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18 pages, 716 KiB

Open AccessFeature PaperArticle

Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams

by Nataliya Shcherbakova, Vincent Gerbaud and Kevin Roger

Entropy 2023, 25(9), 1329; https://doi.org/10.3390/e25091329 - 13 Sep 2023

Viewed by 2183

Abstract

Phase diagrams are powerful tools to understand the multi-scale behaviour of complex systems. Yet, their determination requires in practice both experiments and computations, which quickly becomes a daunting task. Here, we propose a geometrical approach to simplify the numerical computation of liquid–liquid ternary [...] Read more.

Phase diagrams are powerful tools to understand the multi-scale behaviour of complex systems. Yet, their determination requires in practice both experiments and computations, which quickly becomes a daunting task. Here, we propose a geometrical approach to simplify the numerical computation of liquid–liquid ternary phase diagrams. We show that using the intrinsic geometry of the binodal curve, it is possible to formulate the problem as a simple set of ordinary differential equations in an extended 4D space. Consequently, if the thermodynamic potential, such as Gibbs free energy, is known from an experimental data set, the whole phase diagram, including the spinodal curve, can be easily computed. We showcase this approach on four ternary liquid–liquid diagrams, with different topological properties, using a modified Flory–Huggins model. We demonstrate that our method leads to similar or better results comparing those obtained with other methods, but with a much simpler procedure. Acknowledging and using the intrinsic geometry of phase diagrams thus appears as a promising way to further develop the computation of multiphase diagrams. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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9 pages, 278 KiB

Open AccessFeature PaperArticle

Thermodynamic Entropy as a Noether Invariant from Contact Geometry

by Alessandro Bravetti, Miguel Ángel García-Ariza and Diego Tapias

Entropy 2023, 25(7), 1082; https://doi.org/10.3390/e25071082 - 19 Jul 2023

Cited by 5 | Viewed by 1719

Abstract

We use a formulation of Noether’s theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of [...] Read more.

We use a formulation of Noether’s theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of the system plus the reservoir are conserved as a consequence thereof. Our results contribute to understanding thermodynamic entropy from a geometric point of view. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

13 pages, 381 KiB

Open AccessArticle

Restricted Phase Space Thermodynamics of Einstein-Power-Yang–Mills AdS Black Hole

by Yun-Zhi Du, Huai-Fan Li, Yang Zhang, Xiang-Nan Zhou and Jun-Xin Zhao

Entropy 2023, 25(4), 687; https://doi.org/10.3390/e25040687 - 19 Apr 2023

Cited by 8 | Viewed by 1645

Abstract

We consider the thermodynamics of the Einstein-power-Yang–Mills AdS black holes in the context of the gauge-gravity duality. Under this framework, Newton’s gravitational constant and the cosmological constant are varied in the system. We rewrite the thermodynamic first law in a more extended form [...] Read more.

We consider the thermodynamics of the Einstein-power-Yang–Mills AdS black holes in the context of the gauge-gravity duality. Under this framework, Newton’s gravitational constant and the cosmological constant are varied in the system. We rewrite the thermodynamic first law in a more extended form containing both the pressure and the central charge of the dual conformal field theory, i.e., the restricted phase transition formula. A novel phenomena arises: the dual quantity of pressure is the effective volume, not the geometric one. That leads to a new behavior of the Van de Waals-like phase transition for this system with the fixed central charge: the supercritical phase transition. From the Ehrenfest’s scheme perspective, we check out the second-order phase transition of the EPYM AdS black hole. Furthermore the effect of the non-linear Yang–Mills parameter on these thermodynamic properties is also investigated. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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22 pages, 376 KiB

Open AccessFeature PaperArticle

Geometric Modeling for Control of Thermodynamic Systems

by Arjan van der Schaft

Entropy 2023, 25(4), 577; https://doi.org/10.3390/e25040577 - 27 Mar 2023

Cited by 2 | Viewed by 1391

Abstract

This paper discusses the way that energy and entropy can be regarded as storage functions with respect to supply rates corresponding to the power and thermal ports of the thermodynamic system. Then, this research demonstrates how the factorization of the irreversible entropy production [...] Read more.

