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Symmetry
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Mathematical Aspects in Non-equilibrium Thermodynamics
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Mathematical Aspects in Non-equilibrium Thermodynamics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (15 June 2022) | Viewed by 13420

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Róbert Kovács

E-Mail Website
Guest Editor
1. Department of Energy Engineering, Faculty of Mechanical Engineering, BME, 1521 Budapest, Hungary
2. Institute for Particle and Nuclear Physics, Theoretical Physics Department, Gravitational Physics Research Group, Wigner Research Centre for Physics, Budapest, Hungary
Interests: non-equilibrium thermodynamics of heat conduction (low temperature, biological applications) and rarefied gases; analytical and numerical methods
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Patrizia Rogolino

E-Mail Website
Guest Editor
MIFT Department, University of Messina, Italy
Interests: non-equilibrium thermodynamics of heat conduction; graded materials; thermoelectricity; analytical and numerical methods
Prof. Dr. Francesco Oliveri

E-Mail Website
Guest Editor
MIFT Department, University of Messina, Italy
Interests: non-local constitutive theories; symmetries of differential equations; analytical and numerical methods

Special Issue Information

Dear Colleagues,

Non-equilibrium thermodynamics is a relatively new field of research and beginning to be increasingly important in engineering applications as well. However, in addition to the many approaches developed in recent decades, all of them raise questions from a mathematical point of view, such as the proper definitions of initial and boundary conditions, analytical and numerical solution methods, as well as the geometrical background, which is a very interesting aspect.

These questions cover numerous topics, such as the mathematical analysis of various thermodynamic approaches, the investigation of the resulting governing equations, symmetry analysis of the solutions, and how the numerical methods can be built on the geometrical background.

The aim of this Special Issue is to offer the possibility to discuss and present up-to-date problems that may not be constrained only to the previously mentioned aspects but must be connected to non-equilibrium thermodynamics.

Thus, in the present Special Issue, we aim to collect open mathematical questions and characteristics for different thermodynamic approaches; therefore, both original and review papers are welcome.

Dr. Kovács Róbert
Prof. Dr. Patrizia Rogolino
Prof. Dr. Francesco Oliveri
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • non-equilibrium thermodynamics
  • analytical and numerical solutions
  • different thermodynamical approaches

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Published Papers (6 papers)

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Editorial

Jump to: Research, Review

2 pages, 157 KiB  
Editorial
Mathematical Aspects in Non-Equilibrium Thermodynamics
by Róbert Kovács, Patrizia Rogolino and Francesco Oliveri
Symmetry 2023, 15(4), 929; https://doi.org/10.3390/sym15040929 - 17 Apr 2023
Cited by 1 | Viewed by 1263
Abstract
Prof [...] Full article
(This article belongs to the Special Issue Mathematical Aspects in Non-equilibrium Thermodynamics)

