Level dynamics: An approach to the study of avoided level crossings and transition to chaos

The Dyson-Pechukas level dynamics has been reformulated and made suitable for studying avoided level crossings and transition to chaos. The N-level dynamics is converted into a many-body problem of one-dimensional Coulomb gas with N-constituent particles having intrinsic excitations. It is shown that local fluctuation of the level distribution is generated by a large number of avoided level crossings. The role played by avoided level crossings in generating chaoticity in level dynamics is similar to the role played by short-range collisions in causing thermalization in many-body dynamics. Furthermore, the effect of level changing rates in producing avoided level crossings is the same as particle velocities in causing particle-particle collisions. A one-dimensional su(2) Hamiltonian has been constructed as an illustration of the level dynamics, showing how the avoided level crossings cause the transition from a regular distribution to the chaotic Gaussian orthogonal ensemble (GOE) distribution of the levels. The existence of the one-dimensional su(2) Hamiltonian which can show both GOE and Poisson level statistics is remarkable and deserves further investigation