This paper discusses the way that energy and entropy can be regarded as storage functions with respect to supply rates corresponding to the power and thermal ports of the thermodynamic system. Then, this research demonstrates how the factorization of the irreversible entropy production leads to quasi-Hamiltonian formulations, and how this can be used for stability analysis. The Liouville geometry approach to contact geometry is summarized, and how this leads to the definition of port-thermodynamic systems is discussed. This notion is utilized for control by interconnection of thermodynamic systems. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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12 pages, 308 KiB

Open AccessArticle

Symplectic Geometry Aspects of the Parametrically-Dependent Kardar–Parisi–Zhang Equation of Spin Glasses Theory, Its Integrability and Related Thermodynamic Stability

by Anatolij K. Prykarpatski, Petro Y. Pukach and Myroslava I. Vovk

Entropy 2023, 25(2), 308; https://doi.org/10.3390/e25020308 - 7 Feb 2023

Cited by 2 | Viewed by 1328

Abstract

A thermodynamically unstable spin glass growth model described by means of the parametrically-dependent Kardar–Parisi–Zhang equation is analyzed within the symplectic geometry-based gradient–holonomic and optimal control motivated algorithms. The finitely-parametric functional extensions of the model are studied, and the existence of conservation laws and [...] Read more.

A thermodynamically unstable spin glass growth model described by means of the parametrically-dependent Kardar–Parisi–Zhang equation is analyzed within the symplectic geometry-based gradient–holonomic and optimal control motivated algorithms. The finitely-parametric functional extensions of the model are studied, and the existence of conservation laws and the related Hamiltonian structure is stated. A relationship of the Kardar–Parisi–Zhang equation to a so called dark type class of integrable dynamical systems, on functional manifolds with hidden symmetries, is stated. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

36 pages, 3506 KiB

Open AccessArticle

Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation

by Frédéric Barbaresco

Entropy 2022, 24(11), 1626; https://doi.org/10.3390/e24111626 - 9 Nov 2022

Cited by 5 | Viewed by 2561

Abstract

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of [...] Read more.

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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18 pages, 381 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

An Overview on Irreversible Port-Hamiltonian Systems

by Hector Ramirez and Yann Le Gorrec

Entropy 2022, 24(10), 1478; https://doi.org/10.3390/e24101478 - 17 Oct 2022

Cited by 7 | Viewed by 2334

Abstract

A comprehensive overview of the irreversible port-Hamiltonian system’s formulation for finite and infinite dimensional systems defined on 1D spatial domains is provided in a unified manner. The irreversible port-Hamiltonian system formulation shows the extension of classical port-Hamiltonian system formulations to cope with irreversible [...] Read more.

A comprehensive overview of the irreversible port-Hamiltonian system’s formulation for finite and infinite dimensional systems defined on 1D spatial domains is provided in a unified manner. The irreversible port-Hamiltonian system formulation shows the extension of classical port-Hamiltonian system formulations to cope with irreversible thermodynamic systems for finite and infinite dimensional systems. This is achieved by including, in an explicit manner, the coupling between irreversible mechanical and thermal phenomena with the thermal domain as an energy-preserving and entropy-increasing operator. Similarly to Hamiltonian systems, this operator is skew-symmetric, guaranteeing energy conservation. To distinguish from Hamiltonian systems, the operator depends on co-state variables and is, hence, a nonlinear-function in the gradient of the total energy. This is what allows encoding the second law as a structural property of irreversible port-Hamiltonian systems. The formalism encompasses coupled thermo-mechanical systems and purely reversible or conservative systems as a particular case. This appears clearly when splitting the state space such that the entropy coordinate is separated from other state variables. Several examples have been used to illustrate the formalism, both for finite and infinite dimensional systems, and a discussion on ongoing and future studies is provided. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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12 pages, 6989 KiB

Open AccessArticle

Topological Invariants of Vapor–Liquid, Vapor–Liquid–Liquid and Liquid–Liquid Phase Diagrams

by Anastasia V. Frolkova

Entropy 2021, 23(12), 1666; https://doi.org/10.3390/e23121666 - 10 Dec 2021

Viewed by 2546

Abstract

The study of topological invariants of phase diagrams allows for the development of a qualitative theory of the processes being researched. Studies of the properties of objects in the same equivalence class may be carried out with the aim of predicting the properties [...] Read more.

