All posts by admin

senior scientist, R&D gigaHedron

Generalization of the fractional Poisson distribution

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

Reflection symmetric Erdelyi-Kober type operators – a quasi-particle interpretation

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

A fractal approach to the dark silicon problem: a comparison of 3D computer architectures — standard slices versus fractal Menger sponge geometry

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

Fractional Calculus – An Introduction for Physicists – 2nd revised and extended Edition

From the cover: 
The book presents a concise introduction to the basic methods and strategies in fractional calculus and enables the reader to catch up with the state of the art in this field as well as to participate and contribute in the development of this exciting research area.
The contents are devoted to the application of fractional calculus to physical problems. The fractional concept is applied to subjects in classical mechanics, group theory, quantum mechanics, nuclear physics, hadron spectroscopy and quantum field theory and it will surprise the reader with new intriguing insights.
This new, extended edition now also covers additional chapters about image processing, folded potentials in cluster physics, infrared spectroscopy and local aspects of fractional calculus. A new feature is exercises with elaborated solutions, which significantly supports a deeper understanding of general aspects of the theory. As a result, this book should also be useful as a supporting medium for teachers and courses devoted to this subject.

Fractional Calculus – An Introduction for Physicists (2nd revised and extended Edition)
by Richard Herrmann,
World Scientific Publishing, Singapore, 
March 2014, 500 pp, 6 x 9 in.
ISBN: 978-9814551076
Order information at World Scientific Publ.:

additional material:  Front matter (Contents etc.) , Chapter 1 (Introduction) , Index
preview:
amazon look inside

buy online: amazon.com.au, amazon.bramazon.caamazon.cn, amazon.comamazon.co.uk, amazon.de, amazon.es, amazon.framazon.inamazon.itamazon.jp, amazon.mx, amazon.nl,  Barnes & NobleBAM!(USA), Blackwell’s(UK), booktopia(AU), fishpond(AU),  powell’s(USA), ….

reviews:

” …a popular book on fractional calculus, which has proven useful to many new researchers in the field. …A very welcome new feature in the  second edition is the inclusion of exercises at the end of every chapter, with  detailed solutions in the back of the book. This book is specifically aimed at physicists, although many of my colleagues outside physics have also found it useful. …The book takes a practical approach, which will be especially appealing to those accustomed to thinking about modeling in terms of differential equations and transforms.” 

M. M. Meerschaert, Statistics and Probability Dept. ,
Michigan State University,
.

 ” …I was pleasantly surprised not only by the amount of material the author masterly presents, but also by the timely inclusion of historical remarks that frame the discussions within aspects that the reader is familiar with…
Richard Herrmann’s
Fractional Calculus is a highly recommended book…

J. Rogel-Salazar, School of Physics, Astronomy and Mathematics,
University of Hertfordshire,
For full details on this review:  Contemporary Physics,  (2015) 56(2) 240
.

“… A valuable addition to the second edition is the exercises-solutions section. … The significant change in size between the two editions within a short period indicates the importance of fractional calculus. … The reviewer strongly recommends this book for beginners as well as specialists in the fields of physics, mathematics and complex adaptive systems.” 

E. Ahmed, Zentralblatt MATH
For full details on this review: Zentralblatt MATH (2014), Zbl 06293341

Towards a geometric interpretation of generalized fractional integrals – Erdelyi-Kober type integrals on R^N as an example

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

On the origin of space

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

Folded potentials in cluster physics – a comparison of Yukawa and Coulomb potentials with Riesz fractional integrals

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

Properties of a fractional derivative Schroedinger type wave equation and a new interpretation of the charmonium spectrum

 

author: R. Herrmann

abstract:
Based on the Caputo fractional derivative the classical, non relativistic Hamiltonian is quantized leading to a fractional Schroedinger type wave equation. The free particle solutions are localized in space. Solutions for the infinite well potential and the radial symmetric ground state solution are presented. It is shown, that the behaviour of these functions may be reproduced with a ordinary Schroeodinger equation with an additional potential, which is of the form V ~ x for $\alpha<1$, corresponding to the confinement potential, introduced phenomenologically to the standard models for non relativistic interpretation of quarkonium-spectra. The ordinary Schroedinger equation is triple factorized and yields a fractional wave equation with internal SU(3) symmetry. The twofold iterated version of this wave equation shows a direct analogy to the fractional Schroedinger equation derived. The angular momentum eigenvalues are calculated algebraically. The resulting mass formula is applied to the charmonium spectrum and reproduces the experimental masses with an accuracy better than 0.1%. Extending the standard charmonium spectrum, three additional particles are predicted and associated with $\Sigma_c^0(2455)$ and Y(4260) observed recently and one X(4965), not yet observed. The root mean square radius for $\Sigma_c^0(2455)$ is calculated to be ~0.3[fm]. The derived results indicate, that a fractional wave equation may be an appropriate tool for a description of quark-like particles.

download: arXiv:math-ph/0510099v4

Continuous differential operators and a new interpretation of the charmonium spectrum

author:
R. Herrmann
abstract:
The definition of the standard differential operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new continuous operators. The resulting Schroedinger type equation generates free particle solutions, which are confined in space. The angular momentum eigenvalues are calculated algebraically. It is shown, that the charmonium spectrum may be classified by the derived angular momentum eigenvalues for stepsize=2/3.
download:
arXiv:nucl-th/0508033v1

Collective spin from the linearization of the Schrödinger equation in multidimensional Riemannian spaces used in collective nuclear models

authors: R. Herrmann, G. Plunien, M. Greiner, W. Scheid

abstract:
The free Schrödinger equation in Riemannian collective spaces with an arbitrary number of dimensions is linearized. The coupling of external fields is discussed. The operators to classify collective spin states are introduced and the collective Pauli equation is derived.

reference:International Journal of Modern Physics A, Volume 4, Issue 18, pp. 4961-4975 (1989).