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Abstract
Random matrix theory (RMT) provides a successful model for quantum systems, whose classical counterpart has chaotic dynamics. It is based on two assumptions: (1) matrix-element independence, and (2) base invariance. The last decade witnessed several attempts to extend RMT to describe quantum systems with mixed regular-chaotic dynamics. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors have presented other versions of the theory that keep base invariance at the expense of allowing correlations between matrix elements. This is achieved by starting from non-extensive entropies rather than the standard Shannon entropy, or by following the basic prescription of the recently suggested concept of superstatistics. The latter concept was introduced as a generalization of equilibrium thermodynamics to describe non-equilibrium systems by allowing the temperature to fluctuate. We here review the superstatistical generalizations of RMT and illustrate their value by calculating the nearest-neighbor-spacing distributions and comparing the results of calculation with experiments on billiards modeling systems in transition from order to chaos.
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References
- ABUL-MAGD, A.Y. 2006. Superstatistics in random matrix theory. Physica, A361, 41.
- ABUL-MAGD, A.Y., DIETZ, B., FRIEDRICH, T. and RICHTER, A. 2008. Spectral fluctuations of billards with mixed dynamics: From time series to super statistics. Phys. Rev., E77: 1-12.
- BALIAN, R. 1958. Random matrices in information theory. Nuovo Cim., B57: 183-193.
- BECK, C. and COHEN, E.G.D. 2003. Superstatistics. Physica, A322: 267-275.
- BECK, C., COHEN, E.G.D. and SWINNEY, H.L. 2005. From time series to superstatistics. Phys. Rev., E72: 35-42.
- BERRY, M.V. 1977. Regular and irregular semiclassical wave functions. J. Phys., A10: 2083-2091.
- BERRY, M.V. and ROBNIK, M. 1984. Semiclassical level spacings when regular and chaotic orbits coexist. J. Phys., A17: 2413-2421.
- BOHIGAS, O., GIANNONI, M.J. and SCHMIT, C. 1984. Characterization of chaotic quantrum spectra and universality of level fluctuation laws. Phys. Rev. Lett., 52: 1-4.
- BUNIMOVICH, L.A. 1974. On ergodic properties of certain billiards. Funct. Anal. Appl., 8: 254-255.
- BUNIMOVICH, L.A. and VENKATUYIRI, S. 1997. On one mechanism of transition to chaos in lattice dynamical systems. Phys. Rep., 290: 81-100.
- CAËR, G.Le and DELANNAY, R. 1999. Some consequences of exchangeability in random-matrix theory. Phys. Rev., E59: 6281-6285.
- CASATI, A., MOLINARI, L. and IZRAILEV, F. 1990. Scaling properties of band random matrices. Phys. Rev. Lett., 64: 1851-1854.
- DEMBOWSKI, C., GRÄF, H.-D., HEINE, A., HESSE, T., REHFELD, H. and RICHTER, A. 2001. First experimental test of a trace formula for billiard systems showing mixed dynamics. Phys. Rev. Lett., 86: 3284-3287.
- DIETZ, B., FRIEDRICH, T., MISKI-OGLU, M., RICHTER, A., SELIGMAN, T.H. and ZAPFE, K. 2006. Nonperiodic echoes from mushroom billiard hats. Phys. Rev., E74: 1-8.
- ECKMANN, J.-P. and RUELLE, D. 1985. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys., 57: 617-619.
- ELNASHAIE, S.S.E.H. and ELSHISHINI, S.S. 1996. Dynamical Modelling, Bifurcation and Chaotic Behavior of Gas-Solid Catalytic Reactions. Gordon and Breach, Amsterdam.
- EVANS, J. and MICHAEL, F. 2002. Towards a non-extensive random matrix theory. e-prints arXiv:cond-mat/0207472 and /0208151.
- GRADSHTEYN, I.S. and RYZHIK, I.M. 1980. Tables of Integrals, Series, and Products. Academic, New York.
- GUHR, T., MÜLLER-GROELING, A. and WEIDENMÜLLER, H.A. 1998. Random matrix theory in quantum physics. Phys. Rep., 299: 189-425.
- GUTIÉRREZ, M., BRACK, M., RICHTER, K. and SUGITA, A. 2007. The effect of pitchfork bifurcations on the spectral statistics of Hamiltonian systems. J. Phys., A40: 1525-1543.
- HUSSEIN, M.S. and PATO, M.P. 1993. Description of chaos-order transition with random matrices within the maximum entropy principle. Phys. Rev. Lett., 70: 1089 -1092.
- LICHTENBERG, A.J. and LIEBERMAN, M.A. 1983. Regular and Stochastic Motion. Applied Mathematical Sciences, Springer, New York.
- MEHTA, M.L. 1991. Random Matrices. 2nd ed., Academic, New York.
- PROSEN, T. and ROBNIK, M. 1994. Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry-Robnik behavior. J. Phys., A27: 8059-8077.
- REIF, F. 1984. Fundamental of Statistical and Thermal Physics. McGraw-Hill, Singapore.
