Global Persistence Exponent for Nonequilibrium Critical Dynamics

Emergence of Memory in Equilibrium versus Nonequilibrium Systems

Experiments often probe observables that correspond to low-dimensional projections of high-dimensional dynamics. In such situations distinct microscopic configurations become lumped into the same observable state. It is well known that correlations between the observable and the hidden degrees of freedom give rise to memory effects. However, how and under which conditions these correlations emerge remain poorly understood. Here we shed light on two fundamentally different scenarios of the emergence of memory in minimal stationary systems, where observed and hidden degrees of freedom either evolve cooperatively or are coupled by a hidden nonequilibrium current. In the reversible setting the strongest memory manifests when the timescales of hidden and observed dynamics overlap, whereas, strikingly, in the driven setting maximal memory emerges under a clear timescale separation. Our results hint at the possibility of fundamental differences in the way memory emerges in equilibrium versus driven systems that may be utilized as a "diagnostic" of the underlying hidden transport mechanism.

Diffusive persistence on disordered lattices and random networks

To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time t (or, in general, that the node remained active or inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law P(t,L)∼t^{-θ} with an exponent θ≃0.186, in the limit of large linear system size L. At the percolation threshold, however, the scaling exponent changes to θ≃0.141, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite-size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power law P(t,L)∼L^{-zθ} with z≃2.86 instead of the value of z=2 normally associated with finite-size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erdős-Rényi networks, no simple power-law scaling behavior exists above the percolation threshold.

Nonequilibrium critical dynamics of the two-dimensional ±J Ising model

The ±J Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value -J with probability p and +J with probability 1-p. It is especially appealing due to its connection to quantum error correcting codes. Here, we investigate the nonequilibrium critical behavior of the two-dimensional ±J Ising model, after a quench from different initial conditions to a critical point T_{c}(p) on the paramagnetic-ferromagnetic (PF) transition line, especially above, below, and at the multicritical Nishimori point (NP). The dynamical critical exponent z_{c} seems to exhibit nonuniversal behavior for quenches above and below the NP, which is identified as a preasymptotic feature due to the repulsive fixed point at the NP, whereas for a quench directly to the NP, the dynamics reaches the asymptotic regime with z_{c}≃6.02(6). We also consider the geometrical spin clusters (of like spin signs) during the critical dynamics. Each universality class on the PF line is uniquely characterized by the stochastic Loewner evolution with corresponding parameter κ. Moreover, for the critical quenches from the paramagnetic phase, the model, irrespective of the frustration, exhibits an emergent critical percolation topology at the large length scales.

Record ages of non-Markovian scale-invariant random walks

How long is needed for an observable to exceed its previous highest value and establish a new record? This time, known as the age of a record plays a crucial role in quantifying record statistics. Until now, general methods for determining record age statistics have been limited to observations of either independent random variables or successive positions of a Markovian (memoryless) random walk. Here we develop a theoretical framework to determine record age statistics in the presence of memory effects for continuous non-smooth processes that are asymptotically scale-invariant. Our theoretical predictions are confirmed by numerical simulations and experimental realisations of diverse representative non-Markovian random walk models and real time series with memory effects, in fields as diverse as genomics, climatology, hydrology, geology and computer science. Our results reveal the crucial role of the number of records already achieved in time series and change our view on analysing record statistics.

Stocks and cryptocurrencies: Antifragile or robust? A novel antifragility measure of the stock and cryptocurrency markets

In contrast with robust systems that resist noise or fragile systems that break with noise, antifragility is defined as a property of complex systems that benefit from noise or disorder. Here we define and test a simple measure of antifragility for complex dynamical systems. In this work we use our antifragility measure to analyze real data from return prices in the stock and cryptocurrency markets. Our definition of antifragility is the product of the return price and a perturbation. We explore different types of perturbations that typically arise from within the system. Our results suggest that for both the stock market and the cryptocurrency market, the tendency among the 'top performers' is to be robust rather than antifragile. It would be important to explore other possible definitions of antifragility to understand its role in financial markets and in complex dynamical systems in general.

