混沌微波粗糙台球中节域的实验研究
Experimental investigation of nodal domains in the chaotic microwave rough billiard
摘要：
本文给出了混沌半圆微波粗糙台球中处于Shnirelman遍历区域的波函数（电场分布）节域的实验研究结果。节域是波函数具有确定符号的区域。在N=435的水平上测量了粗台球的波函数Psi_N。在这种方式的依赖节域的数量\aleph_N的水平数$N$被发现，我们表明，在限制N->infty的实验数据的最小二乘拟合揭示了渐近节域数aleph_N/N = 0.058 +- 0.006，接近理论预测aleph_N/N +- 0.062。我们还发现节点域的面积s和周长l的分布具有幂指数n_s ~ s^{-tau}和n_l ~ l^{-tau '}，其中标度指数分别为τ= 1.99 ± 0.14和τ'= 2.13 ± 0.23。这些结果与逾渗理论的预测是一致的。最后，我们证明，对于更高级别的数N = 220-435，有符号面积分布在理论极限Sigma_{A} = 0.0386 N^{-1}附近振荡。
英文摘要：
We present the results of experimental study of nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains are regions where a wave function has a definite sign. The wave functions Psi_N of the rough billiard were measured up to the level number N=435. In this way the dependence of the number of nodal domains \aleph_N on the level number $N$ was found. We show that in the limit N->infty a least squares fit of the experimental data reveals the asymptotic number of nodal domains aleph_N/N = 0.058 +- 0.006 that is close to the theoretical prediction aleph_N/N +- 0.062. We also found that the distributions of the areas s of nodal domains and their perimeters l have power behaviors n_s ~ s^{-tau} and n_l ~ l^{-tau'}, where scaling exponents are equal to τ= 1.99 +- 0.14 and τ'=2.13 +- 0.23, respectively. These results are in a good agreement with the predictions of percolation theory. Finally, we demonstrate that for higher level numbers N = 220-435 the signed area distribution oscillates around the theoretical limit Sigma_{A} = 0.0386 N^{-1}.
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