A unified approach to resolving the entropy production paradox

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

doi:10.1515/jnetdy-2012-0008

Cited by 8 publications

(18 citation statements)

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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%

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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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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time Evolution of Relative Entropies for Anomalous Diffusion

Prehl

Boldt

Essex

et al. 2013

Entropy

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“…Both one-parameter families of probability densities produce well-defined entropy and entropy production rates within the domain [5,[15][16][17][18][19][20][21]. These PDFs share scaling properties inherited from their differential equations.…”

Section: Introductionmentioning

confidence: 97%

“…for 1 ≤ γ ≤ 2 and P being a probability density function (PDF), had implications beyond anomalous diffusion [5,[16][17][18]: Diffusion and wave propagation are often conceived as representatives of different and distinct worlds, both mathematically and physically. But the regime that (1) represents is not merely anomalous in terms of diffusion.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric Fractional Diffusion and Entropy Production

Prehl

Boldt

Hoffmann

et al. 2016

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Abstract:The discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.

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