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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Thermodynamics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 9495

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Prof. Dr. Bhimsen Shivamoggi

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Departments of Mathematics and Physics, University of Central Florida, 4000 Central Florida Blvd, Orlando, FL 32816, USA
Interests: fluid dynamics; plasma physics

Special Issue Information

Dear Colleagues,

The entropy of a system provides a measure of missing information (or randomness) in the system. The concept of entropy was introduced by Clausius in 1865 to reformulate the second law of thermodynamics in a more elegant way. In a revolutionary stroke, in 1870 Boltzmann explained how entropy can indeed be used to understand the macroscopic world via the underlying molecular dynamics.

In general, fluid flows are out of equilibrium, so it is not formally possible to ascribe the concept of entropy to fluid flows. However, one typically circumvents this issue by assuming the fluid to be locally close to equilibrium. In the same vein, fully developed turbulence (FDT) in fluids is a dissipative dynamical system with enormous strongly interacting degrees of freedom in a state of strong departure from absolute statistical equilibrium. So, equilibrium statistical mechanics is not formally applicable to FDT, and equilibrium states are not realizable in FDT. Nevertheless, as Kraichnan ingeniously pointed out in 1964, equilibrium states prove to be useful to FDT because they indicate the direction toward which the nonlinear interactions in conjunction with a selective rapid viscous decay of high-wavenumber modes drive the system and produce an energy cascade.

In dealing with the statistical properties of systems of critical points with long-range interactions, Tsallis pointed out perceptively in 1988 that it is useful to generalize the Boltzmann–Gibbs entropy to the non-extensive regime. The Tsallis entropy has been shown to provide alternate perspectives to some aspects of FDT (as well as a wide variety of problems in physics).

The entropy concept has also proved useful when applied to superfluids. In the standard model of superfluid helium II given by Landau in 1941, the superfluid below the lambda point (2.17 K) is taken to be an inviscid, irrotational fluid with thermal excitations modeled by a normal fluid moving on that underlying superfluid. This system supports pressure waves of ordinary sound in which the normal fluid and superfluid components move in phase. On the other hand, this system also supports a new compression wave of entropy called the “second sound” in which the normal fluid and superfluid components move out of phase and the overall mass density remains nearly constant.

Prof. Dr. Bhimsen Shivamoggi
Guest Editor

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Keywords

boltzmann–gibbs entropy
dissipative dynamical system
energy cascade
entropy wave
equilibrium statistical mechanics
fluids
second sound
statistical equilibrium
superfluids
tsallis entropy
turbulence

Published Papers (5 papers)

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Research

17 pages, 8181 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

The Entropy Density Behavior across a Plane Shock Wave

by Rosa M. Velasco and Francisco J. Uribe

Entropy 2023, 25(6), 906; https://doi.org/10.3390/e25060906 - 7 Jun 2023

Cited by 1 | Viewed by 1205

Abstract

Entropy density behavior poses many problems when we study non-equilibrium situations. In particular, the local equilibrium hypothesis (LEH) has played a very important role and is taken for granted in non-equilibrium problems, no matter how extreme they are. In this paper we would like to calculate the Boltzmann entropy balance equation for a plane shock wave and show its performance for Grad’s 13-moment approximation and the Navier–Stokes–Fourier equations. In fact, we calculate the correction for the LEH in Grad’s case and discuss its properties. Full article

(This article belongs to the Special Issue Entropy in Fluids)

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20 pages, 3911 KiB

Open AccessArticle

Novel Detection of Atmospheric Turbulence Profile Using Mie-Scattering Lidar Based on Non-Kolmogorov Turbulence Theory

by Jiandong Mao, Yingnan Zhang, Juan Li, Xin Gong, Hu Zhao and Zhimin Rao

Entropy 2023, 25(3), 477; https://doi.org/10.3390/e25030477 - 9 Mar 2023

Cited by 3 | Viewed by 1613

Abstract

Turbulence can cause effects such as light intensity fluctuations and phase fluctuations when a laser is transmitted in the atmosphere, which has serious impacts on a number of optical engineering application effects and on climate improvement. Therefore, accurately obtaining real-time turbulence intensity information using lidar-active remote sensing technology is of great significance. In this paper, based on residual turbulent scintillation theory, a Mie-scattering lidar method was developed to detect atmospheric turbulence intensity. By extracting light intensity fluctuation information from a Mie-scattering lidar return signal, the atmospheric refractive index structure constant,

C_{n}^{2}

, representing the atmospheric turbulence intensity, could be obtained. Specifically, the scintillation effect on the detection path was analyzed, and the probability density distribution of the light intensity of the Mie-scattering lidar return signal was studied. It was verified that the probability density of logarithmic light intensity basically follows a normal distribution under weak fluctuation conditions. The

