The superdiffusion entropy production paradox in the space-fractional case for extended entropies

Prehl, Janett; Essex, Christopher; Hoffmann, Karl Heinz

doi:10.1016/j.physa.2009.09.009

Cited by 23 publications

(35 citation statements)

References 8 publications

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“…In this paper, we employ the statistical concept of entropy that goes back to Shannon and was introduced by him in the theory of communication and transmission of information (see [2]). The entropy of the processes governed by the time-and space-fractional diffusion equations has been discussed in [3][4][5][6], respectively. It is worth mentioning that according to [3,5] the entropy production rates for the time-and the space-fractional diffusion equations depend on the derivative order α of the time-or space-fractional derivative, respectively, and increase with increasing of α from 1 (diffusion) to 2 (wave propagation) that results in the so called entropy production paradox.…”

Section: Introductionmentioning

confidence: 99%

Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

Luchko

2016

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Abstract:In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time-and space-fractional diffusion equation with the quotient of the orders of the time-and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the α-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation.

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Section: Introductionmentioning

confidence: 99%

Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

Luchko

2016

Axioms

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“…First-order backwards finite difference formula is shown as follows: (19) whose MacLaurin series is given in Equation (20). (20) where T is the sampling period, v is the fractional order, and N is power series of expansion equation.…”

Section: Fractional-order Benchmark Modelmentioning

confidence: 99%

“…Some systems may have fractional-order dynamic characteristics, even if each unit has integer-order dynamic characteristics [14]. What's more, applying fractional calculus to entropy theory has become a hotspot research domain [15][16][17][18][19][20]; the fractional entropy could be used in the formulation of algorithms for image segmentation where traditional Shannon entropy has presented limitations [16]. In an analysis of the past ten years of trends and results in the fractional calculus application to dynamic problems of solid mechanics, the method of mechanical system dynamics analysis based on fractional calculus has gradually become one of main methods in the dynamics analysis of engineering [21].…”

Section: Introductionmentioning

confidence: 99%

Genetic Algorithm-Based Identification of Fractional-Order Systems

Zhou

Cao

Chen

2013

Entropy

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Abstract:Fractional calculus has become an increasingly popular tool for modeling the complex behaviors of physical systems from diverse domains. One of the key issues to apply fractional calculus to engineering problems is to achieve the parameter identification of fractional-order systems. A time-domain identification algorithm based on a genetic algorithm (GA) is proposed in this paper. The multi-variable parameter identification is converted into a parameter optimization by applying GA to the identification of fractional-order systems. To evaluate the identification accuracy and stability, the time-domain output error considering the condition variation is designed as the fitness function for parameter optimization. The identification process is established under various noise levels and excitation levels. The effects of external excitation and the noise level on the identification accuracy are analyzed in detail. The simulation results show that the proposed method could identify the parameters of both commensurate rate and non-commensurate rate fractional-order systems from the data with noise. It is also observed that excitation signal is an important factor influencing the identification accuracy of fractional-order systems.

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“…There is a wide array of families of PDFs induced by different processes, from Lévy distributions to PDFs based on generalizations of generalized hypergeometric functions. These arise in actual physical applications, such as anomalous diffusion processes [e.g., Hoffmann et al, 1998;Prehl et al, 2010]. Chaotic systems even have fractal attractors, and thus, PDFs can even be fractal functions.…”

Section: Locked Into the Laboratory Regimementioning

confidence: 99%

Does laboratory‐scale physics obstruct the development of a theory for climate?

Essex

2013

JGR Atmospheres

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[1] Despite best efforts, there still is no physical theory for climate based on the physics of the laboratory regime (i.e., fluid mechanics, radiative transfer, classical thermodynamics, etc.). This paper builds from previous discussions on how laboratory-regime assumptions may lock our current theoretical efforts into the laboratory regime and how we might get around this problem. Using ultralong time photographic exposures (known as solargraphs) for inspiration, it draws into question classical thinking about intensive thermodynamic variables for theoretical climate purposes while using the fact that physical flows remain well defined even for climate regimes. These flows are characterized here as "generalized wind." A simple example based on radiative energy and entropy transfer illustrates how these generalized wind fields can partially replace what is lost in moving away from laboratory-regime physics. These winds are shown to carry the dynamics in a modified form of radiation-like fluid dynamics that, together with radiation, might be possible to close in climate regimes.Citation: Essex, C. (2013), Does laboratory-scale physics obstruct the development of a theory for climate?,

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The superdiffusion entropy production paradox in the space-fractional case for extended entropies

Cited by 23 publications

References 8 publications

Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

Genetic Algorithm-Based Identification of Fractional-Order Systems

Does laboratory‐scale physics obstruct the development of a theory for climate?

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