Generalized Maxwell Relations in Thermodynamics with Metric Derivatives

Weberszpil, José; Chen, Wen

doi:10.3390/e19080407

Cited by 26 publications

(14 citation statements)

References 38 publications

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“…A large selection of different entropic forms was proposed, which luckily have some mutual connections, so that it is often more a matter of taste than correctness to apply one or another. The Tsallis entropic form works with a new formalism that generalizes the significant classical statistical based terms by a q -factor and, thus, leaves well-established formulas of equilibrium thermodynamics preserved and in a generalization to non-equilibrium thermodynamics practically unaltered, if the basic functions are q -generalized (e.g., by the q -exponential function, the q -logarithmic function or the q -deformed fractional derivative [ 54 ]). We introduced the weakly q -deformed Escort-Gaussian probability distribution, which is the right choice for a Ritz-Galerkin truncated system of the NSE that develops to a high-degree dynamical coupled system of 2 N oscillators or N eddies, respectively.…”

Section: Discussion Of Results Conclusion and Outlookmentioning

confidence: 99%

“…Only a few examples can be given here. For example, Weberszpil and Chen [ 54 ] apply fractional q -deformed derivatives to the second law of thermodynamics and generalize, for example, the Maxwell relations to be adapted to non-equilibrium thermodynamics. Authors, as Hamza, Krim and Mohamed, work on medical image registration by maximizing a Tsallis entropy-based divergence (see [ 55 , 56 ]).…”

Section: Introductionmentioning

confidence: 99%

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Tsallis Extended Thermodynamics Applied to 2-d Turbulence: Lévy Statistics and q-Fractional Generalized Kraichnanian Energy and Enstrophy Spectra

Egolf

Hutter

2018

Entropy

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Abstract:The extended thermodynamics of Tsallis is reviewed in detail and applied to turbulence. It is based on a generalization of the exponential and logarithmic functions with a parameter q. By applying this nonequilibrium thermodynamics, the Boltzmann-Gibbs thermodynamic approach of Kraichnan to 2-d turbulence is generalized. This physical modeling implies fractional calculus methods, obeying anomalous diffusion, described by Lévy statistics with q < 5/3 (sub diffusion), q = 5/3 (normal or Brownian diffusion) and q > 5/3 (super diffusion). The generalized energy spectrum of Kraichnan, occurring at small wave numbers k, now reveals the more general and precise result k −q . This corresponds well for q = 5/3 with the Kolmogorov-Oboukov energy spectrum and for q > 5/3 to turbulence with intermittency. The enstrophy spectrum, occurring at large wave numbers k, leads to a k −3q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan's correct k −3 enstrophy spectrum. The theory reveals in a natural manner a generalized temperature of turbulence, which in the non-equilibrium energy transfer domain decreases with wave number and shows an energy equipartition law with a constant generalized temperature in the equilibrium enstrophy transfer domain. The article contains numerous new results; some are stated in form of eight new (proven) propositions.

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Section: Discussion Of Results Conclusion and Outlookmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tsallis Extended Thermodynamics Applied to 2-d Turbulence: Lévy Statistics and q-Fractional Generalized Kraichnanian Energy and Enstrophy Spectra

Egolf

Hutter

2018

Entropy

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“…Technology needs to be developed to detect such thermodynamic effects due to the presence of life, prior to taking samples for analysis, thereby potentially harming and destroying what is under examination. The application of difficult and sophisticated methods including fractal, multifractal, and thermodynamic approaches would require some time to develop and optimize [82][83][84][85][86][87][88][89]. Anthropomorphism not intended, in terms of intelligence, scanning for the extraterrestrial equivalent of neurons (from an external vantage) is a daunting task.…”

Section: Thermodynamics and Lifementioning

confidence: 99%

Astrovirology, Astrobiology, Artificial Intelligence: Extra-Solar System Investigations

Shapshak

2019

Global Virology III: Virology in the 21st Century

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This chapter attempts to encompass and tackle a large problem in Astrovirology and Astrobiology. There is a huge anthropomorphic prejudice that although life is unlikely, the just-right Goldilocks terrestrial conditions mean that the just-right balance of minerals and basic small molecules inevitably result in life as we know it throughout our solar system, galaxy, and the rest of the universe. Moreover, when such conditions on planets such as ours may not be quite right for the origin of life, it is popularly opined that asteroids and comets magically produce life or at the very least, the important, if not crucial components of terrestrial life so that life then blooms, when their fragments cruise the solar system, stars, and galaxies, and plummet onto appropriately bedecked planets and moons. It is no longer extraordinary to detect extraterrestrial solar systems. Moreover, since extra-solar system space exploration has commenced, this provides the problem of detecting life with enhanced achievability. Small organisms, which replicate outside of a living cell or host, would not be catalogued as viruses. How about viruses that cohabit with life? On the Earth, viruses are a major, if underestimated, condition of life-will that be the case elsewhere? Detection of extra-solar system viruses, if they exist, requires finding life, since viruses necessitate life to replicate. (It should be noted, though, that viruses could be detected through various types of portable ultra-microscopes, including Electron Microscopes (EM) (scanning and transmission) as well as Atomic Force Microscopes (AFM).) However, extra-solar system detection of life does not oblige that viruses exist ubiquitously. Viruses are important potential components of biospheres because of their multiple interactions and influence on evolution, although viruses are small and obligatory parasitic. In addition, nanotechnology-living or replicating nano-synthetic machine organisms might also be present out there, and require consideration as well. An imposing caveat is that, if found, could some extraterrestrial viruses and synthetic nanotechnological microorganisms infect humans? This chapter is dedicated to Ettore Majorana, who changed the view of the universe.

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“…J. Weberszpil & W. Chen [ 38 ] discuss “ Generalized Maxwell Relations in Thermodynamics with Metric Derivatives ”. These are equations, which establish relations between pressure, specific volume, entropy, temperature, and thermodynamic potentials such as internal energy, Helmholtz free energy, enthalpy, and Gibbs free energy.…”

Section: The 12 Contributions Published In This Special Issuementioning

confidence: 99%