Quantum Szilard engine for the fractional power-law potentials

AbstractIn this study, we consider the quantum Szilárd engine with a single particle under the fractional power-law potential. We suggest that such kind of the Szilárd engine works a Stirling-like cycle. We obtain energy eigenvalues and canonical partition functions for the degenerate and non-degenerate cases in this cycle process. By using these quantities we numerically compute work and efficiency for this thermodynamic cycle for various power-law potentials with integer and non-integer exponents. We show that the presented simple engine also yields positive work and efficiency. We discuss the importance of fractional dynamics in physics and finally, we conclude that fractional calculus should be included in the fields of quantum information and thermodynamics.

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Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach

Probabilistic Engineering Mechanics ◽

10.1016/j.probengmech.2010.06.010 ◽

2011 ◽

Vol 26 (1) ◽

pp. 101-108 ◽

Cited By ~ 10

Author(s):

Giulio Cottone ◽

Mario Di Paola ◽

Salvatore Butera

Keyword(s):

Nonlinear Systems ◽

Fractional Calculus ◽

Power Law ◽

Nonlinear Term ◽

Stochastic Dynamics ◽

Fractional Power

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And I say to myself: “What a fractional world!”

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-011-0037-1 ◽

2011 ◽

Vol 14 (4) ◽

Cited By ~ 22

Author(s):

J. Tenreiro Machado

Keyword(s):

Global Warming ◽

Fourier Transform ◽

Fractional Calculus ◽

Complex Systems ◽

Power Law ◽

Deoxyribonucleic Acid ◽

Warming Trend ◽

Fractional Dynamics ◽

The Fourier Transform ◽

Musical Rhythms

AbstractThis paper discusses several complex systems in the perspective of fractional dynamics. For prototype systems are considered the cases of deoxyribonucleic acid decoding, financial evolution, earthquakes events, global warming trend, and musical rhythms. The application of the Fourier transform and of the power law trendlines leads to an assertive representation of the dynamics and to a simple comparison of their characteristics. Moreover, the gallery of different systems, both natural and man made, demonstrates the richness of phenomena that can be described and studied with the tools of fractional calculus.

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REVIEW OF SOME PROMISING FRACTIONAL PHYSICAL MODELS

International Journal of Modern Physics B ◽

10.1142/s0217979213300053 ◽

2013 ◽

Vol 27 (09) ◽

pp. 1330005 ◽

Cited By ~ 75

Author(s):

VASILY E. TARASOV

Keyword(s):

Fractional Calculus ◽

Power Law ◽

Particle Interaction ◽

Discrete Systems ◽

Physical Models ◽

Long Term Memory ◽

Fractional Dynamics ◽

Fractal Media ◽

Systems With Memory ◽

With Memory

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of non-integer orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for future development of fractional dynamics. We suggest a short introduction to fractional calculus as a theory of integration and differentiation of noninteger order. Some applications of integro-differentiations of fractional orders in physics are discussed. Models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented. Quantum analogs of fractional derivatives and model of open nano-system systems with memory are also discussed.

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Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽

10.1515/phys-2020-0158 ◽

2020 ◽

Vol 18 (1) ◽

pp. 594-612 ◽

Cited By ~ 3

Author(s):

Abdon Atangana ◽

Emile Franc Doungmo Goufo

Keyword(s):

Fractional Calculus ◽

Power Law ◽

Initial Value Problems ◽

Differential Operators ◽

Integral Operators ◽

Initial Value ◽

Complex Processes ◽

Singular Kernels ◽

Fractional Differential ◽

Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

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General Fractional Dynamics

Mathematics ◽

10.3390/math9131464 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1464

Author(s):

Vasily E. Tarasov

Keyword(s):

Fractional Calculus ◽

Exact Solutions ◽

Arbitrary Order ◽

Nonlinear Dynamical Systems ◽

Fractional Integrals ◽

Fractional Dynamics ◽

Fractional Systems ◽

Nonlinear Dynamical ◽

Fractional Differential ◽

Solutions Of Equations

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

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Physical insight into fractional power dependence of saturation current on gate voltage in advanced short channel MOSFETs (alpha-power law model)

Proceedings of the International Symposium on Low Power Electronics and Design ◽

10.1109/lpe.2002.1029504 ◽

2003 ◽

Cited By ~ 1

Author(s):

Hyunsik Im ◽

M. Song ◽

T. Hiramoto ◽

T. Sakurai

Keyword(s):

Power Law ◽

Gate Voltage ◽

Fractional Power ◽

Alpha Power ◽

Power Law Model ◽

Power Dependence ◽

Saturation Current ◽

Short Channel ◽

Physical Insight ◽

Insight Into

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Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation

Mathematical Problems in Engineering ◽

10.1155/2016/7845874 ◽

2016 ◽

Vol 2016 ◽

pp. 1-15 ◽

Cited By ~ 5

Author(s):

J. F. Gómez Aguilar ◽

T. Córdova-Fraga ◽

J. Tórres-Jiménez ◽

R. F. Escobar-Jiménez ◽

V. H. Olivares-Peregrino ◽

...

Keyword(s):

Fractional Calculus ◽

Disordered Systems ◽

Fourier Method ◽

Mean Square Displacement ◽

Fractional Power ◽

Transport Processes ◽

Diffusion Equations ◽

Alternative Representation ◽

Dimensional Parameters ◽

Full Solution

The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

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AC Loss Modeling in Ba0.5Sr0.5TiO3 Using Dielectric Relaxation

MRS Proceedings ◽

10.1557/proc-833-g1.10 ◽

2004 ◽

Vol 833 ◽

Cited By ~ 1

Author(s):

Nadia K. Pervez ◽

Jiwei Lu ◽

Susanne Stemmer ◽

Robert A. York

Keyword(s):

Dielectric Loss ◽

Dielectric Relaxation ◽

Power Law ◽

Fractional Power ◽

Ac Loss ◽

Dielectric Susceptibility ◽

Network Analyzer ◽

Q Factor ◽

Relaxation Model ◽

Q Factors

ABSTRACTIn universal relaxation, a material's complex dielectric susceptibility follows a fractional power law f1-n where 0 < n < 1 over multiple decades of frequency. In a variety of materials, including Ba0.5Sr0.5Ti03, dielectric relaxation has been observed to follow this universal relaxation model with values of n close to 1. In this work we have shown that the universal relaxation model can be used to calculate dielectric loss even when n is very close to 1. Our calculated Q-factors agree with measured values at 1 MHz; this agreement suggests that this technique may be used for higher frequencies where network analyzer measurements and electrode parasitics complicate Q-factor determination.

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On Fractional Hamilton Formulation Within Caputo Derivatives

Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2007-34812 ◽

2007 ◽

Author(s):

Dumitru Baleanu ◽

Sami I. Muslih ◽

Eqab M. Rabei

Keyword(s):

Fractional Calculus ◽

Fractional Derivatives ◽

Variational Principles ◽

Hamiltonian Dynamics ◽

Fractional Integration ◽

Fractional Dynamics ◽

Classical Dynamics ◽

Lagrange Equations ◽

Caputo Fractional Derivatives ◽

Non Locality

The fractional Lagrangian and Hamiltonian dynamics is an important issue in fractional calculus area. The classical dynamics can be reformulated in terms of fractional derivatives. The fractional variational principles produce fractional Euler-Lagrange equations and fractional Hamiltonian equations. The fractional dynamics strongly depends of the fractional integration by parts as well as the non-locality of the fractional derivatives. In this paper we present the fractional Hamilton formulation based on Caputo fractional derivatives. One example is treated in details to show the characteristics of the fractional dynamics.

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