flow and partition the Q-R plane into four different regions depending on the topology of the flow; out of these four regions two correspond to vorticity-dominated regions of the flow and two correspond to strain-dominated ones. We obtain Q and R along the trajectories of tracers and heavy inertial particles and find out the time t_{pers} for which they remain in one of the four regions of the Q-R plane. We find that the PDFs of t_{pers} display exponentially decaying tails for all four regions for tracers and heavy inertial particles. From these PDFs we extract characteristic time scales, which help us to quantify the time that such particles spend in vortical or strain-dominated regions of the flow.

Perturbative expansion for the maximum of fractional Brownian motion

Brownian motion is the only random process which is Gaussian, scale invariant, and Markovian. Dropping the Markovian property, i.e., allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst exponent H. For H=1/2, Brownian motion is recovered. We develop a perturbative approach to treat the nonlocality in time in an expansion in ɛ=H-1/2. This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: the maximum m of the process and the time t_{max} at which this maximum is reached, as well as their joint distribution. We test our analytical predictions with extensive numerical simulations for different values of H. They show excellent agreement, even for H far from 1/2.

Minority-spin dynamics in the nonhomogeneous Ising model: Diverging time scales and exponents

We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins 0≤x≤1. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for x≤0.5, persistence for the up (minority) spins shows the behavior P_{min}(t)∼t^{-γ}exp[-(t/τ)^{δ}] with time t, while for the down (majority) spins, P_{maj}(t) approaches a finite value. We find that the timescale τ diverges as (x_{c}-x)^{-λ}, where x_{c}=0.5 and λ≃2.31. The exponent γ varies as θ_{2d}+c_{0}(x_{c}-x)^{β} where θ_{2d}≃0.215 is very close to the persistence exponent in two dimensions; β≃1. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form E(x)=f[(x-x_{c}/x_{c})L^{1/ν}], with ν≈1.47. This result suggests that τ∼L^{z[over ̃]}, where z[over ̃]=λ/ν=1.57±0.11 is an exponent not explored earlier.

Distribution of zeros in the rough geometry of fluctuating interfaces

We study numerically the correlations and the distribution of intervals between successive zeros in the fluctuating geometry of stochastic interfaces, described by the Edwards-Wilkinson equation. For equilibrium states we find that the distribution of interval lengths satisfies a truncated Sparre-Andersen theorem. We show that boundary-dependent finite-size effects induce nontrivial correlations, implying that the independent interval property is not exactly satisfied in finite systems. For out-of-equilibrium nonstationary states we derive the scaling law describing the temporal evolution of the density of zeros starting from an uncorrelated initial condition. As a by-product we derive a general criterion of the von Neumann's type to understand how discretization affects the stability of the numerical integration of stochastic interfaces. We consider both diffusive and spatially fractional dynamics. Our results provide an alternative experimental method for extracting universal information of fluctuating interfaces such as domain walls in thin ferromagnets or ferroelectrics, based exclusively on the detection of crossing points.

Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory

Fractional Brownian motion is a non-Markovian Gaussian process X_{t}, indexed by the Hurst exponent H. It generalizes standard Brownian motion (corresponding to H=1/2). We study the probability distribution of the maximum m of the process and the time t_{max} at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting H=1/2+ϵ. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of H test these analytical predictions and show excellent agreement, even for large ϵ.

Nature versus nurture: predictability in low-temperature Ising dynamics

Consider a dynamical many-body system with a random initial state subsequently evolving through stochastic dynamics. What is the relative importance of the initial state ("nature") versus the realization of the stochastic dynamics ("nurture") in predicting the final state? We examined this question for the two-dimensional Ising ferromagnet following an initial deep quench from T=∞ to T=0. We performed Monte Carlo studies on the overlap between "identical twins" raised in independent dynamical environments, up to size L=500. Our results suggest an overlap decaying with time as t(-θ)(h) with θ(h)=0.22 ± 0.02; the same exponent holds for a quench to low but nonzero temperature. This "heritability exponent" may equal the persistence exponent for the two-dimensional Ising ferromagnet, but the two differ more generally.