- RELAÑO, A., GÓMEZ, J.M.G., MOLINA, R.A., RETAMOSA, J. and FALEIRO, E. 2002. Quantum chaos and 1/f noise. Phys. Rev. Lett., 89: 1-4.
- ROBNIK, M.J. 1983. Classical dynamics of a family of billiards with analytic boundaries. Physica, A16: 3971 -3986.
- ROSENZWEIG, N. and PORTER, C.E. 1960. Repulsion of energy level in complex atomic spectra. Phys. Rev., 120: 1698-1714.
References
ABUL-MAGD, A.Y. 2006. Superstatistics in random matrix theory. Physica, A361, 41.
ABUL-MAGD, A.Y., DIETZ, B., FRIEDRICH, T. and RICHTER, A. 2008. Spectral fluctuations of billards with mixed dynamics: From time series to super statistics. Phys. Rev., E77: 1-12.
BALIAN, R. 1958. Random matrices in information theory. Nuovo Cim., B57: 183-193.
BECK, C. and COHEN, E.G.D. 2003. Superstatistics. Physica, A322: 267-275.
BECK, C., COHEN, E.G.D. and SWINNEY, H.L. 2005. From time series to superstatistics. Phys. Rev., E72: 35-42.
BERRY, M.V. 1977. Regular and irregular semiclassical wave functions. J. Phys., A10: 2083-2091.
BERRY, M.V. and ROBNIK, M. 1984. Semiclassical level spacings when regular and chaotic orbits coexist. J. Phys., A17: 2413-2421.
BOHIGAS, O., GIANNONI, M.J. and SCHMIT, C. 1984. Characterization of chaotic quantrum spectra and universality of level fluctuation laws. Phys. Rev. Lett., 52: 1-4.
BUNIMOVICH, L.A. 1974. On ergodic properties of certain billiards. Funct. Anal. Appl., 8: 254-255.
BUNIMOVICH, L.A. and VENKATUYIRI, S. 1997. On one mechanism of transition to chaos in lattice dynamical systems. Phys. Rep., 290: 81-100.
CAËR, G.Le and DELANNAY, R. 1999. Some consequences of exchangeability in random-matrix theory. Phys. Rev., E59: 6281-6285.
CASATI, A., MOLINARI, L. and IZRAILEV, F. 1990. Scaling properties of band random matrices. Phys. Rev. Lett., 64: 1851-1854.
DEMBOWSKI, C., GRÄF, H.-D., HEINE, A., HESSE, T., REHFELD, H. and RICHTER, A. 2001. First experimental test of a trace formula for billiard systems showing mixed dynamics. Phys. Rev. Lett., 86: 3284-3287.
DIETZ, B., FRIEDRICH, T., MISKI-OGLU, M., RICHTER, A., SELIGMAN, T.H. and ZAPFE, K. 2006. Nonperiodic echoes from mushroom billiard hats. Phys. Rev., E74: 1-8.
ECKMANN, J.-P. and RUELLE, D. 1985. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys., 57: 617-619.
ELNASHAIE, S.S.E.H. and ELSHISHINI, S.S. 1996. Dynamical Modelling, Bifurcation and Chaotic Behavior of Gas-Solid Catalytic Reactions. Gordon and Breach, Amsterdam.
EVANS, J. and MICHAEL, F. 2002. Towards a non-extensive random matrix theory. e-prints arXiv:cond-mat/0207472 and /0208151.
GRADSHTEYN, I.S. and RYZHIK, I.M. 1980. Tables of Integrals, Series, and Products. Academic, New York.
GUHR, T., MÜLLER-GROELING, A. and WEIDENMÜLLER, H.A. 1998. Random matrix theory in quantum physics. Phys. Rep., 299: 189-425.
GUTIÉRREZ, M., BRACK, M., RICHTER, K. and SUGITA, A. 2007. The effect of pitchfork bifurcations on the spectral statistics of Hamiltonian systems. J. Phys., A40: 1525-1543.
HUSSEIN, M.S. and PATO, M.P. 1993. Description of chaos-order transition with random matrices within the maximum entropy principle. Phys. Rev. Lett., 70: 1089 -1092.
LICHTENBERG, A.J. and LIEBERMAN, M.A. 1983. Regular and Stochastic Motion. Applied Mathematical Sciences, Springer, New York.
MEHTA, M.L. 1991. Random Matrices. 2nd ed., Academic, New York.
PROSEN, T. and ROBNIK, M. 1994. Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry-Robnik behavior. J. Phys., A27: 8059-8077.
REIF, F. 1984. Fundamental of Statistical and Thermal Physics. McGraw-Hill, Singapore.
RELAÑO, A., GÓMEZ, J.M.G., MOLINA, R.A., RETAMOSA, J. and FALEIRO, E. 2002. Quantum chaos and 1/f noise. Phys. Rev. Lett., 89: 1-4.
ROBNIK, M.J. 1983. Classical dynamics of a family of billiards with analytic boundaries. Physica, A16: 3971 -3986.
ROSENZWEIG, N. and PORTER, C.E. 1960. Repulsion of energy level in complex atomic spectra. Phys. Rev., 120: 1698-1714.