Abstract
We combine the framework of the inverse problem theory and the Monte Carlo approach to formulate exact mathematical models that enable estimates of the uncertainty distributions for modeling predictions. We illustrate and discuss in particular what we refer to as the ``NO GO property.'' When the uncertainties of data constituting the input to the parameter adjustment procedures exceed certain critical value(s), even an exact modeling looses its stochastic reliability; its further use may provide ``acceptably looking'' rms deviations in the fitting zone with very likely meaningless, because they are unstable, predictions outside of it. We examine confidence intervals for intraneous (inside of the adjustment zone) and extraneous (outside of the adjustment zone) predictions and we demonstrate that ``satisfactory'' rms deviations in the intraneous modeling regime offer generally null certitude about the quality of extraneous predictions. Even though not entirely unknown, this property requires strong emphasizing since ignoring it has lead to misleading conclusions and confusing messages in the literature. We generalize our considerations to the realistic nuclear mean-field simulations of the properties of the nucleonic mean-field energies in spherical nuclei. We predict quantitatively the deterioration with increasing mass of the nucleonic-energy confidence intervals in superheavy nuclei. We show a strong dependence of those confidence intervals on the quantum characteristic of nucleonic states and provide detailed illustrations. In particular we demonstrate that, in the realistic predictions for the superheavy nuclei with the phenomenological Woods-Saxon Hamiltonian for up to $Z\ensuremath{\approx}114$ or so, one obtains relatively stable predictions of the single-particle spectra with $N\ensuremath{\le}180$, while approaching the NO GO zone of this model for further increasing neutron numbers. Thus the main area of today's interest within the instrumental reach for the superheavy nuclei studies remains, according to these estimates for the Woods-Saxon modeling, within the stability zone.
Highlights
Model prediction capacities and estimates of modeling uncertainties—the subjects of increasing importance in particular in contemporary nuclear physics—are of strong interest in many subfields of physics and technological applications as well as in a quickly developing subfield of applied mathematics: the inverse problem theory
We have studied the application of the Monte Carlo technique to produce realistic estimates of the confidence intervals of the nucleonic mean-field energies and, more generally, the uncertainty probability distributions
Illustrative examples show that from the fit quality in the intraneous regime strictly nothing can be said with certainty about the predictions for the extraneous modus operandi, for which case dedicated tests must be programmed
Summary
Model prediction capacities and estimates of modeling uncertainties—the subjects of increasing importance in particular in contemporary nuclear physics—are of strong interest in many subfields of physics and technological applications as well as in a quickly developing subfield of applied mathematics: the inverse problem theory. In contemporary nuclear structure calculations, in particular, for exotic nuclei, extensive use is made of modeling of the total nuclear energies, potential barriers, quasiparticle excitations, rotational-band properties, etc These quantities are frequently used in comparisons of the mean-field theory predictions with experiment. In the case of nuclear physics applications, we refer to “intraneous predictions” when performing the calculations for unknown nuclei or observables inside of the known areas (as if interpolating) and extraneous otherwise (as if extrapolating) The latter may refer to exotic, e.g., superheavy nuclei, usually (far) away from the known nuclear areas on the (Z, N ) plane, or to the deeper and deeper bound nucleonic levels, further and further away from the known ones used for the fitting of the Hamiltonian parameters
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