level 7
quicker1
楼主
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://www.pnas.org/content/98/19/10531.full
2010年05月16日 07点05分
1
+ 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://www.pnas.org/content/98/19/10531.full