Time-dependent Monte Carlo simulations of critical and Lifshitz points of the axial-next-nearest-neighbor Ising model

In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: <M>(m(0)=1)~t(-β/νz), which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent z, evaluated from the behavior of the ratio F(2)(t)=<M(2)>(m(0)=0)/<M>(2)(m(0)=1)~t(3/z), along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent θ that characterizes the initial slip of magnetization and the global persistence exponent θ(g) associated with the probability P(t) that magnetization keeps its signal up to time t. Our estimates for the dynamic critical exponents at the Lifshitz point are z=2.34(2) and θ(g)=0.336(4), values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the z axis, i.e., J(2)=0), i.e., z≈2.07 and θ(g)≈0.38. We also present estimates for the static critical exponents β and ν, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.

Non-Gaussianity and dynamical trapping in locally activated random walks

We propose a minimal model of locally activated diffusion, in which the diffusion coefficient of a one-dimensional Brownian particle is modified in a prescribed way-either increased or decreased-upon each crossing of the origin. Such a local mobility decrease arises in the formation of atherosclerotic plaques due to diffusing macrophage cells accumulating lipid particles. We show that spatially localized mobility perturbations have remarkable consequences on diffusion at all scales, such as the emergence of a non-Gaussian multipeaked probability distribution and a dynamical transition to an absorbing static state. In the context of atherosclerosis, this dynamical transition can be viewed as a minimal mechanism that causes macrophages to aggregate in lipid-enriched regions and thereby to the formation of atherosclerotic plaques.

Perturbation theory for fractional Brownian motion in presence of absorbing boundaries

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations (x(t(1))x(t(2)))=D(t(1)(2H)+t(2)(2H)-|t(1)-t(2)|(2H)), where H, with 0<H<1, is called the Hurst exponent. For H=1/2, x(t) is a Brownian motion, while for H≠1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(+)(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(+)(x,t)~t(-H)R(+)(x/t(H)). Our objective is to compute the scaling function R(+)(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H=1/2+ε, to calculate the scaling function R(+)(y) to first order in ε. We find that R(+)(y) behaves as R(+)(y)~y(ϕ) as y→0 (near the absorbing boundary), while R(+)(y)~y(γ)exp(-y(2)/2) as y→∞, with ϕ=1-4ε+O(ε(2)) and γ=1-2ε+O(ε(2)). Our ε-expansion result confirms the scaling relation ϕ=(1-H)/H proposed in Zoia, Rosso, and Majumdar [Phys. Rev. Lett. 102, 120602 (2009)]. We verify our findings via numerical simulations for H=2/3. The tools developed here are versatile, powerful, and adaptable to different situations.

First-passage and first-exit times of a Bessel-like stochastic process

We study a stochastic process X(t) which is a particular case of the Rayleigh process and whose square is the Bessel process, with various applications in physics, chemistry, biology, economics, finance, and other fields. The stochastic differential equation is dX(t)=(nD/X(t))dt+√(2D)dW(t), where W(t) is the Wiener process. The drift term can arise from a logarithmic potential or from taking X(t) as the norm of a multidimensional random walk. Due to the singularity of the drift term for X(t)=0, different natures of boundary at the origin arise depending on the real parameter n: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behavior is observed in the case of a regular boundary.

Persistence in advection of a passive scalar

We consider the persistence phenomenon in advected passive scalar equation in one dimension. The velocity field is random with the v(k,omega)v(-k,-omega) approximately mid R:kmid R:;{-(2+alpha)} . In the presence of the nonlinearity the complete Green's function becomes G;{-1}=-iomega+Dk2+Sigma . We determine Sigma self-consistently from the correlation function which gives Sigma approximately k;{beta} , with beta=(1-alpha)2 . The effect of the nonlinear term in the equation in the O(;{2}) is to replace the diffusion term due to molecular viscosity by an effective term of the form Sigma_{0}k;{beta} . The stationary correlator for this system is [sech(T2)];{1beta} . Using the self-consistent theory we have determined the relation between beta and alpha . Finally, the independent interval approximation (IIA) method is used to determine the persistent exponent.

