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Entropy
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The First Half Century of Finite-Time Thermodynamics
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The First Half Century of Finite-Time Thermodynamics

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

Deadline for manuscript submissions: 31 October 2025 | Viewed by 2994

Special Issue Editors

Prof. Dr. Bjarne Andresen

E-Mail Website
Guest Editor
Niels Bohr Institute, University of Copenhagen, Jagtvej 155 A, DK-2200 Copenhagen, Denmark
Interests: finite-time thermodynamics (FTT); application of FTT ideas in a wide range of areas: quantum, chemical, and biological systems; global optimization theory; thermodynamic geometry; thermodynamics at extreme time and length scales; thermodynamic stability
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Peter Salamon

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA
Interests: finite-time thermodynamics; geometrical thermodynamics; biomathematics; optimization and mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In 2025, it will be 50 years since the appearance of F. L. Curzon and B. Ahlborn’s seminal paper, entitled “Efficiency of a Carnot Engine at Maximum Power Output”. They introduced the small and simple idea to limit the duration of a thermodynamic process, and this time it caught fire. Finite-time thermodynamics (FTT) was born. The idea quickly spread from thermodynamics to physics, chemistry, economics, and eventually all corners of optimization. Finite-time papers are now counted in the thousands, with new uses being picked up constantly.

However, with this anniversary Special Issue, rather than focusing on the history, we primarily want to look ahead. We aim to compile the most imaginative papers based on the concept of a finite process time. This covers all disciplines, not just thermodynamics but all natural sciences, economics, and anywhere else that the finite-time concept comes into play in its many guises. We aim to cover existing applications as well as innovative ideas for new directions. We also aim to include related new constructs such as thermodynamic geometry, quantum mechanical measures, and information geometry.

What is new in FTT and what are your visions for another half century from now? We hope to receive many inspiring contributions.

Prof. Dr. Bjarne Andresen
Prof. Dr. Peter Salamon
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. Entropy 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 2600 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

  • finite-time thermodynamics
  • thermodynamic geometry
  • control thermodynamics
  • optimization
  • thermodynamic bounds
  • optimal control modeling
  • efficiency
  • wide range of applicability of the concepts in chemistry, physics, quantum, engineering, economics, biology, etc.

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Related Special Issue

Published Papers (5 papers)

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Editorial

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3 pages, 515 KiB  
Editorial
Maximum Power Efficiency
by Boye Ahlborn and Frank Curzon
Entropy 2025, 27(7), 714; https://doi.org/10.3390/e27070714 - 1 Jul 2025
Viewed by 153
Abstract
New research often starts with vague, dream-like ideas, conversed on over coffee in the free flow of animated discussions about physics, the growing up of one’s children, politics, and the success of the local ice hockey team [...] Full article
(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)