Research

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13 pages, 413 KiB  
Article
The Role of the Second Law of Thermodynamics in Continuum Physics: A Muschik and Ehrentraut Theorem Revisited
by Vito Antonio Cimmelli and Patrizia Rogolino
Symmetry 2022, 14(4), 763; https://doi.org/10.3390/sym14040763 - 7 Apr 2022
Cited by 7 | Viewed by 1912
Abstract
In continuum physics, constitutive equations model the material properties of physical systems. In those equations, material symmetry is taken into account by applying suitable representation theorems for symmetric and/or isotropic functions. Such mathematical representations must be in accordance with the second law of [...] Read more.
In continuum physics, constitutive equations model the material properties of physical systems. In those equations, material symmetry is taken into account by applying suitable representation theorems for symmetric and/or isotropic functions. Such mathematical representations must be in accordance with the second law of thermodynamics, which imposes that, in any thermodynamic process, the entropy production must be nonnegative. This requirement is fulfilled by assigning the constitutive equations in a form that guaranties that second law of thermodynamics is satisfied along arbitrary processes. Such an approach, in practice regards the second law of thermodynamics as a restriction on the constitutive equations, which must guarantee that any solution of the balance laws also satisfy the entropy inequality. This is a useful operative assumption, but not a consequence of general physical laws. Indeed, a different point of view, which regards the second law of thermodynamics as a restriction on the thermodynamic processes, i.e., on the solutions of the system of balance laws, is possible. This is tantamount to assuming that there are solutions of the balance laws that satisfy the entropy inequality, and solutions that do not satisfy it. In order to decide what is the correct approach, Muschik and Ehrentraut in 1996, postulated an amendment to the second law, which makes explicit the evident (but rather hidden) assumption that, in any point of the body, the entropy production is zero if, and only if, this point is a thermodynamic equilibrium. Then they proved that, given the amendment, the second law of thermodynamics is necessarily a restriction on the constitutive equations and not on the thermodynamic processes. In the present paper, we revisit their proof, lighting up some geometric aspects that were hidden in therein. Moreover, we propose an alternative formulation of the second law of thermodynamics, which incorporates the amendment. In this way we make this important result more intuitive and easily accessible to a wider audience. Full article
(This article belongs to the Special Issue Mathematical Aspects in Non-equilibrium Thermodynamics)
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12 pages, 450 KiB  
Article
Integrability of the Multi-Species TASEP with Species-Dependent Rates
by Eunghyun Lee
Symmetry 2021, 13(9), 1578; https://doi.org/10.3390/sym13091578 - 27 Aug 2021
Cited by 3 | Viewed by 1431
Abstract
Assume that each species l has its own jump rate bl in the multi-species totally asymmetric simple exclusion process. We show that this model is integrable in the sense that the Bethe ansatz method is applicable to obtain the transition probabilities for [...] Read more.
Assume that each species l has its own jump rate bl in the multi-species totally asymmetric simple exclusion process. We show that this model is integrable in the sense that the Bethe ansatz method is applicable to obtain the transition probabilities for all possible N-particle systems with up to N different species. Full article
(This article belongs to the Special Issue Mathematical Aspects in Non-equilibrium Thermodynamics)
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23 pages, 793 KiB  
Article
Shock Structure and Relaxation in the Multi-Component Mixture of Euler Fluids
by Damir Madjarević, Milana Pavić-Čolić and Srboljub Simić
Symmetry 2021, 13(6), 955; https://doi.org/10.3390/sym13060955 - 27 May 2021
Cited by 11 | Viewed by 2143
Abstract
The shock structure problem is studied for a multi-component mixture of Euler fluids described by the hyperbolic system of balance laws. The model is developed in the framework of extended thermodynamics. Thanks to the equivalence with the kinetic theory approach, phenomenological coefficients are [...] Read more.
The shock structure problem is studied for a multi-component mixture of Euler fluids described by the hyperbolic system of balance laws. The model is developed in the framework of extended thermodynamics. Thanks to the equivalence with the kinetic theory approach, phenomenological coefficients are computed from the linearized weak form of the collision operator. Shock structure is analyzed for a three-component mixture of polyatomic gases, and for various combinations of parameters of the model (Mach number, equilibrium concentrations and molecular mass ratios). The analysis revealed that three-component mixtures possess distinguishing features different from the binary ones, and that certain behavior may be attributed to polyatomic structure of the constituents. The multi-temperature model is compared with a single-temperature one, and the difference between the mean temperatures of the mixture are computed. Mechanical and thermal relaxation times are computed along the shock profiles, and revealed that the thermal ones are smaller in the case discussed in this study. Full article
(This article belongs to the Special Issue Mathematical Aspects in Non-equilibrium Thermodynamics)
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15 pages, 813 KiB  
Article
Cyclic Control Optimization Algorithm for Stirling Engines
by Raphael Paul and Karl Heinz Hoffmann
Symmetry 2021, 13(5), 873; https://doi.org/10.3390/sym13050873 - 13 May 2021
Cited by 23 | Viewed by 3881
Abstract
The ideal Stirling cycle describes a specific way to operate an equilibrium Stirling engine. This cycle consists of two isothermal and two isochoric strokes. For non-equilibrium Stirling engines, which may feature various irreversibilities and whose dynamics is characterized by a set of coupled [...] Read more.
The ideal Stirling cycle describes a specific way to operate an equilibrium Stirling engine. This cycle consists of two isothermal and two isochoric strokes. For non-equilibrium Stirling engines, which may feature various irreversibilities and whose dynamics is characterized by a set of coupled ordinary differential equations, a control strategy that is based on the ideal cycle will not necessarily yield the best performance—for example, it will not generally lead to maximum power. In this paper, we present a method to optimize the engine’s piston paths for different objectives; in particular, power and efficiency. Here, the focus is on an indirect iterative gradient algorithm that we use to solve the cyclic optimal control problem. The cyclic optimal control problem leads to a Hamiltonian system that features a symmetry between its state and costate subproblems. The symmetry manifests itself in the existence of mutually related attractive and repulsive limit cycles. Our algorithm exploits these limit cycles to solve the state and costate problems with periodic boundary conditions. A description of the algorithm is provided and it is explained how the control can be embedded in the system dynamics. Moreover, the optimization results obtained for an exemplary Stirling engine model are discussed. For this Stirling engine model, a comparison of the optimized piston paths against harmonic piston paths shows significant gains in both power and efficiency. At the maximum power point, the relative power gain due to the power-optimal control is ca. 28%, whereas the relative efficiency gain due to the efficiency-optimal control at the maximum efficiency point is ca. 10%. Full article
(This article belongs to the Special Issue Mathematical Aspects in Non-equilibrium Thermodynamics)
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Review

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13 pages, 275 KiB  
Review
Recent Advances on Boundary Conditions for Equations in Nonequilibrium Thermodynamics
by Wen-An Yong and Yizhou Zhou
Symmetry 2021, 13(9), 1710; https://doi.org/10.3390/sym13091710 - 16 Sep 2021
Cited by 4 | Viewed by 1766
Abstract
This paper is concerned with modeling nonequilibrium phenomena in spatial domains with boundaries. The resultant models consist of hyperbolic systems of first-order partial differential equations with boundary conditions (BCs). Taking a linearized moment closure system as an example, we show that the structural [...] Read more.
This paper is concerned with modeling nonequilibrium phenomena in spatial domains with boundaries. The resultant models consist of hyperbolic systems of first-order partial differential equations with boundary conditions (BCs). Taking a linearized moment closure system as an example, we show that the structural stability condition and the uniform Kreiss condition do not automatically guarantee the compatibility of the models with the corresponding classical models. This motivated the generalized Kreiss condition (GKC)—a strengthened version of the uniform Kreiss condition. Under the GKC and the structural stability condition, we show how to derive the reduced BCs for the equilibrium systems as the classical models. For linearized problems, the validity of the reduced BCs can be rigorously verified. Furthermore, we use a simple example to show how thus far developed theory can be used to construct proper BCs for equations modeling nonequilibrium phenomena in spatial domains with boundaries. Full article
(This article belongs to the Special Issue Mathematical Aspects in Non-equilibrium Thermodynamics)
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