Accurate statistics of a flexible polymer chain in shear flow

We present exact and analytically accurate results for the problem of a flexible polymer chain in shear flow. Under such a flow the polymer tumbles, and the probability distribution of the tumbling times tau of the polymer decays exponentially as approximately exp(-alphatau/tau_{0}) (where tau_{0} is the longest relaxation time). We show that for a Rouse chain this nontrivial constant alpha can be calculated in the limit of a large Weissenberg number (high shear rate) and is in excellent agreement with our simulation result of alpha approximately 0.324. We also derive exactly the distribution functions for the length and the orientational angles of the end-to-end vector R of the polymer.

Crossing intervals of non-Markovian Gaussian processes

We review the properties of time intervals between the crossings at a level M of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the independent interval approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large |M|. In addition, the small-time behavior of the interval distributions and the persistence is calculated analytically, for any M. The IIA is found to reproduce most of these exact results, and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.

Fluctuations in the zeros of differentiable Gaussian processes

The stochastic point processes formed by the zero crossings or extremal points of differentiable, stationary Gaussian processes are studied as a function of their autocorrelation function. The properties of these point processes are mapped to the space formed by the parameters appearing in the autocorrelation function, their adopted form being sensitive to the structure of the autocorrelation function principally in the vicinity of the origin. The distribution for the number of zeros occurring in an asymptotically large interval are approximately negative-binomial or binomial depending upon whether the relative variance or Fano factor is greater or less than unity. The correlation properties of the zeros are such that they are repelled from each other or are "antibunched" if the autocorrelation function of the Gaussian process is characterized by a single scale size, but occur in clusters if more than one characteristic scale size is present. The intervals between zeros can be interpreted in terms of the autocorrelation function of the zeros themselves. When bunching occurs the interval density becomes bimodal, indicating the interval sizes within and between the clusters. The interevent periods are statistically dependent on one another with densities whose asymptotic behavior is governed by that of the autocorrelation function of the Gaussian process at large delay times. Poisson distributed fluctuations of the zeros occur only exceptionally but never form a Poisson process.

Statistical behavior of domain systems

We study the statistical behavior of two out of equilibrium systems. The first one is a quasi-one-dimensional gas with two species of particles under the action of an external field which drives each species in opposite directions. The second one is a one-dimensional spin system with nearest-neighbor interactions also under the influence of an external driving force. Both systems show a dynamical scaling with domain formation. The statistical behavior of these domains is compared with models based on the coalescing random walk and the interacting random walk. We find that the scaling domain size distribution of the gas and the spin systems is well-fitted by the Wigner surmise, which lead us to explore a possible connection between these systems and the circular orthogonal ensemble of random matrices. However, the study of the correlation function of the domain edges shows that the statistical behavior of the domains in both gas and spin systems is not completely well-described by a circular orthogonal ensemble, nor it is by other models proposed such as the coalescing random walk and the interacting random walk. Nevertheless, we find that a simple model of independent intervals describes more closely the statistical behavior of the domains formed in these systems.

Persistence of a Rouse polymer chain under transverse shear flow

We consider a single Rouse polymer chain in two dimensions in the presence of a transverse shear flow along the x direction and calculate the persistence probability P0(t) that the x coordinate of a bead in the bulk of the chain does not return to its initial position up to time t. We show that the persistence decays at late times as a power law P0(t) approximately t{-theta} with a nontrivial exponent theta. The analytical estimate of theta=0.359... obtained using an independent interval approximation is in excellent agreement with the numerical value theta approximately 0.360+/-0.001.

