S. Ayik

On stochastic approaches of nuclear dynamics

Abstract 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 Greens 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.

Fluctuations of Single Particle Density in Nuclear Collisions

Abstract In order to incorporate fluctuations into the extended TDHF, a new approach is proposed. The evolution of the single-particle density is considered as a “generalized Langevin process” in which the correlated part of the two-body collisions acts as a “random force”. In the semi-classical approximation, the correlation function of the random force is calculated. A possible algorithm for the numerical solution is discussed.

Transport theory of fluctuation phenomena in nuclear collisions

Abstract An extension of the single-particle transport theory is proposed to allow calculation of fluctuation phenomena in nuclear collisions. Instead of neglecting the high-order correlations in the equation of motion for the single-particle density, they are retained and treated statistically. As a result, correlations appear as a “random force” in the equation of motion, in much as the same way as in a Langevin equation for a stochastic processes. It turns out, consistent with the fluctuation-dissipation theorem of statistical mechanics, that the correlation function of the “random force”, which measures the local fluctuations in density, is entirely determined by the average properties of density. The resultant “stochastic transport equation” describes a diffusive evolution of trajectories of density in an abstract space of all single-particle densities. A preliminary numerical calculation is presented, and implications are discussed.

Mean-field theory and statistical treatment of residual interactions

The time-dependent Hartree-Fock theory is generalized in order to include two-body collisions. Using a method similar to the projection method of quantum statistics an exact and closed equation is derived for the one-particle density matrix which takes into account both the mean field and the two-body residual interactions. In the limit of long mean free path of the particles, the exact equation for the one-particle density matrix is simplified by introducing some general statistical properties of the residual interactions. An equation is obtained for the expectation values of operators which includes both the mean-field contributions and the correlations due to the residual interactions. The effects of the correlations on the fluctuations of collective variables are discussed. It is shown that the statistical approximations introduced do not destroy the energy and the momentum conservations.

Mechanical and chemical spinodal instabilities in finite quantum systems

Self consistent quantum approaches are used to study the instabilities of finite nuclear systems. The frequencies of multipole density fluctuations are determined as a function of dilution and temperature, for several isotopes. The spinodal region of the phase diagrams is determined and it appears that instabilities are reduced by finite size effects. The role of surface and volume instabilities is discussed. It is indicated that the important chemical effects associated with mechanical disruption may lead to isospin fractionation.

Time-dependent shell-model theory of dissipative heavy-ion collisions

A transport theory is formulated within a time-dependent shell-model approach. Time averaging of the equations for macroscopic quantities lead to irreversibility and justifies weak-coupling limit and Markov approximation for the (energy-conserving) one- and two-body collision terms. Two coupled equations for the occupation probabilities of dynamical single-particle states and for the collective variable are derived and explicit formulas for transition rates, dynamical forces, mass parameters and friction coefficients are given. The applicability of the formulation in terms of characteristic quantities of nuclear systems is considered in detail and some peculiarities due to memory effects in the initial equilibration process of heavy-ion collisions are discussed.

Microscopic transport theory of heavy-ion collisions: IV. Transport coefficients including angular momentum

We investigate the transfer of relative angular momentum into internal angular momentum of the fragments in deeply inelastic heavy-ion collisions. The previous studies of drift and diffusion coefficients for mass transfer and energy dissipation are extended to include the dissipation of relative angular momentum. In a single-particle model the transport coefficients are obtained in closed form. We discuss their dependence on initial relative angular momentuml, dissipated angular momentum, mass asymmetry and total mass. Using thel-dependent interaction times obtained earlier we calculate mean value and variance of the dissipated angular momentum. For sufficiently long interaction times, the mean value reaches the sticking limit. The large fluctuations, however, do not allow for a classical description of angular momentum dissipation. These results are consistent with recentγ-multiplicity experiments.

Quantal effects on growth of instabilities in nuclear matter

Abstract The growth rates of unstable modes in nuclear matter are investigated using a quantal dispersion relation. Calculations carried out with a realistic finite range potential show that unstable modes with wave number larger than the Fermi momentum are strongly suppressed, due to the quantal effect, in the mean field evolution. The observed consequence is the shift of the most important modes towards longer wave lengths.

One-body energy dissipation in fusion reactions from mean-field theory

Information on dissipation in the entrance channel of heavy-ion collisions is extracted by the macroscopic reduction procedure of time-dependent Hartree-Fock theory. The method gives access to a fully microscopic description of the friction coefficient associated with the transfer of energy from the relative motion toward intrinsic degrees of freedom. The reduced friction coefficient exhibits a universal behavior, i.e., almost independent of systems investigated, whose order of magnitude is comparable with the calculations based on linear response theory. Similarly to nucleus-nucleus potential, especially close to the Coulomb barrier, there are sizable dynamical effects on the magnitude and form factor of the friction coefficient.

Fluctuation and dissipation dynamics in fusion reactions from a stochastic mean-field approach

By projecting the stochastic mean-field dynamics on a suitable collective path during the entrance channel of heavy-ion collisions, expressions for transport coefficients associated with relative distance are extracted. These transport coefficients, which have forms similar to those familiar from the nucleon exchange model, are evaluated by carrying out time-dependent Hartree-Fock simulations. The calculations provide an accurate description of the magnitude and form factor of transport coefficients associated with one-body dissipation and fluctuation mechanisms.