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 ${\tau }_s\left(R\right)$. The distance dependence of the time constant is shown to be ${\tau }_s\left(R\right)={\tau }_{s0}\exp [-R∕w\left(T\right)]$, where $w^2\left(T\right)$ 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 $S_{\mathrm{in}}(t,R)$, 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 $T=300\mathrm{K}$, and $0.5<\lambda <1$ for $T=460\mathrm{K}$. 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 $\sim {10}^3$. Preliminary analysis indicates that the scaling effect due to the true correlation time is relevant in the parameter space of the experimental observations.

• Received 11 December 2006

DOI:https://doi.org/10.1103/PhysRevE.76.021601