Nonequilibrium relaxations within the ground-state manifold in the antiferromagnetic Ising model on a triangular lattice

We present an extensive Monte Carlo simulation study on the nonequilibrium kinetics of triangular antiferromagnetic Ising model within the ground state ensemble which consists of sectors, each of which is characterized by a unique value of the string density p through a dimer covering method. Building upon our recent work [Phys. Rev. E 68, 066127 (2003)] where we considered the nonequilibrium relaxation observed within the dominant sector with p=2/3, we here focus on the nonequilibrium kinetics within the minor sectors with p<2/3. The initial configurations are chosen as those in which the strings are straight and evenly distributed. In the minor sectors, we observe a characteristic spatial anisotropy in both equilibrium and nonequilibrium spatial correlations. We observe emergence of a critical relaxation region (in the spatial and temporal domain) which grows as p deviates from p=2/3. Spatial anisotropy appears in the equilibrium spatial correlation with the characteristic length scale xi(e,V)(p) diverging with vanishing string density as xi(e,V)(p) approximately p(-2) along the vertical direction, while along the horizontal direction the spatial length scale diverges as xi(e,H) approximately p(-1). Analytic forms for the anisotropic equilibrium correlation functions are given. We also find that the spin autocorrelation function A(t) shows a simple scaling behavior A(t)=A(t/tau(A)(p)), where the time scale tau(A)(p) shows a power-law divergence with vanishing p as tau(A)(p) approximately p(-phi) with phi approximately or equal to 4. These features can be understood in terms of random walk nature of the fluctuations of the strings within the typical separation between neighboring strings.

Probability distribution of the maximum of a smooth temporal signal

We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated with the probability that the process remains below a nonzero level M. When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f(t)=X(0)X(t).

Finite-size effect in persistence in random walks

We have investigated the random walk problem in a finite system and studied the crossover induced in the persistence probability by the system size. Analytical and numerical work show that the scaling function is an exponentially decaying function. We consider two cases of trapping, one by a box of size L and the other by a harmonic trap. Our analytic calculations are supported by numerical works. We also present numerical results on the harmonically trapped randomly accelerated particle and the randomly accelerated particle with viscous drag.

Semiflexible polymers in shear flow

The dynamics of semiflexible polymers under the influence of shear flow is studied analytically. Power laws are derived for various conformational and dynamical quantities which are in agreement with experimental findings. In particular, the tumbling motion is analyzed and expressions are provided for the probability distributions of the orientation angles and the tumbling time. The calculations explain the similarities in the behavior of flexible and semiflexible polymers as well as free-draining and nondraining systems.

Effect of shear on persistence in coarsening systems

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.

Persistence in nonequilibrium surface growth

Persistence probabilities of the interface height in ( 1+1 ) - and ( 2+1 ) -dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady-state regimes. For the MBE universality class, the positive and negative persistence exponents in the steady-state are found to be theta(S)(+) =0.66+/-0.02 and theta(S)(-) =0.78+/-0.02, respectively, in ( 1+1 ) dimensions, and theta(S)(+) =0.76+/-0.02 and theta(S)(-) =0.85+/-0.02, respectively, in ( 2+1 ) dimensions. The noise reduction technique is applied on some of the ( 1+1 ) -dimensional models in order to obtain accurate values of the persistence exponents. We show analytically that a relation between the steady-state persistence exponent and the dynamic growth exponent, found earlier to be valid for linear models, should be satisfied by the smaller of the two steady-state persistence exponents in the nonlinear models. Our numerical results for the persistence exponents are consistent with this prediction. We also find that the steady-state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state. The dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form.

Nonequilibrium critical dynamics of the triangular antiferromagnetic Ising model

We investigate the nonequilibrium critical dynamics of antiferromagnetic Ising model on a two-dimensional triangular lattice via dynamic Monte Carlo simulation employing spin-flip kinetics. Macroscopic degeneracy of the ground state originating from geometric frustration fundamentally affects the nonequilibrium dynamics of the system. In particular, the defects and the loose spins (whose flip costs no energy) play key roles in the dynamics. The long-time evolution is characterized by a critical dynamic scaling with a growing length scale xi(t). With random initial configurations, xi(t) exhibits a subdiffusive growth in time, xi(t) approximately t(1/z) with 1/z approximately 0.43, while xi(t) shows a diffusive growth with z=2 for the relaxation within the dominant sector of the ground-state manifold. The nonequilibrium critical dynamics therefore exhibits an interesting initial-state dependence. Persistence and the two-time temporal properties are also discussed.

