Mar
26
2015

**author: **R. Herrmann

**abstract: **

A generalization of the Poisson distribution based on the generalized Mittag-Leffler function Eα,β(λ) is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter β is suggested.

**reference: **Fract. Calc. Appl. Anal. (2016) **19**(4) 832-842

Jul
17
2014

**author: **R. Herrmann

**abstract: **

Reflection symmetric Erdelyi-Kober type fractional integral operators are used to construct fractional quasi-particle generators. The eigenfunctions and eigenvalues of these operators are given analytically.

A set of fractional creation- and annihilation-operators is defined and the properties of the corresponding free Hamiltonian are investigated. Analogue to the classical approach for interacting multi-particle systems the results are interpreted as a fractional quantum model for a description of residual interactions of pairing type.

**reference: **Fract. Calc. Appl. Anal. (2014) **17**(4) 1215-1228

Apr
07
2014

**author: **R. Herrmann

**abstract: **

The dark silicon problem, which limits the power-growth of future computer generations, is interpreted as a heat energy transport problem when increasing the energy emitting surface area within a given volume. A comparison of two 3D-configuration models, namely a standard slicing and a fractal surface generation within the Menger sponge geometry is presented. It is shown, that for iteration orders n>3 the fractal model shows increasingly better thermal behavior. As a consequence cooling problems may be minimized by using a fractal architecture.

Therefore the Menger sponge geometry is a good example for fractal architectures applicable not only in computer science, but also e.g. in chemistry when building chemical reactors, optimizing catalytic processes or in sensor construction technology building highly effective sensors for toxic gases or water analysis.

**download:** arXiv: arXiv:1404.1891[cs.ET]

**reference: **Chaos Solitons Fractals (2015) **70**, 38-41

Jan
22
2014

**author: **R. Herrmann

**abstract: **

A family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on R^N . Based on this geometric view, several extensions are discussed.

**download:** arXiv: arXiv:1401.6051

**reference: **Fract. Calc. Appl. Anal. (2014) **17**(2) 361-370

Jul
24
2013

**author: **R. Herrmann

**abstract: **

Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a model is presented, which describes space generation as a dynamic process, where the dimension $d$ of space evolves smoothly with time in the range 0 <= d(t) <=3, where the lower and upper boundaries of dimension are derived from first principles. It is demonstrated, that a minimum threshold for the space dimension is necessary to establish an interaction with external probe particles. A possible application in cosmology is suggested.

**download:** arXiv: arXiv:1308.4587

**reference**: Cent. Eur. J. Phys. (2013) **11**(10) 1212-1220

May
03
2013

**author:**

R. Herrmann

**abstract:**

In cluster physics a single particle potential to determine the microscopic part of the total energy of a collective configuration is necessary to calculate the shell- and pairing effects. In this paper we investigate the properties of the Riesz fractional integrals and compare their properties with the standard Coulomb and Yukawa potentials commonly used. It is demonstrated, that Riesz potentials may serve as a promising extension of standard potentials and may be reckoned as a smooth transition from Coulomb to Yukawa like potentials, depending of the fractional parameter $\alpha$. For the macroscopic part of the total energy the Riesz potentials treat the Coulomb-, symmetry- and pairing-contributions from a generalized point of view, since they turn out to be similar realizations of the same fractional integral at distinct $\alpha$ values.

**download:** arXiv: arXiv:1305.0890

**reference**: J. Phys. A.: Math. Theor. (2013) **46**(40) 405203

Apr
03
2013

**author:**

R. Herrmann

**abstract:**

For the Riesz fractional derivative besides the well known integral representation two new differential representations are presented, which emphasize the local aspects of a fractional derivative. The consequences for a valid solution of the fractional Schroedinger equation are discussed.

**download:** arXiv: arXiv:1303.2939

Mar
17
2013

**author:**

R. Herrmann

**abstract:**

Based on the Riemann- and Caputo definition of the fractional derivative we tabulate the lowest n=31 energy levels and generated graphs of the occupation probability of the fractional quantum mechanical harmonic oscillator with a precision of 32 digits for 0.50 < \alpha < 2.00, which corresponds to the transition from U(1) to SO(3).

**reference: **Gam. Ori. Chron. Phys. (2013) **1**(1) 13-176

Mar
15
2013

**author:**

R. Herrmann

**abstract:**

Based on the Riesz definition of the fractional derivative the fractional Schroedinger equation with an infinite well potential is investigated. First it is shown analytically, that the solutions of the free fractional Schroedinger equation are not eigenfunctions, but good approximations for large k and for $\alpha \approx 2$. The first lowest eigenfunctions are then calculated numerically and an approximate analytic formula for the level spectrum is derived.

**download: **arxiv: arXiv:1210.4410[math-ph]

**reference:** Gam. Ori. Chron. Phys. (2013) **1**(1) 1-12

Nov
14
2012

**author:**

R. Herrmann

**abstract:**

For the Riemannian space, built from the collective coordinates used within nuclear models, an additional interaction with the metric is investigated, using the collective equivalent to Einstein’s curvature scalar. The coupling strength is determined using a fit with the AME2003 ground state masses. An extended finite-range droplet model including curvature is introduced, which generates significant improvements for light nuclei and nuclei in the trans-fermium region.

**download:** arxiv: arXiv:0801.0298 [nucl-th] [physics.gen-ph]

**reference: **International Journal of Modern Physics E (2012) **21** 1250103