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Table Of Integrals Series And Products

The Gibbs–Jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form S and of physically appropriate constraints. For instance, for the usual canonical ensemble, one adopts (i) S 1 =−k∑ i p i ln p i , (ii) ∑ipi=1, and (iii) ∑ i p i e i =U 1 ({ei}≡ eigenvalues of the Hamiltonian; U1≡ internal energy). Equilibrium consists in optimizing S1 with regard to {pi} in the presence of constraints (ii) and (iii). Within the recently introduced nonextensive statistics, (i) is generalized into Sq=k[1−∑ipiq]/[q−1] (q→1 reproduces S1), (ii) is maintained, and (iii) is generalized in a manner which might involve piq. In the present effort, we analyze the consequences of some special choices for (iii), and their formal and practical implications for the various physical systems that have been studied in the literature. To illustrate some mathematically relevant points, we calculate the specific heat respectively associated with a nondegenerate two-level system as well as with the classical and quantum harmonic oscillators.

The role of constraints within generalized nonextensive statistics

Energy levels of A = 21−44 nuclei (V)

The experimental and theoretical evidence for intrinsic reflection-asymmetric shapes in nuclei is reviewed. The theoretical methods discussed cover a wide spectrum, from mean-field theory and its extensions to algebraic and cluster approaches. The experimental data for nuclear ground states and at low and high spin, cited as evidence for reflection asymmetry, are collected and categorized. The extensive data on electric dipole transition moments and their theoretical interpretation are surveyed, along with available data on electric octupole moments. The evidence for reflection-asymmetric molecular states in light nuclei is summarized. The application of reflection-asymmetric theories to descriptions of the fission barrier, bimodal fission, superdeformation, and hyperdeformations is reviewed, and some other perspectives in the wider context of nuclear physics are also given. [S0034-6861(96)00102-X]

Intrinsic reflection asymmetry in atomic nuclei

A non-linear, generalized Fokker-Planck (GFP) equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by Tsallis in his generalization of Statistical Mechanics. In the case of a constant or linearly varying drift, the time dependent solutions of the GFP equation are seen to obey a suitable form of the celebrated H-theorem. The temporal changes of Tsallis' entropies are seen to be given in terms of the Fisher's information measure. Particular time-dependent solutions of the GFP equation of the maximum (Tsallis') entropy form are obtained.

Non-extensive statistical mechanics and generalized Fokker-Planck equation

Stellar polytropes and Tsallis' entropy

Lutsko and Boon have shown via elegant theoretical reasoning (EPL, 95 (2011) 20006), that Tsallis' thermostatistics is affected by divergence problems. We explicitly verify such fact in trying to compute the nonextensive q-partition function for the harmonic oscillator in more than two dimensions. One can see that it indeed diverges. The appeal to the so-called q-Laplace transform, where the q-exponential function plays the role of the ordinary exponential, is seen to overcome the serious problem envisaged by Lutsko and Boon.

https://iopscience.iop.org/article/10.1209/0295-5075/104/60003/pdf

Possible divergences in Tsallis' thermostatistics

Pairing vibrational states and the generator coordinate method

The Surface Delta Interaction (SDI) is employed to calculate the ernergy of gamma-vibrational and theK=0, 1, 2, and 3 Octupole vibrational band heads in the even-mass Transuranic nuclei. The quasi-particle Random Phase (RPA) and the quasi-particle Tamm Dancov approximations (TDA) are utilized to solve the Hamiltonian. The resulting energies and gamma transitions are compared with the Pairing plus Quadrupole (PQF) or Octupole (POF) force model and the available experimental data. The agreement between theory and experiment is satisfatory. Only theK=0, 1, and 2 Octupole bands are collective. TheK=3 negative parity state is practicaly a pure two quasi-particle state.

The surface delta interaction in the transuranic nuclei

The Lipkin model is extended to yield a non-trivial Hartree-Fock (HF) solution for any coupling constant of the two-body force In this model the multi-configuration Hartree-Fock (MCHF) approach is compared with the exact solution and with the HF solution The results show a strong superiority of the MCHF method over the HF approximation

A. Plastino

Papers

Stellar polytropes and Tsallis' entropy

Possible divergences in Tsallis' thermostatistics

Pairing vibrational states and the generator coordinate method

The surface delta interaction in the transuranic nuclei

Application of the multi-configuration Hartree-Fock theory to a simple model