On stochastic approaches of nuclear dynamics

We present recent developments of stochastic descriptions of nuclear dynamics. We focus on the newly introduced microscopic descriptions, such as stochastic extensions of currently used kinetic equations, as well as on more phenomenological, macroscopic approaches. We show to what extent these stochastic descriptions may offer a proper picture of nuclear dynamics both in strongly out of equilibrium situations, such as the ones encountered in energetic heavy-ion collisions or in closer to equilibrium situations such as the deexcitation of hot nuclei by thermal fission. In Section 1 we present a pedestrian introduction to the stochastic description of dynamical systems. We start from the elementary Brownian motion and introduce the Langevin and Fokker-Planck descriptions of the motion on that occasion. A few words are then spent to discuss the numerical methods developed for simulating stochastic equations. Section 2 of the paper is devoted to a formal introduction and discussion of both macroscopic and microscopic stochastic descriptions of nuclear dynamics. After a brief introduction reminding general concepts of equilibrium statistical physics we focus on microscopic descriptions of the many-body problem. We introduce here the Boltzmann Langevin equation which will provide a basis for many subsequent discussions. After having discussed the obtention of this equation from various points of view (from density matrix and Green's function techniques in particular), we consider reduced versions of this equation as well as a Fokker-Planck alternative. Section 3 is devoted to an analysis of fission by means of Langevin or Fokker-Planck-like approaches. We mainly discuss phenomenological approaches and spend some time in a detailed presentation of the ingredients entering these models. We present results obtained in these dynamical calculations when a proper account of particle evaporation is included for describing the fission of hot nuclei. Critical comparisons with experimental data are also provided. In Section 4 we focus on the application of the Boltzmann Langevin Equation to various situations encountered in energetic nuclear collisions. We first remind some typical examples for which this stochastic approach is both necessary and well suited. Typical applications are nuclear multifragmentation and subthreshold particle production, such as in particular kaon production. We discuss possible simulations of this equation and present some results in realistic calculations of collisions. We particularly focus on the dynamics of collective variables such as the quadrupole moment of the momentum distribution. We finally discuss other numerical simulations developed in the field. The last section before conclusion is devoted to extensions presently developed in the field of microscopic stochastic descriptions of nuclear dynamics. We present as a first step a relativistic version of the theory, then focus on fluid dynamics reductions. We finally discuss in some detail the recently introduced Stochastic time-dependent Hartree-Fock theory, which could provide new interesting developments.