Virial expansion coefficients in the harmonic approximation

Abstract

The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

“…Thus, when D > 2 we have: for p ∈ (0, p * ) the region of R + where the Laguerre polynomials exhibit the cosine asymptotics contributes with the dominant part in the integral (11). For p = p * the transition cosine-Bessel regime determines the asymptotics of N n,l (D, p * ), and for p > p * the Bessel regime plays the main role.…”

“…D ∈ [0, 2). The weigthed L p -norms of Laguerre polynomials N n,l (D, p), given by (11), have the following asymptotical (n → ∞) values:…”

“…In addition, there are two asymptotical regimes corresponding to the transition regions, cosine-Bessel and cosineAiry. If these regimes dominate in integral (11), then the asymptotics of N n (D, p) has a factor ln n besides the power law in n. It is also curious to mention that if these regimes dominate then gamma factors in constant C(β, p) in (15) for the oscillatory cosine regime explode. For the cosineBessel regime it happens for β + 1 − p/2 = 0, and for the cosine-Airy regime it happens for 1 − p/2 = 0.…”

“…Indeed they have been used in a great diversity of fields from fractional and quantum statistics [2,3] up to quantum many-body physics [4,5,6,7,8,9,10,11,12] and black-holes thermodynamics [13,14], and they have been applied to gain insight a e-mail: dehesa@ugr.es into numerous quantum phenomena and systems ranging from heat transport [15] and entanglement [16,17] to Keppler systems [18], quantum dots [19,6,20], neural networks [21], cold atomic gases [22,23] and systems with ontological states [24]. Let us also remark that the oscillator wavefunctions saturate the various mathematical realizations of the quantum uncertainty principle of Heisenberg and entropic types, which are based on the variance and its moment generalizations (Heisenberg-like uncertainty relations) [25,26] and the Shannon entropy [27,28], Rényi entropy [29,30] and the Fisher information [31,25] (entropic uncertainty relations), respectively.…”

Entropic measures of Rydberg‐like harmonic states

“…Let us just highlight that the oscillator wave functions saturate the most important mathematical realizations of the quantum uncertainty principle such as the Heisenberg‐like uncertainty relations, which are based on the variance and/or higher‐order moments, and the entropic uncertainty relations based on the Shannon entropy, the Rényi entropy, or the Fisher information . Furthermore, they have been used in numerous scientific fields ranging from quantum many‐body physics, heat transport, quantum entanglement, Keppler systems, quantum dots and cold atomic gases to fractional and quantum statistics, and black‐holes thermodynamics . However, the information‐theoretic properties of the three (or higher)‐dimensional harmonic oscillator are not yet settled down in spite of numerous efforts (see e.g., Refs.…”