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Entropies: Between Information Geometry and Kinetics

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A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (16 February 2020) | Viewed by 44979

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Prof. Dr. Alexander Gorban

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Guest Editor

Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
Interests: neural networks; chemical and biological kinetics; human adaptation to hard living conditions; methods and technologies of collective thinking
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Miroslav Grmela

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Guest Editor

Department of Chemical Engineering, Polytechnique Montréal, Montreal, QC, Canada
Interests: multiscale nonequilibrium thermodynamics; kinetic theory; mechanics of complex fluids; differential geometry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The history of entropy is a wonderful combination of physical, probabilistic, and geometrical ideas. Chemical and biological problems and modern data analysis further enrich this beautiful combination. This issue has an ambitious goal: To collect highly interdisciplinary papers that combine more than one subject and are related to geometric, statistical, and physical ideas of entropy. Geometric ideas in thermodynamics, statistical physics, and kinetics, geometry of data analytics in high dimensions, ideas of statistical physics in geometry, and mathematical backgrounds of all these approaches are very welcome. We are looking for new ideas and methods to support emerging and fast developing areas, like Artificial Intelligence and Smart Materials, and for solutions to classical problems such as efficient model reduction in kinetic systems. We also encourage authors to supplement articles with real-world applications whenever possible.

Prof. Dr. Alexander N. Gorban
Prof. Dr. Miroslav Grmela
Guest Editors

Manuscript Submission Information

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Keywords

entropy
thermodynamics
kinetics
geometry
information processing
measure concentration
high dimensionality
model reduction
data analysis

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Published Papers (11 papers)

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28 pages, 4982 KiB

Open AccessEditor’s ChoiceArticle

Theory, Analysis, and Applications of the Entropic Lattice Boltzmann Model for Compressible Flows

by Nicolò Frapolli, Shyam Chikatamarla and Ilya Karlin

Entropy 2020, 22(3), 370; https://doi.org/10.3390/e22030370 - 24 Mar 2020

Cited by 16 | Viewed by 5443

Abstract

The entropic lattice Boltzmann method for the simulation of compressible flows is studied in detail and new opportunities for extending operating range are explored. We address limitations on the maximum Mach number and temperature range allowed for a given lattice. Solutions to both these problems are presented by modifying the original lattices without increasing the number of discrete velocities and without altering the numerical algorithm. In order to increase the Mach number, we employ shifted lattices while the magnitude of lattice speeds is increased in order to extend the temperature range. Accuracy and efficiency of the shifted lattices are demonstrated with simulations of the supersonic flow field around a diamond-shaped and NACA0012 airfoil, the subsonic, transonic, and supersonic flow field around the Busemann biplane, and the interaction of vortices with a planar shock wave. For the lattices with extended temperature range, the model is validated with the simulation of the Richtmyer–Meshkov instability. We also discuss some key ideas of how to reduce the number of discrete speeds in three-dimensional simulations by pruning of the higher-order lattices, and introduce a new construction of the corresponding guided equilibrium by entropy minimization. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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27 pages, 5113 KiB

Open AccessArticle

Robust and Scalable Learning of Complex Intrinsic Dataset Geometry via ElPiGraph

by Luca Albergante, Evgeny Mirkes, Jonathan Bac, Huidong Chen, Alexis Martin, Louis Faure, Emmanuel Barillot, Luca Pinello, Alexander Gorban and Andrei Zinovyev

Entropy 2020, 22(3), 296; https://doi.org/10.3390/e22030296 - 4 Mar 2020

Cited by 35 | Viewed by 7459

Abstract

Multidimensional datapoint clouds representing large datasets are frequently characterized by non-trivial low-dimensional geometry and topology which can be recovered by unsupervised machine learning approaches, in particular, by principal graphs. Principal graphs approximate the multivariate data by a graph injected into the data space with some constraints imposed on the node mapping. Here we present ElPiGraph, a scalable and robust method for constructing principal graphs. ElPiGraph exploits and further develops the concept of elastic energy, the topological graph grammar approach, and a gradient descent-like optimization of the graph topology. The method is able to withstand high levels of noise and is capable of approximating data point clouds via principal graph ensembles. This strategy can be used to estimate the statistical significance of complex data features and to summarize them into a single consensus principal graph. ElPiGraph deals efficiently with large datasets in various fields such as biology, where it can be used for example with single-cell transcriptomic or epigenomic datasets to infer gene expression dynamics and recover differentiation landscapes. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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29 pages, 2360 KiB

