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Extension Through Time Smoothing of the Time Dependent Mean Field Theory

  • Balian, R.
  • Veneroni, M.
Publication Date
Oct 01, 1980
DOI: 10.1016/0003-4916(81)90156-1
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The mean field theory is considered from the viewpoint of the “contraction of the description” for complicated systems of fermions. Keeping only some small number of variables, considered as “simple” or collective, may or may not produce irreversibility. The single-particle density ϱ still contains non-collective aspects, presumably associated with high frequencies. The mechanism which generates these high frequencies from an initial slow motion is illustrated by considering the Landau-Zener-Stueckelberg model. Since it does not appear consistent to keep such rapid oscillations, time-smoothing is proposed as a possible procedure for erasing them. An evolution equation for the density ϱ is thus obtained by “naively” time-averaging the TDHF equation. The resulting evolution, however, is time-reversal invariant, and it may be unstable with respect to the initial conditions. To cure these defects, a different, iterative, type of time-smoothing is introduced, which prevents from the outset the development of high frequencies. This adds to the TDHF equation a term containing the RPA kernel and the smoothing parameter. Irreversibility occurs in the sense of an increase of entropy. Depending on the nature of the two-body interaction and on the initial conditions, the “mechanical energy” associated with the slow motion may either decrease with time, dissipating into fast motions, or increase. The occupation numbers evolve according to a balance equation similar to that of Pauli and they show a tendency towards equalization. The iterative time-smoothing, which involves a loss-of-memory mechanism, may also be viewed as an approximate way to account for the destructive coupling between the correlations and the rapid single-particle motions. Indeed it is recovered through the simulation of the residual interaction by a random noise.

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