It is argued that phase transitions in finite systems are likely to exhibit level statistics ascribed to quantum chaos. Apparent counterexamples become chaotic under generic perturbation. The mathematical mechanism for this to occur lies in the abundance of exceptional points in the transitional region. Exceptional points are singularities in the complex \ensuremath{\lambda} plane of the spectrum ${\mathit{E}}_{\mathit{n}}$ (\ensuremath{\lambda}), where \ensuremath{\lambda} is the parameter that effects the transition. Examples are given for illustration. It is shown that the quality of the mean field is unaffected outside the transitional region whether or not the transitional region is chaotic. Related properties of the eigenvectors are also discussed.It is argued that phase transitions in finite systems are likely to exhibit level statistics ascribed to quantum chaos. Apparent counterexamples become chaotic under generic perturbation. The mathematical mechanism for this to occur lies in the abundance of exceptional points in the transitional region. Exceptional points are singularities in the complex \ensuremath{\lambda} plane of the spectrum ${\mathit{E}}_{\mathit{n}}$(\ensuremath{\lambda}), where \ensuremath{\lambda} is the parameter that effects the transition. Examples are given for illustration. It is shown that the quality of the mean field is unaffected outside the transitional region whether or not the transitional region is chaotic. Related properties of the eigenvectors are also discussed.

Physical Review AApr 1, 199175 citations

Quantum mechanics and quantum chaos

Robert C. Bishop

Institute of Physics eBooksMay 1, 20250 citations

Chaos in Classical and Quantum Mechanics

Martin C. Gutzwiller

Interdisciplinary applied mathematicsJan 1, 19903,916 citations

Causal Wave Mechanics and the Advent of Complexity. II. Dynamic uncertainty in quantum systems and the correspondence principle

Andrei P. Kirilyuk

The intrinsic multivaluedness of interaction process, revealed in Part I of this series of papers, is interpreted as the origin of the true dynamical (in particular, quantum) chaos. The latter is causally deduced as unceasing series of transitions, dynamically probabilistic by their origin, between the equally real, but incompatible 'realisations' (modes of interaction) of a system. The obtained set of realisations form the causally derived, intrinsically complete "space of events" providing the crucial extension of the notion of probability and the method of its first-principle calculation. The fundamental dynamic uncertainty thus revealed is specified for Hamiltonian quantum systems and applied to quantum chaos description in periodically perturbed systems. The ordinary semiclassical transition in our quantum-mechanical results leads to exact reproduction of the main features of chaotic behaviour of the system known from classical mechanics, which permits one to "re-establish" the correspondence principle for chaotic systems (inevitably lost in any their conventional, single-valued description). The causal dynamical randomness in the extended quantum mechanics is not restricted, however, to semiclassical conditions and generically occurs also in essentially quantum regimes, even though partial "quantum suppression of chaos" does exist and is specified in our description, as well as other particular types of the quantum (truly) chaotic behaviour.The intrinsic multivaluedness of interaction process, revealed in Part I of this series of papers, is interpreted as the origin of the true dynamical (in particular, quantum) chaos. The latter is causally deduced as unceasing series of transitions, dynamically probabilistic by their origin, between the equally real, but incompatible 'realisations' (modes of interaction) of a system. The obtained set of realisations form the causally derived, intrinsically complete "space of events" providing the crucial extension of the notion of probability and the method of its first-principle calculation. The fundamental dynamic uncertainty thus revealed is specified for Hamiltonian quantum systems and applied to quantum chaos description in periodically perturbed systems. The ordinary semiclassical transition in our quantum-mechanical results leads to exact reproduction of the main features of chaotic behaviour of the system known from classical mechanics, which permits one to "re-establish" the correspondence principle for chaotic systems (inevitably lost in any their conventional, single-valued description). The causal dynamical randomness in the extended quantum mechanics is not restricted, however, to semiclassical conditions and generically occurs also in essentially quantum regimes, even though partial "quantum suppression of chaos" does exist and is specified in our description, as well as other particular types of the quantum (truly) chaotic behaviour.

arXiv:quant-ph/9511035Nov 23, 19950 citations

{\mathit{E}}_{\mathit{n}}

(\ensuremath{\lambda}), where \ensuremath{\lambda} is the parameter that effects the transition. Examples are given for illustration. It is shown that the quality of the mean field is unaffected outside the transitional region whether or not the transitional region is chaotic. Related properties of the eigenvectors are also discussed.