Open AccessArticle

Universal Gorban’s Entropies: Geometric Case Study

by Evgeny M. Mirkes

Entropy 2020, 22(3), 264; https://doi.org/10.3390/e22030264 - 25 Feb 2020

Cited by 1 | Viewed by 3326

Abstract

Recently, A.N. Gorban presented a rich family of universal Lyapunov functions for any linear or non-linear reaction network with detailed or complex balance. Two main elements of the construction algorithm are partial equilibria of reactions and convex envelopes of families of functions. These new functions aimed to resolve “the mystery” about the difference between the rich family of Lyapunov functions (f-divergences) for linear kinetics and a limited collection of Lyapunov functions for non-linear networks in thermodynamic conditions. The lack of examples did not allow to evaluate the difference between Gorban’s entropies and the classical Boltzmann–Gibbs–Shannon entropy despite obvious difference in their construction. In this paper, Gorban’s results are briefly reviewed, and these functions are analysed and compared for several mechanisms of chemical reactions. The level sets and dynamics along the kinetic trajectories are analysed. The most pronounced difference between the new and classical thermodynamic Lyapunov functions was found far from the partial equilibria, whereas when some fast elementary reactions became close to equilibrium then this difference decreased and vanished in partial equilibria. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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24 pages, 8242 KiB

Open AccessArticle

Thermomass Theory in the Framework of GENERIC

by Ben-Dian Nie, Bing-Yang Cao, Zeng-Yuan Guo and Yu-Chao Hua

Entropy 2020, 22(2), 227; https://doi.org/10.3390/e22020227 - 18 Feb 2020

Cited by 7 | Viewed by 2644

Abstract

Thermomass theory was developed to deal with the non-Fourier heat conduction phenomena involving the influence of heat inertia. However, its structure, derived from an analogy to fluid mechanics, requires further mathematical verification. In this paper, General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, which is a geometrical and mathematical structure in nonequilibrium thermodynamics, was employed to verify the thermomass theory. At first, the thermomass theory was introduced briefly; then, the GENERIC framework was applied in the thermomass gas system with state variables, thermomass gas density ρ_h and thermomass momentum m_h, and the time evolution equations obtained from GENERIC framework were compared with those in thermomass theory. It was demonstrated that the equations generated by GENERIC theory were the same as the continuity and momentum equations in thermomass theory with proper potentials and eta-function. Thermomass theory gives a physical interpretation to the GENERIC theory in non-Fourier heat conduction phenomena. By combining these two theories, it was found that the Hamiltonian energy in reversible process and the dissipation potential in irreversible process could be unified into one formulation, i.e., the thermomass energy. Furthermore, via the framework of GENERIC, thermomass theory could be extended to involve more state variables, such as internal source term and distortion matrix term. Numerical simulations investigated the influences of the convective term and distortion matrix term in the equations. It was found that the convective term changed the shape of thermal energy distribution and enhanced the spreading behaviors of thermal energy. The distortion matrix implies the elasticity and viscosity of the thermomass gas. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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13 pages, 1066 KiB

Open AccessArticle

Learning the Macroscopic Flow Model of Short Fiber Suspensions from Fine-Scale Simulated Data

by Minyoung Yun, Clara Argerich Martin, Pierre Giormini, Francisco Chinesta and Suresh Advani

Entropy 2020, 22(1), 30; https://doi.org/10.3390/e22010030 - 24 Dec 2019

Cited by 4 | Viewed by 2608

Abstract

Fiber–fiber interaction plays an important role in the evolution of fiber orientation in semi-concentrated suspensions. Flow induced orientation in short-fiber reinforced composites determines the anisotropic properties of manufactured parts and consequently their performances. In the case of dilute suspensions, the orientation evolution can be accurately described by using the Jeffery model; however, as soon as the fiber concentration increases, fiber–fiber interactions cannot be ignored anymore and the final orientation state strongly depends on the modeling of those interactions. First modeling frameworks described these interactions from a diffusion mechanism; however, it was necessary to consider richer descriptions (anisotropic diffusion, etc.) to address experimental observations. Even if different proposals were considered, none of them seem general and accurate enough. In this paper we do not address a new proposal of a fiber interaction model, but a data-driven methodology able to enrich existing models from data, that in our case comes from a direct numerical simulation of well resolved microscopic physics. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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40 pages, 3058 KiB

Open AccessArticle

Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid

by Mark Dostalík, Vít Průša and Karel Tůma

Entropy 2019, 21(12), 1219; https://doi.org/10.3390/e21121219 - 13 Dec 2019

Cited by 8 | Viewed by 3165

Abstract

Using a Lyapunov type functional constructed on the basis of thermodynamical arguments, we investigate the finite amplitude stability of internal steady flows of viscoelastic fluids described by the Giesekus model. Using the functional, we derive bounds on the Reynolds and the Weissenberg number that guarantee the unconditional asymptotic stability of the corresponding steady internal flow, wherein the distance between the steady flow field and the perturbed flow field is measured with the help of the Bures–Wasserstein distance between positive definite matrices. The application of the theoretical results is documented in the finite amplitude stability analysis of Taylor–Couette flow. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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15 pages, 660 KiB