Persistence of manifolds in nonequilibrium critical dynamics

We study the persistence probability P(t) that, starting from a random initial condition, the magnetization of a d'-dimensional manifold of a d-dimensional spin system at its critical point does not change sign up to time t. For d'>0 we find three distinct late-time decay forms for P(t): exponential, stretched exponential, and power law, depending on a single parameter zeta=(D-2+eta)/z, where D=d-d' and eta,z are standard critical exponents. In particular, we predict that for a line magnetization in the critical d=2 Ising model, P(t) decays as a power law while, for d=3, P(t) decays as a power of t for a plane magnetization but as a stretched exponential for a line magnetization. Numerical results are consistent with these predictions.

Global persistence exponent of the two-dimensional Blume-Capel model

The global persistence exponent theta(g) is calculated for the two-dimensional Blume-Capel model following a quench to the critical point from both disordered states and such with small initial magnetizations. Estimates are obtained for the nonequilibrium critical dynamics on the critical line and at the tricritical point. Ising-like universality is observed along the critical line and a different value theta(g)=1.080(4) is found at the tricritical point.

Fraction of uninfected walkers in the one-dimensional Potts model

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-->Phi) 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 known to obey a power-law decay U(t) approximately t(-phi(q)), with a nontrivial exponent phi(q) [C. Monthus, Phys. Rev. E 54, 4844 (1996); S. N. Majumdar and S. J. Cornell, ibid. 57, 3757 (1998)]. We probe the numerical values of phi(q) to a higher degree of accuracy than previous simulations and relate the exponent phi(q) to the persistence exponent theta(q) [B. Derrida, V. Hakim, and V. Pasquier, Phys. Rev. Lett. 75, 751 (1995)], through the relation phi(q)=gamma(q)theta(q) where gamma is an exponent introduced in [S. J. O'Donoghue and A. J. Bray, preceding paper, Phys. Rev. E 65, 051113 (2002)]. 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 rho(t) approximately 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) approximately t(-theta) with theta approximately 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) approximately t(-phi), where phi approximately 1.13 when infection occurs between like particles only, and phi approximately 1.93 when we also include cross-species contamination. We find that the relation between phi and theta in this model can also be characterized by an exponent gamma, where similarly, phi=gamma(theta).

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=nDeltaT. For a Gaussian stationary process the persistence (no crossing) probability decays as exp(-theta(D)T)=[rho(a)](n) for large n, where a=exp(-DeltaT/2) and the discrete persistence exponent theta(D) is given by theta(D)=(ln rho)/(2 ln a). Using the "independent interval approximation," we show how theta(D) varies with DeltaT for small DeltaT 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, 015101(R) (2001)] to determine rho(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 rho(a) for this case.

Persistence in the one-dimensional A+B--> Ø reaction-diffusion model

The persistence properties of a set of random walkers obeying the A+B--> Ø 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) approximately const+t(-straight theta), where straight theta is the persistence exponent (type I persistence). We argue that, for a density of particles rho>>1, this nontrivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, partial differential(t)straight phi= partial differential(xx)straight phi, where straight theta approximately 0.1207 [S. N. Majumdar, C. Sire, A. J. Bray, and S. J. Cornell, Phys. Rev. Lett. 77, 2867 (1996)]. In the case of an initial low density, rho(0)<<1, we find straight theta approximately 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) approximately exp(-constxrho(1/2)(0)t(1/4)), provided rho(0)<<1. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.

Persistence in a stationary time series

We study the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time. We show that under certain conditions, the persistence of such a continuous process reduces to the persistence of a corresponding discrete sequence obtained from the measurement of the process only at integer times. We then construct a specific sequence for which the persistence can be computed even though the sequence is non-Markovian. We show that this may be considered as a limiting case of persistence in the diffusion process on a hierarchical lattice.

