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Analysis of Boltzmann-Langevin Dynamics in Nuclear Matter

Sakir Ayik B, Ph. Chomaz, M. Colonna, Jørgen Randrup

Published Nov 23, 1994• 2 citations
OpenAlex

Abstract

The Boltzmann-Langevin dynamics of harmonic modes in nuclear matter is analyzed within linear-response theory, both with an elementary treatment and by utilizing the frequency-dependent response function. It is shown how the source terms agitating the modes can be obtained from the basic BL correlation kernel by a simple projection onto the associated dual basis states, which are proportional to the RPA amplitudes and can be expressed explicitly. The source terms for the correlated agitation of any two such modes can then be extracted directly, without consideration of the other modes. This facilitates the analysis of collective modes in unstable matter and makes it possible to asses the accuracy of an approximate projection technique employed previously.

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The Boltzmann-Langevin dynamics of harmonic modes in nuclear matter is analyzed within linear-response theory, both with an elementary treatment and by utilizing the frequency-dependent response function. It is shown how the source terms agitating the modes can be obtained from the basic BL correlation kernel by a simple projection onto the associated dual basis states, which are proportional to the RPA amplitudes and can be expressed explicitly. The source terms for the correlated agitation of any two such modes can then be extracted directly, without consideration of the other modes. This facilitates the analysis of collective modes in unstable matter and makes it possible to asses the accuracy of an approximate projection technique employed previously.
In the spirit of the linear response theory, a simple derivation is given of a Langevin equation for collective modes of a macroscopic system. Unlike in the usual treatment, we separate a weak interaction of individual particles with collective modes which is allowed to depend on time. In nonequilibrium cases, we have shown that a damping constant of the collective modes can be expressed in terms of a time correlation function, in which its average is taken with respect to a nonequilibrium density matrix rather than a usual canonical density matrix. A microscopic expression for a Langevin random force is obtained in terms of fluctuations of a density matrix.
The dynamics of fermionic many-body systems is investigated in the framework of Boltzmann--Langevin (BL) stochastic one-body approaches. Within the recently introduced Boltzmann--Langevin one-body (BLOB) model, we examine the interplay between mean-field effects and two-body correlations, of stochastic nature, for nuclear matter at moderate temperature and in several density conditions, corresponding to stable or mechanically unstable situations. Numerical results are compared with analytic expectations for the fluctuation amplitude of isoscalar and isovector densities, probing the link to the properties of the employed effective interaction; namely, symmetry energy (for isovector modes) and incompressibility (for isoscalar modes). For unstable systems, clusterization is observed. The associated features are compared with analytical results for the typical length and timescales characterizing the growth of unstable modes in nuclear matter and for the isotopic variance of the emerging fragments. We show that the BLOB model is generally better suited than simplified approaches previously introduced to solve the BL equation, and it is therefore more advantageous in applications to open systems, such as heavy-ion collisions.
We address the dynamics of damped collective modes in terms of first and second moments. The modes are introduced in a self-consistent fashion with the help of a suitable application of linear response theory. Quantum effects in the fluctuations are governed by diffusion coefficients D_{μν}. The latter are obtained through a fluctuation dissipation theorem generalized to allow for a treatment of unstable modes. Numerical evaluations of the D_{μν} are presented. We discuss briefly how this picture may be used to describe global motion within a locally harmonic approximation.
Aiming at exploring microscopic dynamics responsible for the dissipative large-amplitude collective motion, the dynamic response and correlation functions are introduced within the general theory of nuclear coupled-master equations. The theory is based on the microscopic theory of nuclear collective dynamics which has been developed within the time-dependent Hartree-Fock (TDHF) theory for disclosing the complex structure of the TDHF manifold. A systematic numerical method for calculating the dynamic response and correlation functions is proposed. By performing numerical calculation for a simple model Hamiltonian, it is pointed out that the dynamic response function gives important information in understanding the large-amplitude dissipative collective motion which is described by an ensemble of trajectories within the TDHF manifold.
We address the dynamics of damped collective modes in terms of first and second moments. The modes are introduced in a self-consistent fashion with the help of a suitable application of linear response theory. Quantum effects in the fluctuations are governed by diffusion coefficients D μν . The latter are obtained through a fluctuation dissipation theorem generalized to allow for a treatment of unstable modes. Numerical evaluations of the D μν are presented. We discuss briefly how this picture may be used to describe global motion within a locally harmonic approximation. Relations to other methods are discussed, like "dissipative tunneling", RPA at finite temperature and generalizations of the "Static Path Approximation".