Survival in equilibrium step fluctuations

Generalized survival in step fluctuations

The properties of the generalized survival probability, that is, the probability of not crossing an arbitrary location R during relaxation, have been investigated experimentally (via scanning tunneling microscope observations) and numerically. The results confirm that the generalized survival probability decays exponentially with a time constant tau(s) (R). The distance dependence of the time constant is shown to be tau(s) (R) = tau(s0) exp[-R/w (T)], where w2 (T) is the material-dependent mean-squared width of the step fluctuations. The result reveals the dependence on the physical parameters of the system inherent in the prior prediction of the time constant scaling with R/L(alpha), with L the system size and alpha the roughness exponent. The survival behavior is also analyzed using a contrasting concept, the generalized inside survival S(in) (t,R), which involves fluctuations to an arbitrary location R further from the average. Numerical simulations of the inside survival probability also show an exponential time dependence, and the extracted time constant empirically shows (R/w)(lambda) behavior, with lambda varying over 0.6 to 0.8 as the sampling conditions are changed. The experimental data show similar behavior, and can be well fit with lambda = 1.0 for T = 300 K, and 0.5 < lambda < 1 for T = 460 K. Over this temperature range, the ratio of the fixed sampling time to the underlying physical time constant, and thus the true correlation time, increases by a factor of approximately 10(3). Preliminary analysis indicates that the scaling effect due to the true correlation time is relevant in the parameter space of the experimental observations.

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.

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.

Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models

We study the probability density function P(h(m), L) of the maximum relative height h(m) in a wide class of one-dimensional solid-on-solid models of finite size L. For all these lattice models, in the large-L limit, a central limit argument shows that, for periodic boundary conditions, P(h(m), L) takes a universal scaling form P(h(m), L) approximately radical(12w(L))(-1) f(h(m)radical(12w(L))(-1), with w(L) the width of the fluctuating interface f(x) and the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using the transfer matrix technique, valid for any L > 0. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a nonuniversal amplitude, and simply given by the derivative of the Airy distribution function f'(x).

Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the "sampling interval" used in the measurement for both "steady-state" and "finite" initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A "deterministic approximation" is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.

Volatility, persistence, and survival in financial markets

We study the temporal fluctuations in time-dependent stock prices (both individual and composite) as a stochastic phenomenon using general techniques and methods of nonequilibrium statistical mechanics. In particular, we analyze stock price fluctuations as a non-Markovian stochastic process using the first-passage statistical concepts of persistence and survival. We report the results of empirical measurements of the normalized qth-order correlation functions fq(t), survival probability S(t), and persistence probability P(t) for several stock market dynamical sets. We analyze both minute-to-minute and higher-frequency stock market recordings (i.e., with the sampling time deltat of the order of days). We find that the fluctuating stock price is multifractal and the choice of deltat has no effect on the qualitative multifractal behavior displayed by the 1/q dependence of the generalized Hurst exponent Hq associated with the power-law evolution of the correlation function fq(t) approximately tHq. The probability S(t) of the stock price remaining above the average up to time t is very sensitive to the total measurement time tm and the sampling time. The probability P(t) of the stock not returning to the initial value within an interval t has a universal power-law behavior P(t) approximately t(-theta), with a persistence exponent theta close to 0.5 that agrees with the prediction theta=1-H2. The empirical financial stocks also present an interesting feature found in turbulent fluids, the extended self-similarity.

Persistence of randomly coupled fluctuating interfaces

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h(1) and h(2) , respectively, each growing over a d -dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h(2) , however, is coupled to h(1) via a quenched random velocity field. In the limit d-->0 , our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t(0) -->infinity , the stochastic process h(2) , at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H2 =1- beta(1) /2 , where beta(1) is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be theta(2)(s) =1- H2 = beta(1) /2 . These analytical results are verified by numerical simulations.

Sampling-time effects for persistence and survival in step structural fluctuations

The effects of sampling rate and total measurement time have been determined for single-point measurements of step fluctuations within the context of first-passage properties. Time dependent scanning tunneling microscopy has been used to evaluate step fluctuations on Ag(111) films grown on mica as a function of temperature (300-410 K) , on screw dislocations on the facets of Pb crystallites at 320 K , and on Al-terminated Si(111) over the temperature range 770-970 K . Although the fundamental time constant for step fluctuations on Ag and Al/Si varies by orders of magnitude over the temperature ranges of measurement, no dependence of the persistence amplitude on temperature is observed. Instead, the persistence probability is found to scale directly with t/delta t where delta t is the time interval used for sampling. Survival probabilities show a more complex scaling dependence, which includes both the sampling interval and the total measurement time t(m) . Scaling with t/delta t occurs only when delta t/ t(m) is a constant. We show that this observation is equivalent to theoretical predictions that the survival probability will scale as delta t/ L(z) , where L is the effective length of a step. This implies that the survival probability for large systems, when measured with fixed values of t(m) or delta t , should also show little or no temperature dependence.

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.

Generalized survival in equilibrium step fluctuations

We investigate the dynamics of a generalized survival probability S(t,R) defined with respect to an arbitrary reference level R (rather than the average) in equilibrium step fluctuations. The exponential decay at large time scales of the generalized survival probability is numerically analyzed. S(t,R) is shown to exhibit simple scaling behavior as a function of system size L, sampling time deltat, and the reference level R. The generalized survival time scale tau(s) (R) associated with S(t,R) is shown to decay exponentially as a function of R.

Spatial persistence and survival probabilities for fluctuating interfaces

We report the results of numerical investigations of the steady-state (SS) and finite-initial-conditions (FIC) spatial persistence and survival probabilities for (1+1) -dimensional interfaces with dynamics governed by the nonlinear Kardar-Parisi-Zhang equation and the linear Edwards-Wilkinson (EW) equation with both white (uncorrelated) and colored (spatially correlated) noise. We study the effects of a finite sampling distance on the measured spatial persistence probability and show that both SS and FIC persistence probabilities exhibit simple scaling behavior as a function of the system size and the sampling distance. Analytical expressions for the exponents associated with the power-law decay of SS and FIC spatial persistence probabilities of the EW equation with power-law correlated noise are established and numerically verified.