Measurement of persistence in 1D diffusion

Using a novel NMR scheme we observed persistence in 1D gas diffusion. Analytical approximations and numerical simulations have indicated that for an initially random array of spins undergoing diffusion, the probability p(t) that the average spin magnetization in a given region has not changed sign (i.e., "persists") up to time t follows a power law t(-straight theta), where straight theta depends on the dimensionality of the system. Using laser-polarized 129Xe gas, we prepared an initial "quasirandom" 1D array of spin magnetization and then monitored the ensemble's evolution due to diffusion using real-time NMR imaging. Our measurements are consistent with analytical and numerical predictions of straight theta approximately 0.12.

Critical dimensions of the diffusion equation

We study the evolution of a random initial field under pure diffusion in various space dimensions. From numerical calculations we find that the persistence properties of the system show sharp transitions at critical dimensions d(1) approximately 26 and d(2) approximately 46. We also give refined measurements of the persistence exponents for low dimensions.

Numerical study of persistence in models with absorbing states

Extensive Monte Carlo simulations are performed in order to evaluate both the local (straight theta(l)) and global (straight theta(g)) persistence exponents in the Ziff-Gulari-Barshad (ZGB) [Phys. Rev. Lett. 56, 2553 (1986)] irreversible reaction model. At the second-order irreversible phase transition (IPT) we find that both the local and the global persistence exhibit power-law behavior with a crossover between two different time regimes. On the other hand, at the ZGB first-order IPT, active sites are short lived and the persistence decays more abruptly; it is not clear whether it shows power-law behavior or not. In order to analyze universality issues, we have also studied another model with absorbing states, the contact process, and evaluated the local persistence exponent in dimensions from 1 to 4. A striking apparent superuniversality is reported: the local persistence exponent seems to coincide in both one- and two-dimensional systems. Some other aspects of persistence in systems with absorbing states are also analyzed.

Persistence in higher dimensions: A finite size scaling study

We show that the persistence probability P(t,L), in a coarsening system of linear size L at a time t, has the finite-size scaling form P(t,L) approximately L(-zstraight theta)f(t/L(z)), where straight theta is the persistence exponent and z is the coarsening exponent. The scaling function f(x) approximately x(-straight theta) for x<<1 and is constant for large x. The scaling form implies a fractal distribution of persistent sites with power-law spatial correlations. We study the scaling numerically for the Glauber-Ising model at dimension d=1 to 4 and extend the study to the diffusion problem. Our finite-size scaling ansatz is satisfied in all these cases providing a good estimate of the exponent straight theta.

Unusual dynamical scaling in the spatial distribution of persistent sites in one-dimensional potts models

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, straight theta), where straight theta(q) is the persistence exponent which describes the fraction P(t) approximately t(-straight theta) of sites which have not changed their state up to time t. When straight theta>1/2, the scaling length t(straight theta) for the interval-size distribution is larger than the coarsening length scale t(1/2) that characterizes spatial correlations of the Potts variables.

Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensional XY model

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) approximately -r(-sigma). The "persistence probability," P0(t), that the particle has not visited the origin up to time t is calculated for a number of cases. For sigma>1, the force is asymptotically irrelevant (with respect to the noise), and the asymptotics of P0(t) are those of a free random walker. For sigma<1, the noise is (dangerously) irrelevant and the asymptotics of P0(t) can be extracted from a weak noise limit within a path-integral formalism employing the Onsager-Machlup functional. The case sigma=1, corresponding to a logarithmic potential, is most interesting because the noise is exactly marginal. In this case, P0(t) decays as a power law, P0(t) approximately t(-straight theta) with an exponent straight 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) approximately r(2) ln(r/a) where a is a microscopic cutoff (the vortex core size). Implications for the nonequilibrium critical dynamics of the system are discussed and compared to numerical simulation results.