Non-Markovian transport equations for nuclear large amplitude motion are derived from the collisional kinetic equation. The memory effects are caused by Fermi surface distortions and depend on the relaxation time. It is shown that nuclear collective motion and nuclear fission are influenced strongly by memory effects at the relaxation time у5ϫ10 Ϫ23 s. In particular, the descent of the nucleus from the fission barrier is accompanied by characteristic shape oscillations. The eigenfrequency and the damping of the shape oscillations depend on the contribution of the memory integral in the equations of motion. The shape oscillations disappear at the short relaxation time regime at →0, which corresponds to the usual Markovian motion in the presence of friction forces. We show that the elastic forces produced by the memory integral lead to a significant delay for the descent of the nucleus from the barrier. Numerical calculations for the nucleus 236 U show that due to the memory effect the saddle-to-scission time grows by a factor of about 3 with respect to the corresponding saddle-to-scission time obtained in liquid drop model calculations with friction forces.

Statistical analysis of distributions of the collective states in the actinide and rare-earth nuclei is performed in terms of the nearest neighbor spacing distribution (NNSD). Several approximations, such as the linear approach to the level repulsion density and that suggested by Brody to the NNSDs were applied for the analysis. We found an intermediate character of the experimental spectra between the order and the chaos for a number of the rare-earth and actinide nuclei. They are more close to the Wigner distribution for energies limited by 3 MeV, and to the Poisson distribution for data including higher excitation energies and higher spins. The latter is in agreement with the theoretical calculations. These features are confirmed by the cumulative distributions, where the Wigner contribution dominates at smaller spacings while the Poisson one is more important at larger spacings. *

Within the generalized (non-Markovian) multidimensional Langevin approach, the time and energy characteristics of symmetric fission of highly excited heavy nuclei are studied. In two-dimensional space of the collective deformation parameters, it is considered a nuclear descent from the top of the fission barrier to the scission point. The distributions of descent times and total kinetic energy of fission fragments are calculated as functions of memory time, measuring the relative size of memory effects in the collective dynamics. We found that the peculiarities of the non-Markovian dynamics at fairly large values of the memory time are reflected in the saturation of the mean time of motion from the saddle to scission with the growth of the strength of memory effects in the system.

The simple one-parameter nearest neighbor-spacing distribution (NNSD) is suggested for statistical analysis of nuclear spectra. This distribution is derived within the Wigner-Dyson approach in the linear approximation for the level repulsion density of quantum states. The obtained NNSD gives the individual information on the Wigner and Poisson contributions in agreement with that of the statistical experimental distributions of collective states in deformed nuclei. Using this NNSD, one finds that the symmetry breaking due to the fixing of projections of the angular momentum of collective states enhances a chaos as a shift of the NNSD from the Poisson to Wigner distribution behavior.

The general problem of dissipation in macroscopic large-amplitude collective motion and its relation to energy diffusion of intrinsic degrees of freedom of a nucleus is studied. By applying the cranking approach to the nuclear many-body system, a set of coupled dynamical equations for the collective classical variable and the quantum mechanical occupancies of the intrinsic nuclear states is derived in the limit of weak coupling of the collective and intrinsic subsystems. Different dynamical regimes of the intrinsic nuclear motion and its consequences on time properties of collective dissipation are discussed.

We study dissipative properties of the motion of a slow nuclear collective variable weakly coupled to a complex quantum environment formed by the fast nucleonic degrees of freedom. The fast quantum mechanical subsystem is treated within the random matrix approach, where the complexity of the nucleonic degrees of freedom's motion can be changed by a parameter from regular to fully chaotic. Classical dynamics is assumed for the slow variable, and the equation of motion is determined from conservation of the total energy of the nuclear many-body system. We show that the macroscopic equation of motion for the collective variable is subject to a memory-dependent friction force, with a retardation defined by the chaoticity of the fast nucleonic environment.

We study classical and quantum chaos for two interacting particles on the plane. This is the simplest nontrivial case which sheds light on chaos in interacting many-body systems. The system consists of a confining one-body potential, assumed to be a deformed harmonic oscillator, and a two-body interaction of Coulomb type. In general, the dynamics is mixed with regular and chaotic trajectories. The relative roles of the one-body field and the two-body interaction are investigated. Chaos sets in as the strength of the two-body interaction increases. However, the degree of chaoticity strongly depends on the shape of the one-body potential and, for some shapes of the harmonic oscillator, the dynamics remains regular for all values of the two-body interaction. Scaling properties are found for the classical as well as for the quantum mechanical problem.

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