Fractional Diffusion and Entropy Production

Hoffmann, Karl Heinz; Essex, Christopher; Schulzky, Christian

doi:10.1515/jnet.1998.23.2.166

Cited by 47 publications

(57 citation statements)

References 13 publications

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“…(15) further below]. The time scaling transform in (4) was also proposed by Hoffmann 18 and Li 3 , referred to as "internal clock", to solve counterintuitive paradox on the entropy production of anomalous diffusion process.…”

Section: Fractal Time-space Transformsmentioning

confidence: 99%

Time–space fabric underlying anomalous diffusion

Chen

2006

Chaos, Solitons & Fractals

363

158

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This study unveils the time-space transforms underlying anomalous diffusion process. PACS numbers: 61.43. Hv, 47.27.Qb, 65.40.Fb, 05.40.Ca, 43.20.Bi Anomalous diffusion is one of the most important concepts in modern physics [1][2][3][4] and is present in extremely diverse engineering fields such as charges transport in amorphous semiconductor, 5 vibration and acoustic dissipation in soft matter, 6 magnetic plasma, 7 polymer dynamics, 8 turbulence 9 and quantum processes 10 among many other problems. 4 However, it is noted that the anomaly is mostly introduced in a descriptive level of statistical representation or phenomenological modeling. The purpose of this

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Section: Fractal Time-space Transformsmentioning

confidence: 99%

Time–space fabric underlying anomalous diffusion

Chen

2006

Chaos, Solitons & Fractals

363

158

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“…There is no reason to expect that pairs of densities would relax to each other when subject to different processes. However, it brings back the issue of different internal quicknesses [40][41][42], although now articulated from the standpoint of relative entropies. Moreover, this paper focuses on the ordering property, which ultimately is about the differences between the densities.…”

Section: Discussionmentioning

confidence: 99%

“…For that the parameter α must vary between one (the (half) wave case) and the (the diffusion case). This bridging regime has been analyzed under different perspectives [40][41][42] and has shown unexpected features. The space-fractional diffusion equation represents a family of processes in the bridge regime that can be ordered by the parameter α, which will be called the bridge ordering.…”

Section: Introductionmentioning

confidence: 99%

Time Evolution of Relative Entropies for Anomalous Diffusion

Prehl

Boldt

Essex

et al. 2013

Entropy

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The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is "nearer" to pure diffusion than another, placing pure wave propagation, desirably, "furthest" from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.

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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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Fractional Diffusion and Entropy Production

Cited by 47 publications

References 13 publications

Time–space fabric underlying anomalous diffusion

Time–space fabric underlying anomalous diffusion

Time Evolution of Relative Entropies for Anomalous Diffusion

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

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