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Random matrices and quantum chaos Thomas Kriecherbauer*, Jens Marklof†,‡, and Alexander Soshnikov§ + Author Affiliations *Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstrasse 39, D-80333 Munich, Germany; †School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom; and §Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633 Next SectionAbstract The theory of random matrices has far-reaching applications in many different areas of mathematics and physics. In this note, we briefly describe the state of the theory and two of the perhaps most surprising appearances of random matrices, namely in the theory of quantum chaos and in the theory of prime numbers. Since the pioneering work of E. Wigner in the 1950s, it has emerged that the statistical properties of many quantum systems can be modeled by random matrices. Wigner's original work was concerned with neutron excitation spectra of heavy nuclei. These are many-particle systems whose interaction, according to Wigner, is so complex that the Hamiltonian representing the system should behave like a large random matrix. It was discovered 30 years later that even simple one-particle quantum systems exhibit random matrix statistics, if the classical limit of the system is chaotic. An example of such a system is the electron in the heart-shaped region of Fig. 1, studied in ref. 1. In Fig. 2, the distribution of energy level spacings for that system is compared with that of the Gaussian Orthogonal Ensemble of random matrices. In this situation, nearby levels seem to repel each other, because the probability of finding small spacings is small. It is, in fact, believed that all generic quantum systems follow random matrix statistics of a suitably chosen ensemble, if the underlying classical dynamics is chaotic. The choice of ensemble depends on the physical symmetries of the system, for instance, time-reversal symmetry (2). If, in contrast, the underlying dynamics is regular, i.e., nonchaotic, the energy levels will not follow random matrix statistics but rather will behave like independent random variables from a Poisson process (3). An example of such a regular system is an electron confined to a circular domain. Its level spacing distribution is shown in Fig. 3, vs. the exponential distribution of a Poisson process. http://tieba.baidu.com/mo/q/checkurl?url=http%3A%2F%2Fwww.pnas.org%2Fcontent%2F98%2F19%2F10531.full&urlrefer=9210acd2f3d7a5fd10a8981d1c8c4745
程序 using System;using System.Collections.Generic;using System.ComponentModel;using System.Data;using System.Linq;using System.Text;using System.Web;using System.Net;using System.Runtime.InteropServices;using System.Threading;using System.IO;namespace test2{    class Program    {        //百度网页的编码格式为GB2312        static Encoding encoding = Encoding.GetEncoding("GB2312");        static String cookieStr;//保存cookie字符串用的        static Thread[] threadGrp;        static int thread_num =15;//线程数目        static int indexer = 0;   //代理索引        static int urlIndex = 0;  //需要访问的链接的索引                    static int normal_sleep =5000;//每次访问之后线程休眠的时间,刷得太快会被封ip        static int failed_sleep =0;        static int counter = 0;        static int majorCounter=0;//记录总共访问了多少次        static bool ifUseProxy = true;//是否启用代理,用了代理访问速度会变慢,需要注意的是,baidu能够认出普通的代理        static void Main(string[] args)        {            //从文件中获取上一次的session cookie或者重新获取session cookie            FileStream fs = new FileStream("C:\\ss.txt",FileMode.OpenOrCreate);            if (fs.Length>0)            {                //如果文件是空或者没有文件,则重新获取session cookie                StreamReader sr = new StreamReader(fs);                cookieStr = sr.ReadToEnd();                //将新的cookie写到文件中,这样做的目的是不必每次启动程序就去登录获取cookie,次数多了之后,会需要验证码
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