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Journal of Statistical Mechanics: Theory and Experiment Volume 2007 July 2007 Create an alert RSS this journal

M Constantin et al J. Stat. Mech. (2007) P07011 doi:10.1088/1742-5468/2007/07/P07011

Persistence and survival in equilibrium step fluctuations

FREE ARTICLE Focus on Dynamics of Non-Equilibrium Systems

M Constantin1, C Dasgupta2, S Das Sarma3, D B Dougherty4 and E D Williams5

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Abstract References Cited By Figures

Figure 1

Figure 1. Schematic plot (red solid line) of the step position h(x, t) as a function of the coordinate x along the average step position at a fixed time t. The blue dashed line indicates the average step position.



Figure 2

Figure 2. Schematic plot of the step-edge position h(x, t) as a function of time t at a fixed value of x, illustrating the definitions of the temporal persistence and survival probabilities. Calculations of the temporal persistence probability P(t) and the temporal survival probability S(t) respectively involve the statistics of the time intervals indicated as `P(t) = 1' (black) and `S(t) = 1' (blue) (the average step position is \bar {h}=0 ). For the choice of the reference position R shown in the plot, the calculation of the generalized temporal survival probability S(t, R) involves the statistics of the time interval indicated as `S(t, R) = 1' (red).



Figure 3

Figure 3. Example of line-scan pseudo-image of step fluctuations. In this figure, the step-edge position is denoted by h and x0 denotes the coordinate in the average step direction.



Figure 4

Figure 4. Double-logarithmic plots of the experimentally obtained persistence probability p(t) for Al/Si(111) surface steps, as a function of time t, for three different temperatures (770, 970 and 870 K, from top to bottom). The plots have been offset vertically from one another for clarity of display (from [21]).



Figure 5

Figure 5. Log–log plots of P(t, s) versus t for high-temperature surface step fluctuations via the AD mechanism, shown for different values of s: s = 1.0, 0.75, 0.5, 0.25, 0, –0.25, –0.5, –0.75 (from the bottom to the top). (a) Equation (2) (main figure) and the Family model (inset); (b) experimental data from STM step images of Al/Si(111) surface at 970 K; and (c) comparison of the various sets of results for θl as a function of s. The error bars shown for the experimental data are obtained from variations of the local slope of the log P(t, s) versus log t plots. Simulation results for two sample sizes are shown to illustrate that the use of small samples leads to an underestimation of θl(s) (from [23]).



Figure 6

Figure 6. The survival probability S(t) and the autocorrelation function C(t) for the Langevin equation of equation (2). The dashed lines are fits of the long-time data to an exponential form. In panels ((a)–(c)), the uppermost plots show the data for C(t). Panel (a): L = 100, δt = 0.625. Panel (b): L = 200, δt = 2.5. Panel (c): L = 400, δt = 10.0 (upper plot) and δt = 1.0 (lower plot). Panel (d): Finite-size scaling of S(t, L, δt). Results for S for 3 different sample sizes with the same value of δt/Lz (z = 2) are plotted versus t/Lz (from [24]).



Figure 7

Figure 7. S(t) and C(t) for two experimental systems. The dashed lines are fits of the long-time data to an exponential form. Panel (a): Al/Si(111) at T = 970 K. Panel (b): Ag(111) at T = 450 K (from [24]).



Figure 8

Figure 8. S(t, R) for the discrete Family model. The dashed lines are fits of the long-time data to an exponential form. The system size is L = 100, the sampling time is δt = 1.0 and the reference level R takes four different values: 0, 1, 2 and 3 (from top to bottom). The inset shows the dependence of the generalized survival timescale τs(R) on the reference level value (up to R = 5). The continuous curve represents a fit to an exponential decay of τs(R) with R (from [25]).



Figure 9

Figure 9. Experimental data (top panel) for the generalized survival probability S(t, R) for Ag(111) steps. The bottom panel shows the variation of the time constant τs(R) with the scaling variable R/W where W is the measured root mean square fluctuation of the step position (from [33]).



Figure 10

Figure 10. Persistence probability, P(t), for the Family model shown for different system sizes with different sampling times. Panel (a): Double-log plot showing three different P(t) versus time t curves corresponding to: L = 200 and δt = 4, L = 400 and δt = 16, L = 800 and δt = 64, from bottom to top, respectively. Panel (b): Finite-size scaling of P(t, L, δt). Results for the persistence probabilities for three different sizes (as in panel (a)) with the same value of δt/Lz (i.e. 1/104) are plotted versus t/δt (z = 2). The dotted (dashed) line is a fit of the data to a power law with an exponent {\simeq } 0.75 ({\simeq } 1.0 ) (from [26]).



Figure 11

Figure 11. Experimentally determined persistence probabilities P(t) for steps on Al/Si(111) are shown in panel (a) as a function of time t for different values of the temperature T and the sampling interval δt. The different curves collapse to the same one (panel (b)) when the persistence probabilities are plotted versus t/δt (from [29]).



Figure 12

Figure 12. The steady-state spatial persistence probability, PSS(x), for EW interfaces, obtained using the discrete Family model. Panel (a): Double-log plots of PSS(x) versus x for a fixed sampling distance δx = 1, using three different values of L, as indicated in the legend. Panel (b): Double-log plots of PSS(x) versus x/δx for a fixed system size, L = 1000, and three different values of δx, as indicated in the legend (from [27]).



Figure 13

Figure 13. Representative spatial persistence and survival probability data. The data were taken at 970 K, from an STM image with pixel size of 0.977 nm. The persistence and survival probabilities are represented by squares and circles respectively. The inset is the same persistence curve using logarithmic scales. The solid green line is a power-law fit to the data over the linear region with the steady-state spatial persistence exponent θSS = 0.59 (from [32]).



Figure 14

Figure 14. Survival probabilities determined from single steps chosen to display measurements at two different pixel sizes and a wide range of step lengths. Solid diamonds, squares, and circles are from (500 nm)2 images and have system lengths of L = 98.9 nm, 170 nm, and 162 nm respectively. Open diamonds, squares, and circles are from (300 nm)2 images and have system lengths of L = 65.8 nm, 154 nm, and 87 nm respectively. The survival probability is plotted as a function of the scaled distance x/L. The solid line represents the theoretical prediction of [31] (from [32]).




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