# Spectral averaging and partition functions

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## Keywords

Partition Function Level Density Spectral Average Spectrum Function Eigenvalue Density
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## References

- 1.
*Theory and Applications of Moment Methods in Many-Fermion Spaces*, ed. by B. J. Dalton*et al.*(Plenum, N.Y. 1980).Google Scholar - 2.J. B. French: In
*Proceedings of Swmmer School on Nuclear Spectroscopy*, ed. by G. F. Bertsch and D. Kurath (Springer, Berlin, 1980).Google Scholar - 3.J. B. French and V. K. B. Kota, Ann. Rev. Nucl. Part. Sci. 32, 35, (1982).Google Scholar
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**A396**, 87c (1983).Google Scholar - 5.There are probably hundreds of papers about level density; see Reference 1 for good reviews. All but a few of the theoretical papers ignore the residual interactions, most of these doing so without apology or explanation. This was essen tial and appropriate in the earliest papers (see (14) for the two fundamental ones) but by no means so far most of the others.Google Scholar
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^{-l}I(E) where I(E) dE = no of states in (E,E+dE); then ∝ I(E) dE = d, ∝ ρ(E) dE = 1. These densities are uniquely defined by their moments, and similarly for other operators. Not necessarily so in infinite spaces, nor would ρ(E) exist.Google Scholar - 9.K. F. Ratcliff, Phys. Rev. C3, 117 (1971).Google Scholar
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- 16.F. S. Chang, J. B. French, and T. H. Thio, Ann. Phys. (N.Y.) 66, 137 (1971). The irreps are described in terms of the column structure, rather than the row structure (which is more usual), of the corresponding Young shapes. Thus (N-
*v,v*] has one column with N-*v*blocks and one with*v*. The more common labeling would be [l^{N-2v},2^{v}].Google Scholar - 17.For the accurate location of the ground state (a problem which does not arise in the NIP case) it is good to do better, by including one or two correction terms in (1) for those configurations which contribute significantly in the groundstate domain.Google Scholar
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- 24.We take advantage of the space available on this last page to indicate how the things discussed above relate to other aspects of statistical nuclear physics. It could reasonably be said that the whole subject is the natural development of ideas introduced with Bohr's compound nucleus, Bethe's level density and Wigner's ensemble treatment of spectral fluctuations, but taking account,
*inter alia*, of things learned in the meantime about: continuum states (including those generated in heavy-ion reactions); departures from statistical equilibrium in compound states; nuclear symmetries; the detailed properties of the free-space nucleon-nucleon interaction, and the effective interaction which follows from it and is used in shell-model analysis; many advances in statistical mechanics. Some of these things refer specifically to the energy region of high excitation, others mainly to the ground-state domain. The ideal theory which would encompass all these and other phenomena, at high energies and low, does not yet exist. Nobody however believes that the microscopic details are individually of consequence at high excitations; either one should measure a lot of them, and subject the set to statistical analysis (as one does with energy-level fluctuations), or one should measure an average over them, as one does, often indirectly, by the statistical theory of reactions. This being agreed to, the high-energy region is left to those interested in the development and application of methods of statistical mechanics; two examples are the combinatorial “exciton” theory of Griffin, used for the treatment of non-equilibrium processes,and the transport theories of Weidenmüller, Agassi, and others for heavy-ion reactions. It seems altogether likely that the procedures described above could be usefully applied to many high-excitation processes, just as it has been to the level density; but eventually one would need its extension to a proper treatment of continuum states. For low excitations the evidence is good to excellent that expectation values (31, 32), and even strength fluctuations (36), extrapolate to the ground state. It is not known whether the same extrapolation extends to the spectrum itself, i.e. whether the fluctuation-free spectrum which derives9 from the smoothed spectrum function would properly capture all the significant information even if the exact symmetries were correctly dealt with. Looked at otherwise the question is whether the ground states (for exact symmetries) are in some way “special”, resisting statistical treatment, as the reviewer at the 1975 Tucson Conference has declared; the question seems open. Probably the way to study this problem is to produce a proper method for statistically extending exact matrix results to larger spaces. Ensemble averaging for these parts of H which do not generate strong collectivities would seem to be of value here.Google Scholar

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© Springer-Verlag 1984