Correlation between the In-Plane Critical Behavior and Out-of-Plane Interaction of Ternary Lipid Membranes

Liquid-liquid phase-separating lipid membranes belong to the 2-D Ising universality class. While their in-plane critical behaviors are well studied, how the behaviors modulate out-of-plane interactions is rarely explored, despite its profound implications for biomembranes and 2-D ferromagnets. Here, we examine how the interlayer interaction, manifested as membrane fusion, is affected by the membranes' critical fluctuations. Remarkably, the critical fluctuations suppress membrane fusion, suggesting a correlation between critical behaviors and interlayer interactions for 2-D Ising systems.

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.

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.

Universality of the local persistence exponent for models in the directed Ising class in one dimension

We investigate local persistence in five different models and their variants in the directed Ising (DI) universality class in one dimension. These models have right-left symmetry. We study Grassberger's models A and B. We also study branching and annihilating random walks with two offspring: the nonequilibrium kinetic Ising model and the interacting monomer-dimer model. Grassberger's models are updated in parallel. This is not the case in other models. We find that the local persistence exponent in all these models is unity or very close to it. A change in the mode of the update does not change the exponent unless the universality class changes. In general, persistence exponents are not universal. Thus it is of interest that the persistence exponent in a range of models in the DI class is the same. Excellent scaling behavior of finite-size scaling is obtained using exponents in the DI class in all models. We also study off-critical scaling in some models and DI exponents yield excellent scaling behavior. We further study graded persistence, which shows similar behavior. However, for a logistic map with delay, which also has the transition in the DI class, there is no transition from nonzero to zero persistence at the critical point. Thus the accompanying transition in persistence and universality of the persistence exponent hold when the underlying model has right-left symmetry.

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.

Nonequilibrium critical dynamics of the two-dimensional Ashkin-Teller model at the Baxter line

We investigate the short-time universal behavior of the two-dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power-law decay of the magnetization. Thus, the dynamic critical exponents θ_{m} and θ_{p}, related to the magnetic and electric order parameters, as well as the persistence exponent θ_{g}, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent z and the static critical exponents β and ν for both order parameters. We propose a refined method to estimate the static exponents that considers two different averages: one that combines an internal average using several seeds with another, which is taken over temporal variations in the power laws. Moreover, we also performed the bootstrapping method for a complementary analysis. Our results show that the ratio β/ν exhibits universal behavior along the critical line corroborating the conjecture for both magnetization and polarization.

Coarsening and persistence in a one-dimensional system of orienting arrowheads: Domain-wall kinetics with A+B→0

We demonstrate the large-scale effects of the interplay between shape and hard-core interactions in a system with left- and right-pointing arrowheads <> on a line, with reorientation dynamics. This interplay leads to the formation of two types of domain walls, >< (A) and <> (B). The correlation length in the equilibrium state diverges exponentially with increasing arrowhead density, with an ordered state of like orientations arising in the limit. In this high-density limit, the A domain walls diffuse, while the B walls are static. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of domain-wall annihilation A+B→0, quite different from the A+A→0 kinetics pertinent to the Glauber-Ising model. The survival probability of a finite set of walls is shown to decay exponentially with time, in contrast to the power-law decay known for A+A→0. In the thermodynamic limit with a finite density of walls, coarsening as a function of time t is studied by simulation. While the number of walls falls as t^{-1/2}, the fraction of persistent arrowheads decays as t^{-θ} where θ is close to 1/4, quite different from the Ising value. The global persistence too has θ=1/4, as follows from a heuristic argument. In a generalization where the B walls diffuse slowly, θ varies continuously, increasing with increasing diffusion constant.

Fractality in persistence decay and domain growth during ferromagnetic ordering: Dependence upon initial correlation

The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.

A statistical physics perspective on alignment-independent protein sequence comparison

Motivation: Within bioinformatics, the textual alignment of amino acid sequences has long dominated the determination of similarity between proteins, with all that implies for shared structure, function and evolutionary descent. Despite the relative success of modern-day sequence alignment algorithms, so-called alignment-free approaches offer a complementary means of determining and expressing similarity, with potential benefits in certain key applications, such as regression analysis of protein structure-function studies, where alignment-base similarity has performed poorly.