The study of topological invariants of phase diagrams allows for the development of a qualitative theory of the processes being researched. Studies of the properties of objects in the same equivalence class may be carried out with the aim of predicting the properties of unexplored objects from this class, or predicting the behavior of a whole system. This paper describes a number of topological invariants in vapor–liquid, vapor–liquid–liquid and liquid–liquid equilibrium diagrams. The properties of some invariants are studied and illustrated. It is shown that the invariant of a diagram with a miscibility gap can be used to distinguish equivalence classes of phase diagrams, and that the balance equation of the singular-point indices, based on the Euler characteristic, may be used to analyze the binodal-surface structure of a quaternary system. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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9 pages, 269 KiB

Open AccessArticle

Geometric Analysis of a System with Chemical Interactions

by Dmitry Gromov and Alexander Toikka

Entropy 2021, 23(11), 1548; https://doi.org/10.3390/e23111548 - 21 Nov 2021

Cited by 2 | Viewed by 1742

Abstract

In this paper, we present some initial results aimed at defining a framework for the analysis of thermodynamic systems with additional restrictions imposed on the intensive parameters. Specifically, for the case of chemical reactions, we considered the states of constant affinity that form [...] Read more.

In this paper, we present some initial results aimed at defining a framework for the analysis of thermodynamic systems with additional restrictions imposed on the intensive parameters. Specifically, for the case of chemical reactions, we considered the states of constant affinity that form isoffine submanifolds of the thermodynamic phase space. Wer discuss the problem of extending the previously obtained stability conditions to the considered class of systems. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

11 pages, 379 KiB

Open AccessArticle

Economic Cycles of Carnot Type

by Constantin Udriste, Vladimir Golubyatnikov and Ionel Tevy

Entropy 2021, 23(10), 1344; https://doi.org/10.3390/e23101344 - 14 Oct 2021

Cited by 2 | Viewed by 2171

Abstract

Originally, the Carnot cycle was a theoretical thermodynamic cycle that provided an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature [...] Read more.

Originally, the Carnot cycle was a theoretical thermodynamic cycle that provided an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference by the application of work to the system. The first aim of this paper is to introduce and study the economic Carnot cycles concerning Roegenian economics, using our thermodynamic–economic dictionary. These cycles are described in both a

Q - P

diagram and a

E - I

diagram. An economic Carnot cycle has a maximum efficiency for a reversible economic “engine”. Three problems together with their solutions clarify the meaning of the economic Carnot cycle, in our context. Then we transform the ideal gas theory into the ideal income theory. The second aim is to analyze the economic Van der Waals equation, showing that the diffeomorphic-invariant information about the Van der Waals surface can be obtained by examining a cuspidal potential. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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6 pages, 234 KiB

Open AccessOpinion

Some Remarks on the Boundary of Thermodynamic Stability

by Alexander Toikka, Georgii Misikov and Maria Toikka

Entropy 2023, 25(7), 969; https://doi.org/10.3390/e25070969 - 23 Jun 2023

Cited by 2 | Viewed by 1081 | Correction

Abstract

In this paper, we have considered some elements of the classical phenomenological theory of thermodynamic stability, which seem controversial and ambiguous. The main focus is on the conditions of the stability boundary; a new version of the derivation of the relations defining the [...] Read more.

In this paper, we have considered some elements of the classical phenomenological theory of thermodynamic stability, which seem controversial and ambiguous. The main focus is on the conditions of the stability boundary; a new version of the derivation of the relations defining the specified boundary is proposed. Although the final results, in general, coincide with the classical relations, the described approach, from our point of view, provides a clearer and more accurate idea of the stability conditions and their boundaries. Full article

(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)

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