Generalized survival in step fluctuations

Abstract

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 Τ_s(R). The distance dependence of the time constant is shown to be Τ_s(R)=Τ_s0exp[-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^α , with L the system size and a 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)^λ behavior, with λ 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 λ=1.0 for T=300 K, and 0.5<λ<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 -10³. Preliminary analysis indicates that the scaling effect due to the true correlation time is relevant in the parameter space of the experimental observations.