Research

Jump to: Editorial

19 pages, 994 KiB  
Article
(Finite-Time) Thermodynamics, Hyperbolicity, Lorentz Invariance: Study of an Example
by Bernard Guy
Entropy 2025, 27(7), 700; https://doi.org/10.3390/e27070700 - 29 Jun 2025
Viewed by 314
Abstract
Our study lies at the intersection of three fields: finite-time thermodynamics, relativity theory, and the theory of hyperbolic conservation laws. Each of these fields has its own requirements and richness, and in order to link them together as effectively as possible, we have [...] Read more.
Our study lies at the intersection of three fields: finite-time thermodynamics, relativity theory, and the theory of hyperbolic conservation laws. Each of these fields has its own requirements and richness, and in order to link them together as effectively as possible, we have simplified each one, reducing it to its fundamental principles. The example chosen concerns the propagation of chemical changes in a very large reactor, as found in geology. We ask ourselves two sets of questions: (1) How do the finiteness of propagation speeds modeled by hyperbolic problems (diffusion is neglected) and the finiteness of the time allocated to transformations interact? (2) How do the finiteness of time and that of resources interact? The similarity in the behavior of the pairs of variables (x, t and resources, resource flows) in Lorentz relativistic transformations allows us to put them on the same level and propose complementary-type relationships between the two classes of finiteness. If times are finite, so are resources, which can be neither zero nor infinite. In hyperbolic problems, a condition is necessary to select solutions with a physical sense among the multiplicity of weak solutions: this is given by the entropy production, which is Lorentz invariant (and not entropy alone). Full article
(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)
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26 pages, 328 KiB  
Article
Finite-Time Thermodynamics: Problems, Approaches, and Results
by Anatoly M. Tsirlin, Alexander I. Balunov and Ivan A. Sukin
Entropy 2025, 27(6), 649; https://doi.org/10.3390/e27060649 - 17 Jun 2025
Viewed by 300
Abstract
In this manuscript, the typical problems of “finite-time thermodynamics”, their general methodology, and the general features of their solutions are considered. We also consider the role of minimal dissipation processes, the properties of the irreversibility index, and the consequences of its existence. A [...] Read more.
In this manuscript, the typical problems of “finite-time thermodynamics”, their general methodology, and the general features of their solutions are considered. We also consider the role of minimal dissipation processes, the properties of the irreversibility index, and the consequences of its existence. A generalization of the Carathéodory theorem for averaged optimization problems corresponding to cyclic processes and the properties of optimal solutions following from it are given. The existence of the irreversibility index for economic macrosystems and their analogies to and differences from thermodynamic systems are proven. Full article
(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)
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11 pages, 1908 KiB  
Article
Thermodynamics of Intrinsic Reaction Coordinate (IRC) Chemical Reaction Pathways
by Frank Weinhold
Entropy 2025, 27(4), 390; https://doi.org/10.3390/e27040390 - 7 Apr 2025
Cited by 1 | Viewed by 681
Abstract
We address the scientific “time” concept in the context of more general relaxation processes toward the Wärmetod of thermodynamic equilibrium. More specifically, we sketch a construction of a conceptual ladder of chemical reaction steps that can rigorously bridge a description from the microscopic [...] Read more.
We address the scientific “time” concept in the context of more general relaxation processes toward the Wärmetod of thermodynamic equilibrium. More specifically, we sketch a construction of a conceptual ladder of chemical reaction steps that can rigorously bridge a description from the microscopic domain of molecular quantum chemistry to the macroscopic materials domain of Gibbsian thermodynamics. This conceptual reformulation follows the pioneering work of Kenichi Fukui (Nobel 1981) in rigorously formulating the intrinsic reaction coordinate (IRC) pathway for controlled description of non-equilibrium passages between reactant and product equilibrium states of an overall material transformation. Elementary chemical reaction steps are thereby identified as the logical building-blocks of an integrated mathematical framework that seamlessly spans the gulf between classical (pre-1925) and quantal (post-1925) scientific conceptions and encompasses both static and dynamic aspects of material change. All modern chemical reaction rate studies build on the apparent infallibility of quantum-chemical solutions of Schrödinger’s wave equation and its Dirac-type relativistic corrections. This infallibility may now be properly accepted as an added“inductive law” of Gibbsian chemical thermodynamics, the only component of 19th-century physics that passed intact through the revolutionary quantum upheavals of 1925. Full article
(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)
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10 pages, 448 KiB  
Article
Revisiting Endoreversible Carnot Engine: Extending the Yvon Engine
by Xiu-Hua Zhao and Yu-Han Ma
Entropy 2025, 27(2), 195; https://doi.org/10.3390/e27020195 - 13 Feb 2025
Cited by 1 | Viewed by 1030
Abstract
Curzon and Ahlborn’s 1975 paper, a pioneering work that inspired the birth of the field of finite-time thermodynamics, unveiled the efficiency at maximum power (EMP) of the endoreversible Carnot heat engine, now commonly referred to as the Curzon–Ahlborn (CA) engine. Historically, despite the [...] Read more.
Curzon and Ahlborn’s 1975 paper, a pioneering work that inspired the birth of the field of finite-time thermodynamics, unveiled the efficiency at maximum power (EMP) of the endoreversible Carnot heat engine, now commonly referred to as the Curzon–Ahlborn (CA) engine. Historically, despite the significance of the CA engine, similar findings had emerged at an earlier time, such as the Yvon engine proposed by J. Yvon in 1955 that shares the exact same EMP, that is, the CA efficiency ηCA. However, the special setup of the Yvon engine has circumscribed its broader influence. This paper extends the Yvon engine model to achieve a level of generality comparable to that of the CA engine. With the power expression of the extended Yvon engine, we directly explain the universality that ηCA is independent of the heat transfer coefficients between the working substance and the heat reservoirs. A rigorous comparison reveals that the extended Yvon engine and CA engine represent the steady-state and cyclic forms of the endoreversible Carnot heat engine, respectively, and are equivalent. Full article
(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)
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