Category Archives: Fractional Calculus

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

Fractional Calculus – An Introduction to Physicists – 1st Edition


OUT OF PRINT

From the cover:
Fractional calculus is undergoing rapid and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics.

This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory.

Fractional Calculus – An Introduction for Physicists
by Richard Herrmann,
World Scientific Publishing, Singapore, February 2011, reprinted 2012, 276 pp, 6 x 9 in.,

ISBN: 978-9814340243

reviews:

“The book is a solid introduction to fractional calculus that contains, in particular an elucidating section on geometric interpretation of fractional operators… the bulk of the book concentrates on aspects of fractional calculus related to symmetries in quantum mechanics…what is covered is presented in an authoritative, solid style and actually provides very entertaining reading…Overall, Fractional Calculus is an affordable and valuable introduction to the field that will appeal to physicists interested in scientific what-ifs…”

Ralf Metzler, Physics Today
For full details on this review, please visit: Physics Today 65(2), (2012) 55–56;
doi: 10.1063/PT.3.1443
.

“The book has the property that derived results are directly compared with experimental findings. As a consequence, the reader is guided and encouraged to apply the fractional calculus approach in her/his research area. 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, please visit: Zentralblatt MATH (2012), Zbl 1232.26006
.

“…the first three chapters actually appear very helpful at the graduate level. Each chapter has a careful precis at the start. There are many analyses illustrating outcomes of fractional analyses…If this [fractional calculus] is the field of your research then this book is essential with numerous references…”

 

J. E. Caroll, Contemporary Physics
For full details on this review, please visit: Contemporary Physics 53(2), (2012), 187–188; doi:10.1080/00107514.2011.648957

Numerical solution of the fractional quantum mechanical harmonic oscillator based on the Riemann and Caputo derivative

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

The fractional Schroedinger equation and the infinite potential well – numerical results using the Riesz derivative

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

Curvature interaction in collective space

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

Infrared spectroscopy of diatomic molecules – a fractional calculus approach

author:
R. Herrmann

abstract:
The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically  solving the fractional Schr\”odinger equation based on the Riemann- and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

download: arxiv: arXiv:1209.1630 [physics.gen-ph]
reference: International Journal of Modern Physics B (2013) 27(6) 1350019

Covariant fractional extension of the modified Laplace-operator used in 3D-shape recovery

author:
R. Herrmann

abstract:
Extending the Liouville-Caputo definition of a fractional derivative to a nonlocal covariant generalization of arbitrary bound operators acting on multidimensional Riemannian spaces an appropriate approach for the 3D-shape recovery of aperture afflicted 2D slide sequences is proposed. We demonstrate, that the step from a local to a nonlocal algorithm yields an order of magnitude in accuracy and the use of the specific fractional approach an additional factor 2 in accuracy of the derived results.

download: arxiv: 1111.1311v1 [cs.CV]
reference: Fract. Calc. Appl. Anal. (2012) 15(2) 332-343

Common aspects of q-deformed Lie algebras and fractional calculus

author:
R. Herrmann

abstract:
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. A new class of fractional q-deformed Lie algebras is proposed, which for the first time allows a smooth transition between different Lie algebras. For the fractional harmonic oscillator, the corresponding fractional q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool to describe e.g. the ground state spectra of even-even nuclei. In addition, the equivalence of rotational and vibrational spectra for fractional q-deformed Lie algebras is shown and the $B_\alpha(E2)$ values for the fractional q-deformed symmetric rotor are calculated. A first interpretation of half integer representations of the fractional rotation group is given in terms of a description of $K=1/2^-$ band spectra of odd-even nuclei.

download: arxiv: arXiv:1007.1084v1 [physics.gen-ph]
reference: Physica A (2010) 389 4613-4622

Fractional quantum numbers deduced from experimental ground state meson spectra

author: R. Herrmann

abstract:
Based on the Caputo definition of the fractional derivative the ground state spectra of mesons are classified as multiplets of the  fractional rotation group. The comparison with the experimental values leads to the conclusion, that quarks carry an up to now unrevealed fractional multiplicative quantum number, which we call fractional hyper charge.

download: arxiv: arXiv:1003.5246v1 [physics.gen-ph]