192
6 Quantum Dynamics and Random Matrix Theory
with a large number of degrees of freedom is thermalized if it’s classical dynamics
is chaotic and its quantum dynamics is governed by the Gaussian Orthogonal
Ensemble. Srednicki (1994) also showed that it is possible to derive the BoseEinstein or Fermi-Dirac distribution if the eigenstates are assumed to symmetric
or anti-symmetric, respectively. The procedure is similar to that used to derive
the Bose-Einstein or Fermi-Dirac distributions directly from the microcanonical
ensemble (Jancel 1963).
6.7 Conclusions
There is now a growing body of work on random matrix theory, as it relates to the
thermalization of quantum systems (called eigenstate thermalization). Two recent
reviews describing the literature on this topic are D’Alessio (2016) and Deutsch
(2018). In the past, there have also been attempts to apply random matrix theory to
quantum chromodynamics. References to these older works and other applications
of the random matrix theory to dynamical systems can be found in the review article
(Guhr et al. 1998).
Now that we have developed the random matrix theory for Hamiltonian systems,
it is useful to see how well it agrees with reality. In the next chapter, we look at the
spectral properties of a variety of systems for which the spectrum has been obtained
either from experiment or numerically, and we compare the behavior of these spectra
to the predictions of random matrix theory.
References
Berry MV (1977b) J Phys A Math Gen 10:2083
Berry MV (1977a) Phil Trans Roy Soc (London) 287:237
Brody TA (1974) Lett Nuovo Cimento 7:482
Brody TA, Flores J, French JB, Mello PA, Pandey A, Wong SSM (1981) Rev Mod Phys 53:385
Brookes BC, Dick WFL (1969) Introduction to statistical methods. Heinemann, London
D’Alessio LD, Kafri Y, Polkovnikov A, Rigol M (2016) Adv. Phys. 65:239
Deutsch JM (2018) Rep Prog Phys 81:082001
Dyson FJ, Mehta ML (1963) J Math Phys 4:489
Gaudin M (1961) Nucl Phys 25:447
Guhr T, Muller-Groeling A, Weidenmuller HA (1998) Phys Rep 299:189
Haller E, Koppel H, Cederbaum LS (1983) Chem Phys Lett 101:215
Jancel R (1963) Foundations of classical and quantum statistical mechanics. Pergamon Press,
Oxford
Li W, Reichl LE, Wu B (2002) Phys Rev E 65:56220
Mehta ML (1960) Nucl Phys 18:420
Meyer SL (1975) Data analysis for scientists and dngineers. John Wiley and Sons, Inc., New York
Porter CE (1965) Statistical theories of spectra: fluctuations. Academic Press, New York
Porter CE, Thomas RG (1956) Phys Rev 104:483
Reichl LE (2016) A modern course in statistical physics, 4th edn. Wiley-VCH, Weinheim
6 Quantum Dynamics and Random Matrix Theory
with a large number of degrees of freedom is thermalized if it’s classical dynamics
is chaotic and its quantum dynamics is governed by the Gaussian Orthogonal
Ensemble. Srednicki (1994) also showed that it is possible to derive the BoseEinstein or Fermi-Dirac distribution if the eigenstates are assumed to symmetric
or anti-symmetric, respectively. The procedure is similar to that used to derive
the Bose-Einstein or Fermi-Dirac distributions directly from the microcanonical
ensemble (Jancel 1963).
6.7 Conclusions
There is now a growing body of work on random matrix theory, as it relates to the
thermalization of quantum systems (called eigenstate thermalization). Two recent
reviews describing the literature on this topic are D’Alessio (2016) and Deutsch
(2018). In the past, there have also been attempts to apply random matrix theory to
quantum chromodynamics. References to these older works and other applications
of the random matrix theory to dynamical systems can be found in the review article
(Guhr et al. 1998).
Now that we have developed the random matrix theory for Hamiltonian systems,
it is useful to see how well it agrees with reality. In the next chapter, we look at the
spectral properties of a variety of systems for which the spectrum has been obtained
either from experiment or numerically, and we compare the behavior of these spectra
to the predictions of random matrix theory.
References
Berry MV (1977b) J Phys A Math Gen 10:2083
Berry MV (1977a) Phil Trans Roy Soc (London) 287:237
Brody TA (1974) Lett Nuovo Cimento 7:482
Brody TA, Flores J, French JB, Mello PA, Pandey A, Wong SSM (1981) Rev Mod Phys 53:385
Brookes BC, Dick WFL (1969) Introduction to statistical methods. Heinemann, London
D’Alessio LD, Kafri Y, Polkovnikov A, Rigol M (2016) Adv. Phys. 65:239
Deutsch JM (2018) Rep Prog Phys 81:082001
Dyson FJ, Mehta ML (1963) J Math Phys 4:489
Gaudin M (1961) Nucl Phys 25:447
Guhr T, Muller-Groeling A, Weidenmuller HA (1998) Phys Rep 299:189
Haller E, Koppel H, Cederbaum LS (1983) Chem Phys Lett 101:215
Jancel R (1963) Foundations of classical and quantum statistical mechanics. Pergamon Press,
Oxford
Li W, Reichl LE, Wu B (2002) Phys Rev E 65:56220
Mehta ML (1960) Nucl Phys 18:420
Meyer SL (1975) Data analysis for scientists and dngineers. John Wiley and Sons, Inc., New York
Porter CE (1965) Statistical theories of spectra: fluctuations. Academic Press, New York
Porter CE, Thomas RG (1956) Phys Rev 104:483
Reichl LE (2016) A modern course in statistical physics, 4th edn. Wiley-VCH, Weinheim