Györgyi G, Moloney N R, Ozogány K, Rácz Z
Institute for Theoretical Physics - HAS Research Groups, Eötvös University, Pázmány sétány 1/a, 1117 Budapest, Hungary.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Feb;75(2 Pt 1):021123. doi: 10.1103/PhysRevE.75.021123. Epub 2007 Feb 28.
Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/f(alpha) power spectrum. For 0<or=alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha-->infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha>1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.
给出了具有1/f(α)功率谱的平稳周期高斯信号（一维界面）的最大相对高度分布的数值和解析结果。对于0≤α<1（相关性衰减 regime），我们观察到随着样本量增加，数学上确立的极限分布（Fisher-Tippett-Gumbel分布）趋近得极其缓慢。对于α>1（强相关性 regime），收敛很快，并且可以通过数值方法构建分布函数的高精度图库。在α→∞的极限情况下以及对于大α，可以通过微扰展开获得解析结果。此外，使用路径积分技术，我们推导了分布函数的一个迹公式，该公式对于α = 2n（偶数整数）有效。从后者我们提取了分布函数的小自变量渐近线，发现其对任意α>1的解析延拓与模拟结果一致。界面的极值和粗糙度统计量的比较揭示了分布函数在小自变量和大自变量渐近线方面的相似性。
