Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpatial persistence and survival probabilities for fluctuating interfaces
64
0
0.0
(
0
)
Added by
Magdalena Constantin
Publication date
2003
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 (KPZ) 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.
rate research
Read More
Numerical and analytic results for the exponent theta describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic roughness exponent beta, with 0 < beta < 1; for beta = 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent theta_S = 1 - beta. The exponent theta_0 for the flat initial condition appears to be nontrivial. We prove that theta_0 to infty for beta to 0, theta_0 geq theta_S for beta < 1/2 and theta_0 leq theta_S for beta > 1/2, and calculate theta_{0,S} perturbatively to first order in an expansion around the Markovian case beta = 1/2. Using the exact result theta_S = 1 - beta, accurate upper and lower bounds on theta_0 can be derived which show, in particular, that theta_0 geq (1 - beta)^2/beta for small beta.
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.
The persistence behavior for fluctuating steps on the $Si(111)$ $(sqrt3 times sqrt3)R30^{0} - Al$ surface was determined by analyzing time-dependent STM images for temperatures between 770 and 970K. The measured persistence probability follows a power law decay with an exponent of $theta=0.77 pm 0.03$. This is consistent with the value of $theta= 3/4$ predicted for attachment/detachment limited step kinetics. If the persistence analysis is carried out in terms of return to a fixed reference position, the measured persistence probability decays exponentially. Numerical studies of the Langevin equation used to model step motion corroborate the experimental observations.
Results of analytic and numerical investigations of first-passage properties of equilibrium fluctuations of monatomic steps on a vicinal surface are reviewed. Both temporal and spatial persistence and survival probabilities, as well as the probability of persistent large deviations are considered. Results of experiments in which dynamical scanning tunneling microscopy is used to evaluate these first-passage properties for steps with different microscopic mechanisms of mass transport are also presented and interpreted in terms of theoretical predictions for appropriate models. Effects of discrete sampling, finite system size and finite observation time, which are important in understanding the results of experiments and simulations, are discussed.
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 STM 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 320K, and on Al-terminated Si(111) over the temperature range 770K - 970K. 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/Dt where Dt 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 tm. Scaling with t/Dt occurs only when Dt/tm is a constant. We show that this observation is equivalent to theoretical predictions that the survival probability will scale as Dt/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 tm or Dt should also show little or no temperature dependence.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
