SKM 2023 – wissenschaftliches Programm

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DY: Fachverband Dynamik und Statistische Physik

DY 6: Statistical Physics: General I

DY 6.8: Vortrag

Montag, 27. März 2023, 12:00–12:15, ZEU 160

A combinatorial approach to the many-body density of levels and the Bethe approximation — •Carolyn Echter, Georg Maier, Juan-Diego Urbina, and Klaus Richter — Institut für Theoretische Physik, Universität Regensburg, Regensburg, Germany

The Bethe formula, originally derived in [1] to estimate the density of levels of heavy nuclei, has become a widely used approximation for the many-body density of levels of a non-interacting fermionic system, appropriate for large numbers of particles and energies in a system-dependent range. Notably, in the case of equally spaced single-particle energy levels, it coincides with the asymptotic result for the number of unrestricted partitions of an integer known from analytic number theory [2]. An explanation is suggested by the combinatorics of distributing integer amounts of energy to particles obeying given statistics. We present a combinatorial derivation of the exact many-body density of levels for various particle statistics in the case of a constant single-particle density of states, thereby adding to existing discussions [3,4] and explaining the asymptotic agreement of Bethe's approximation with number theoretical partition functions. We compare numerically with semiclassical results and make suggestions towards a bosonic analogue of the Bethe formula based on our observations.

[1] H. A. Bethe, Phys. Rev. 50, 332-41 (1936). [2] G. H. Hardy, S. Ramanujan, Proc. London Math. Soc. (2) 17, 75-115 (1918). [3] F. C. Auluck, D. S. Kothari, Math. Proc. Camb. Philos. Soc. 42, 272-77 (1946). [4] A. Comtet, P. Leboeuf, S. N. Majumdar, Phy. Rev. Lett. 98, 070404 (2007).