Results: Here, we offer a fresh, statistical physics-based perspective focusing on the question of alignment-free comparison, in the process adapting results from ‘first passage probability distribution’ to summarize statistics of ensemble averaged amino acid propensity values. In this article, we introduce and elaborate this approach.

Contact: d.r.flower@aston.ac.uk

Persistence of Brownian motion in a shear flow

The persistence of a Brownian particle in a shear flow is investigated. The persistence probability P(t), which is the probability that the particle does not return to its initial position up to time t, is known to obey a power law P(t)∝t(-θ). Since the displacement of a particle along the flow direction due to convection is much larger than that due to Brownian motion, we define an alternative displacement in which the convection effect is removed. We derive theoretically the two-time correlation function and the persistence exponent θ of this displacement. The exponent has different values at short and long times. The theoretical results are compared with experiment and a good agreement is found.

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.

Persistence of a Brownian particle in a time-dependent potential

We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times, leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential decay and an algebraic decay. Analytical calculations and numerical simulations show that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is determined by the exponent of the decay. Finally, using the two-time correlation function for the position of the Brownian particle, we construct the persistence probability for the Brownian walker in such a scenario.

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.

Finite-size effects in dynamics of zero-range processes

The finite-size effects prominent in zero-range processes exhibiting a condensation transition are studied by using continuous-time Monte Carlo simulations. We observe that, well above the thermodynamic critical point, both static and dynamic properties display fluidlike behavior up to a density ρc(L), which is the finite-size counterpart of the critical density ρc=ρc(L→∞). We determine this density from the crossover behavior of the average size of the largest cluster. We then show that several dynamical characteristics undergo a qualitative change at this density. In particular, the size distribution of the largest cluster at the moment of relocation, the persistence properties of the largest cluster, and the correlations in its motion are studied.

Local and global persistence exponents of two quenched continuous-lattice spin models

Local and global persistence exponents associated with zero-temperature quenched dynamics of two-dimensional XY model and three-dimensional Heisenberg model have been estimated using numerical simulations. The method of block persistence has been used to find the global and local exponents simultaneously (in a single simulation). Temperature universality of both the exponents for three-dimensional Heisenberg model has been confirmed by simulating the stochastic (with noise) version of the equation of motion. The noise amplitudes added were small enough to retain the dynamics below criticality. In the second part of our work we have studied scaling associated with correlated persistence sites in the three-dimensional Heisenberg model in the later stages of the dynamics. The relevant length scale associated with correlated persistent sites was found to behave in a manner similar to the dynamic length scale associated with the phase ordering dynamics.

Longest excursion of stochastic processes in nonequilibrium systems

We consider the excursions, i.e., the intervals between consecutive zeros, of stochastic processes that arise in a variety of nonequilibrium systems and study the temporal growth of the longest one l_{max}(t) up to time t. For smooth processes, we find a universal linear growth l_{max}(t) approximately Q_{infinity}t with a model dependent amplitude Q_{infinity}. In contrast, for nonsmooth processes with a persistence exponent theta, we show that l_{max}(t) has a linear growth if theta < theta_{c} while l_{max}(t) approximately t;{1-psi} if theta > theta_{c}. The amplitude Q_{infinity} and the exponent psi are novel quantities associated with nonequilibrium dynamics. This behavior is obtained by exact analytical calculations for renewal and multiplicative processes and numerical simulations for other systems such as the coarsening dynamics in Ising model as well as the diffusion equation with random initial conditions.

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.

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.

Global persistence exponent in critical dynamics: finite-size-induced crossover

We extend the definition of a global order parameter to the case of a critical system confined between two infinite parallel plates separated by a distance L. For a quench to the critical point we study the persistence property of the global order parameter and show that there is a crossover behavior characterized by a nonuniversal exponent which depends on the ratio of the system size to a dynamic length scale.

