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Time-dependent Green’s functions approach to nuclear reactions

Arnau Rios, Paweł Danielewicz

Published Nov 21, 2012• 0 citations
OpenAlex

Abstract

Abstract. Nonequilibrium Green’s functions represent underutilized means of studying the time evolution of quantum manybody systems. In view of a rising computer power, an effort is underway to apply the Green’s functions formalism to the dynamics of central nuclear reactions. As the first step, mean-field evolution for the density matrix for colliding slabs is studied in one dimension. Strategy to extend the dynamics to correlations is described. Keywords: Green’s functions, Kadanoff-Baym equations, time-dependent Hartree-Fock, nuclear reactions PACS: 24.10.-i, 24.10.Cn

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Abstract. Nonequilibrium Green’s functions represent underutilized means of studying the time evolution of quantum manybody systems. In view of a rising computer power, an effort is underway to apply the Green’s functions formalism to the dynamics of central nuclear reactions. As the first step, mean-field evolution for the density matrix for colliding slabs is studied in one dimension. Strategy to extend the dynamics to correlations is described. Keywords: Green’s functions, Kadanoff-Baym equations, time-dependent Hartree-Fock, nuclear reactions PACS: 24.10.-i, 24.10.Cn
Computational difficulties aside, nonequilibrium Green's functions appear ideally suited for investigating the dynamics of central nuclear reactions. Many particles actively participate in those reactions. At the two energy extremes for the collisions, the limiting cases of the Green's function approach have been successful: the time-dependent Hartree-Fock theory at low energy and Boltzmann equation at high. The strategy for computational adaptation of the Green's function to central reactions is discussed. The strategy involves, in particular, incremental progression from one to three dimensions to develop and assess approximations, discarding of far-away function elements, use of effective interactions and preparation of initial states for the reactions through adiabatic switching. At this stage we concentrate on inclusion of correlations in one dimension, where relatively few approximations are needed, and we carry out reference calculations that can benchmark approximations needed for more dimensions. We switch on short-range interactions generating the correlations adiabatically in the Kadanoff-Baym equations to arrive at correlated ground states for uniform matter. As the energy of the correlated matter does not quite match the expectations for nuclear matter we add mean field to arrive at the match in energy. From there on, we move to finite systems. In switching on the correlations we observe emergence of extended tails in momentum distributions and evolution of single particle occupations away from 1 and 0.
Semiclassical time-dependent approaches are nowadays able to describe energetic central collisions of heavy isotopes with minimal assumptions. Simulations in the mean-field picture have provided insight on both nuclear structure and low-energy many-body reaction mechanisms. In this context, nonequilibrium Greens functions techniques have the potential to improve the description of the time evolution of nuclear systems by introducing effects beyond the mean-field. We describe a first attempt to use the Kadanoff-Baym dynamics for one-dimensional nuclear systems, with a particular emphasis on the process of correlation buildup.
The time‐dependent Green’s functions formalism provides a consistent description of the time evolution of quantum many‐body systems, either in the mean‐field approximation or in more sophisticated correlated approaches. We describe an attempt to apply this formalism to the mean‐field dynamics of symmetric reactions for one‐dimensional nuclear slabs. We pay particular attention to the off‐diagonal elements of the Green’s functions in real space representation. Their importance is quantified by means of an elimination scheme based on a super‐operator cut‐off field and their relevance for the global time evolution is assessed. The Wigner function and its structure in the mean‐field approximation is also discussed.
Nonequilibrium Green's functions represent a promising tool for describing central nuclear reactions. Even at the single-particle level, though, the Green's functions contain more information that computers may handle in the foreseeable future. In this study, we explore slab collisions in one dimension, first in the mean field approximation and demonstrate that only function elements close to the diagonal in arguments are relevant, in practice, for the reaction calculations. This bodes well for the application of the Green's functions to the reactions. Moreover we demonstrate that an initial state for a reaction calculation may be generated through adiabatic transformation of interactions. Finally, we report on our progress in incorporating correlations into the dynamic calculations.