Open AccessArticle

Data-Driven GENERIC Modeling of Poroviscoelastic Materials

by Chady Ghnatios, Iciar Alfaro, David González, Francisco Chinesta and Elias Cueto

Entropy 2019, 21(12), 1165; https://doi.org/10.3390/e21121165 - 28 Nov 2019

Cited by 13 | Viewed by 2863

Abstract

Biphasic soft materials are challenging to model by nature. Ongoing efforts are targeting their effective modeling and simulation. This work uses experimental atomic force nanoindentation of thick hydrogels to identify the indentation forces are a function of the indentation depth. Later on, the atomic force microscopy results are used in a GENERIC general equation for non-equilibrium reversible–irreversible coupling (GENERIC) formalism to identify the best model conserving basic thermodynamic laws. The data-driven GENERIC analysis identifies the material behavior with high fidelity for both data fitting and prediction. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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33 pages, 410 KiB

Open AccessFeature PaperArticle

Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

by Oğul Esen, Miroslav Grmela, Hasan Gümral and Michal Pavelka

Entropy 2019, 21(9), 907; https://doi.org/10.3390/e21090907 - 18 Sep 2019

Cited by 18 | Viewed by 2765

Abstract

Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. [...] Read more.

T^{*} T^{*} Q

) of (both compressible and incompressible) Euler’s fluid and Vlasov’s plasma are derived. Poisson mappings relating the momentum realizations with the usual field equations are constructed as duals of injective Lie algebra homomorphisms. The geometric pathway is then used to construct the evolution equations for 10-moments kinetic theory. This way the entire Grad hierarchy (including entropic fields) can be constructed in a purely geometric way. This geometric way is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

11 pages, 841 KiB

Open AccessArticle

A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion

by Xiaomin Duan, Huafei Sun and Xinyu Zhao

Entropy 2019, 21(5), 531; https://doi.org/10.3390/e21050531 - 25 May 2019

Cited by 2 | Viewed by 3267

Abstract

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group

S E (3)

, so the Riemannian mean based on the Lie group [...] Read more.

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group

S E (3)

, so the Riemannian mean based on the Lie group structures of

S E (3)

reflects the characteristics of change-points. Once a change-point occurs, the distance between the current point and the Riemannian mean of its neighbor points should be a local maximum. A gradient descent algorithm is proposed to calculate the Riemannian mean. Using the Baker–Campbell–Hausdorff formula, the first-order approximation of the Riemannian mean is taken as the initial value of the iterative procedure. The performance of our method was evaluated by numerical examples and manipulator experiments. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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Review

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14 pages, 2541 KiB

Open AccessEditor’s ChoiceReview

New Invariant Expressions in Chemical Kinetics

by Gregory S. Yablonsky, Daniel Branco, Guy B. Marin and Denis Constales

Entropy 2020, 22(3), 373; https://doi.org/10.3390/e22030373 - 24 Mar 2020

Cited by 6 | Viewed by 3383

Abstract

This paper presents a review of our original results obtained during the last decade. These results have been found theoretically for classical mass-action-law models of chemical kinetics and justified experimentally. In contrast with the traditional invariances, they relate to a special battery of kinetic experiments, not a single experiment. Two types of invariances are distinguished and described in detail: thermodynamic invariants, i.e., special combinations of kinetic dependences that yield the equilibrium constants, or simple functions of the equilibrium constants; and “mixed” kinetico-thermodynamic invariances, functions both of equilibrium constants and non-thermodynamic ratios of kinetic coefficients. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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16 pages, 885 KiB

Open AccessReview

High-Dimensional Brain in a High-Dimensional World: Blessing of Dimensionality

by Alexander N. Gorban, Valery A. Makarov and Ivan Y. Tyukin

Entropy 2020, 22(1), 82; https://doi.org/10.3390/e22010082 - 9 Jan 2020

Cited by 34 | Viewed by 7135

Abstract

High-dimensional data and high-dimensional representations of reality are inherent features of modern Artificial Intelligence systems and applications of machine learning. The well-known phenomenon of the “curse of dimensionality” states: many problems become exponentially difficult in high dimensions. Recently, the other side of the coin, the “blessing of dimensionality”, has attracted much attention. It turns out that generic high-dimensional datasets exhibit fairly simple geometric properties. Thus, there is a fundamental tradeoff between complexity and simplicity in high dimensional spaces. Here we present a brief explanatory review of recent ideas, results and hypotheses about the blessing of dimensionality and related simplifying effects relevant to machine learning and neuroscience. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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