Short-time dynamics of polypeptides

The authors study the short-time dynamics of helix-forming polypeptide chains using an all-atom representation of the molecules and an implicit solvation model to approximate the interaction with the surrounding solvent. The results confirm earlier observations that the helix-coil transition in proteins can be described by a set of critical exponents. The high statistics of the simulations allows the authors to determine the exponent values with increased precision and support universality of the helix-coil transition in homopolymers and (helical) proteins.

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.

Global persistence exponent of the double-exchange model

We obtained the global persistence exponent for a continuous spin model on the simple cubic lattice with double-exchange interaction by using two different methods. First, we estimated the exponent theta(g) by following the time evolution of probability P(t) that the order parameter of the model does not change its sign up to time t[P(t) approximately t(-theta(g)]. Afterwards, that exponent was estimated through the scaling collapse of the universal function L(theta(g)(z)P(t) for different lattice sizes. Our results for both approaches are in very good agreement with each other.

Competition between two kinds of correlations in literary texts

A theory of additive Markov chains with long-range memory is used to describe the correlation properties of coarse-grained literary texts. The complex structure of the correlations in the texts is revealed. Anti-persistent correlations at small distances, L approximately < 300, and persistent ones at L approximately > 300 define this non-trivial structure. For some concrete examples of literary texts, the memory functions are obtained and their power-law behavior at long distances is disclosed. This property is shown to be a cause of self-similarity of texts with respect to the decimation procedure.

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.

Temporal and spatial persistence of combustion fronts in paper

The spatial and temporal persistence, or first-return distributions are measured for slow-combustion fronts in paper. The stationary temporal and (perhaps less convincingly) spatial persistence exponents agree with the predictions based on the front dynamics, which asymptotically belongs to the Kardar-Parisi-Zhang universality class. The stationary short-range and the transient behavior of the fronts are non-Markovian, and the observed persistence properties thus do not agree with the predictions based on Markovian theory. This deviation is a consequence of additional time and length scales, related to the crossovers to the asymptotic coarse-grained behavior.

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.

Monte Carlo simulations of short-time critical dynamics with a conserved quantity

With Monte Carlo simulations, we investigate short-time critical dynamics of the three-dimensional antiferromagnetic Ising model with a globally conserved magnetization m(s) (not the order parameter). From the power law behavior of the staggered magnetization (the order parameter), its second moment and the autocorrelation, we determine all static and dynamic critical exponents as well as the critical temperature. The universality class of m(s)=0 is the same as that without a conserved quantity, but the universality class of nonzero m(s) is different.

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.

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.

Short-time dynamics of an ising system on fractal structures

The short-time critical relaxation of an Ising model on a Sierpinski carpet is investigated using Monte Carlo simulation. We find that when the system is quenched from high temperature to the critical temperature, the evolution of the order parameter and its persistence probability, the susceptibility, and the autocorrelation function all show power-law scaling behavior at the short-time regime. The results suggest that the spatial heterogeneity and the fractal nature of the underlying structure do not influence the scaling behavior of the short-time critical dynamics. The critical temperature, dynamic exponent z, and other equilibrium critical exponents beta and nu of the fractal spin system are determined accurately using conventional Monte Carlo simulation algorithms. The mechanism for short-time dynamic scaling is discussed.

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.

Persistence exponents in a three-dimensional symmetric binary fluid mixture

The persistence exponent, straight theta, is defined by N(F) approximately t(-straight theta), where t is the time since the start of the coarsening process and the "no-flip fraction," N(F), is the number of points that have not seen a change of "color" since t=0. Here we investigate numerically the persistence exponent for a binary fluid system where the coarsening is dominated by hydrodynamic transport. We find that N(F) follows a power law decay (as opposed to exponential) with the value of straight theta somewhat dependent on the domain growth rate (L approximately t(alpha), where L is the average domain size), in the range straight theta=1.23+/-0.1 (alpha=2/3) to straight theta=1.37+/-0.2 (alpha=1). These alpha values correspond to the inertial and viscous hydrodynamic regimes, respectively.