C_{n}^{2}

profile based on Kolmogorov turbulence theory was retrieved using a layered, iterative method through the scintillation index. The method for detecting Kolmogorov turbulence intensity was applied to the detection of the non-Kolmogorov turbulence intensity. Through detection using the scintillation index, the corresponding

{\tilde{C}}_{n}^{2}

profile could be calculated. The detection of the

{\tilde{C}}_{n}^{2}

and

C_{n}^{2}

profiles were compared with the Hufnagel–Valley (HV) night model in the Yinchuan area. The results show that the detection results are consistent with the overall change trend of the model. In general, it is feasible to detect a non-Kolmogorov turbulence profile using Mie-scattering lidar. Full article

(This article belongs to the Special Issue Entropy in Fluids)

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13 pages, 333 KiB

Open AccessArticle

Further Properties of Tsallis Entropy and Its Application

by Ghadah Alomani and Mohamed Kayid

Entropy 2023, 25(2), 199; https://doi.org/10.3390/e25020199 - 19 Jan 2023

Cited by 5 | Viewed by 2085

Abstract

The entropy of Tsallis is a different measure of uncertainty for the Shannon entropy. The present work aims to study some additional properties of this measure and then initiate its connection with the usual stochastic order. Some other properties of the dynamical version of this measure are also investigated. It is well known that systems having greater lifetimes and small uncertainty are preferred systems and that the reliability of a system usually decreases as its uncertainty increases. Since Tsallis entropy measures uncertainty, the above remark leads us to study the Tsallis entropy of the lifetime of coherent systems and also the lifetime of mixed systems where the components have lifetimes which are independent and further, identically distributed (the iid case). Finally, we give some bounds on the Tsallis entropy of the systems and clarify their applicability. Full article

(This article belongs to the Special Issue Entropy in Fluids)

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21 pages, 6755 KiB

Open AccessArticle

Novel Simulation and Analysis of Mie-Scattering Lidar for Detecting Atmospheric Turbulence Based on Non-Kolmogorov Turbulence Power Spectrum Model

by Yingnan Zhang, Jiandong Mao, Juan Li and Xin Gong

Entropy 2022, 24(12), 1764; https://doi.org/10.3390/e24121764 - 1 Dec 2022

Cited by 3 | Viewed by 1567

Abstract

The Mie-scattering lidar can detect atmospheric turbulence intensity by using the return signals of Gaussian beams at different heights. The power spectrum method and Zernike polynomial method are used to simulate the non-Kolmogorov turbulent phase plate, respectively, and the power spectrum method with faster running speed is selected for the subsequent simulation. In order to verify the possibility of detecting atmospheric turbulence by the Mie-scattering lidar, some numerical simulations are carried out. The power spectrum method is used to simulate the propagation of the Gaussian beam from the Mie-scattering lidar in a vertical path. The propagation characteristics of the Gaussian beam using a non-Kolmogorov turbulence model are obtained by analyzing the intensity distribution and spot drift effect. The simulation results show that the scintillation index of simulation is consistent with the theoretical value trend, and the accuracy is very high, indicating that the method of atmospheric turbulence detection using Mie-scattering lidar is effective. The simulation plays a guiding role for the subsequent experimental platform construction and equipment design. Full article

(This article belongs to the Special Issue Entropy in Fluids)

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39 pages, 8985 KiB

Open AccessArticle

Statistical Equilibrium Principles in 2D Fluid Flow: From Geophysical Fluids to the Solar Tachocline

by Peter B. Weichman and John Bradley Marston

Entropy 2022, 24(10), 1389; https://doi.org/10.3390/e24101389 - 29 Sep 2022

Cited by 1 | Viewed by 1761

Abstract

An overview is presented of several diverse branches of work in the area of effectively 2D fluid equilibria which have in common that they are constrained by an infinite number of conservation laws. Broad concepts, and the enormous variety of physical phenomena that can be explored, are highlighted. These span, roughly in order of increasing complexity, Euler flow, nonlinear Rossby waves, 3D axisymmetric flow, shallow water dynamics, and 2D magnetohydrodynamics. The classical field theories describing these systems bear some resemblance to perhaps more familiar fluctuating membrane and continuous spin models, but the fluid physics drives these models into unconventional regimes exhibiting large scale jet and eddy structures. From a dynamical point of view these structures are the end result of various conserved variable forward and inverse cascades. The resulting balance between large scale structure and small scale fluctuations is controlled by the competition between energy and entropy in the system free energy, in turn highly tunable through setting the values of the conserved integrals. Although the statistical mechanical description of such systems is fully self-consistent, with remarkable mathematical structure and diversity of solutions, great care must be taken because the underlying assumptions, especially ergodicity, can be violated or at minimum lead to exceedingly long equilibration times. Generalization of the theory to include weak driving and dissipation (e.g., non-equilibrium statistical mechanics and associated linear response formalism) could provide additional insights, but has yet to be properly explored. Full article

(This article belongs to the Special Issue Entropy in Fluids)

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