{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T09:00:20Z","timestamp":1789981220307,"version":"4.0.1"},"reference-count":29,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,9,20]],"date-time":"2026-09-20T00:00:00Z","timestamp":1789862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Department of Physics at UMBC"},{"DOI":"10.13039\/100000925","name":"John Templeton Foundation","doi-asserted-by":"publisher","award":["63626"],"award-info":[{"award-number":["63626"]}],"id":[{"id":"10.13039\/100000925","id-type":"DOI","asserted-by":"publisher"}]},{"award":["63626"],"award-info":[{"award-number":["63626"]}],"id":[{"id":"https:\/\/ror.org\/035tnyy05","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Given a particular heat engine cycle, what is the optimal working medium that results in the highest efficiency? While one might jump to the conclusion that it must surely be the ideal gas, the situation is actually more intricate. Starting with a general Helmholtz potential that depends polynomially on molar volume and temperature, we derive exact expressions for the ideal Stirling, Otto, and Brayton cycles. We find that for the thermodynamic systems described by our ansatz for the Helmholtz potential, the maximal efficiency is achieved if the working medium is described by a fundamental relation linear in temperature. This includes the ideal gas, but also classical harmonic oscillators and phenomenological models of the rubber band.<\/jats:p>","DOI":"10.3390\/e28091033","type":"journal-article","created":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T08:03:47Z","timestamp":1789977827000},"page":"1033","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Ideal Heat Engine Cycles at Maximal Efficiency\u2014The Ideal Gas and Beyond"],"prefix":"10.3390","volume":"28","author":[{"given":"Gregory","family":"Behrendt","sequence":"first","affiliation":[{"name":"Department of Physics, University of Maryland, College Park, MD 20742, USA"},{"name":"Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0504-6932","authenticated-orcid":false,"given":"Sebastian","family":"Deffner","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA"},{"name":"Quantum Science Institute, University of Maryland, Baltimore County, Baltimore, MD 21250, USA"},{"name":"National Quantum Laboratory, College Park, MD 20740, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,20]]},"reference":[{"key":"ref_1","unstructured":"Callen, H. (1985). Thermodynamics and an Introduction to Thermostastistics, Wiley."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Deffner, S. (2018). Efficiency of Harmonic Quantum Otto Engines at Maximal Power. Entropy, 20.","DOI":"10.3390\/e20110875"},{"key":"ref_3","first-page":"311","article-title":"Endoreversible thermodynamics","volume":"22","author":"Hoffmann","year":"1997","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1119\/1.10023","article-title":"Efficiency of a Carnot engine at maximum power output","volume":"43","author":"Curzon","year":"1975","journal-title":"Am. J. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1119\/1.15071","article-title":"Thermal efficiency at maximum work output: New results for old heat engines","volume":"55","author":"Leff","year":"1987","journal-title":"Am. J. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1016\/S0360-5442(96)00159-4","article-title":"Analysis of the Stirling heat engine at maximum power conditions","volume":"22","author":"Erbay","year":"1997","journal-title":"Energy"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/0360-5442(94)90111-2","article-title":"Power optimization of an endoreversible stirling cycle with regeneration","volume":"19","author":"Blank","year":"1994","journal-title":"Energy"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1016\/S0360-5442(00)00023-2","article-title":"Finite time thermodynamic analysis of endoreversible Stirling heat engine with regenerative losses","volume":"25","author":"Kaushik","year":"2000","journal-title":"Energy"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1088\/1367-2630\/8\/5\/083","article-title":"Irreversible performance of a quantum harmonic heat engine","volume":"8","author":"Rezek","year":"2006","journal-title":"New J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"150603","DOI":"10.1103\/PhysRevLett.105.150603","article-title":"Efficiency at Maximum Power of Low-Dissipation Carnot Engines","volume":"105","author":"Esposito","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"203006","DOI":"10.1103\/PhysRevLett.109.203006","article-title":"Single-Ion Heat Engine at Maximum Power","volume":"109","author":"Abah","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"030602","DOI":"10.1103\/PhysRevLett.112.030602","article-title":"Nanoscale Heat Engine Beyond the Carnot Limit","volume":"112","author":"Abah","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"123203","DOI":"10.1088\/1742-5468\/ab4e92","article-title":"Approaching Carnot efficiency at maximum power in linear response regime","volume":"2019","year":"2019","journal-title":"J. Stat. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"19001","DOI":"10.1209\/0295-5075\/acad9c","article-title":"Boosting thermodynamic performance by bending space-time","volume":"141","author":"Ferketic","year":"2023","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Behrendt, G., and Deffner, S. (2025). Endoreversible Stirling Cycles: Plasma Engines at Maximal Power. Entropy, 27.","DOI":"10.3390\/e27080807"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"042126","DOI":"10.1103\/PhysRevE.100.042126","article-title":"Collective performance of a finite-time quantum Otto cycle","volume":"100","author":"Kloc","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"012110","DOI":"10.1103\/PhysRevE.101.012110","article-title":"Bosons outperform fermions: The thermodynamic advantage of symmetry","volume":"101","author":"Myers","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Myers, N.M., McCready, J., and Deffner, S. (2021). Quantum Heat Engines with Singular Interactions. Symmetry, 13.","DOI":"10.3390\/sym13060978"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"105001","DOI":"10.1088\/1367-2630\/ac2756","article-title":"Quantum Otto engines at relativistic energies","volume":"23","author":"Myers","year":"2021","journal-title":"New J. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"040312","DOI":"10.1103\/PRXQuantum.2.040312","article-title":"Thermodynamics of Statistical Anyons","volume":"2","author":"Myers","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"025001","DOI":"10.1088\/1367-2630\/ac47cc","article-title":"Boosting engine performance with Bose\u2013Einstein condensation","volume":"24","author":"Myers","year":"2022","journal-title":"New J. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Myers, N.M., Pe\u00f1a, F.J., Cort\u00e9s, N., and Vargas, P. (2023). Multilayer Graphene as an Endoreversible Otto Engine. Nanomaterials, 13.","DOI":"10.20944\/preprints202304.0113.v1"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Pe\u00f1a, F.J., Myers, N.M., \u00d3rdenes, D., Albarr\u00e1n-Arriagada, F., and Vargas, P. (2023). Enhanced Efficiency at Maximum Power in a Fock\u2013Darwin Model Quantum Dot Engine. Entropy, 25.","DOI":"10.20944\/preprints202302.0490.v1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1515\/jnet-2020-0039","article-title":"Endoreversible Otto Engines at Maximal Power","volume":"45","author":"Smith","year":"2020","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_25","unstructured":"Borgnakke, C. (2025). Fundamentals of Thermodynamics, John Wiley & Sons."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Patel, R.C., Bass, D.C., Dukuze, G.P., Andrade, A., and Combs, C.S. (2022). Analysis and Development of a Small-Scale Supercritical Carbon Dioxide (sCO2) Brayton Cycle. Energies, 15.","DOI":"10.3390\/en15103580"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"128586","DOI":"10.1016\/j.applthermaleng.2025.128586","article-title":"A technology review of pumped thermal energy storage based on CO2 cycles","volume":"281","author":"Chai","year":"2025","journal-title":"Appl. Therm. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2087","DOI":"10.1103\/PhysRevA.21.2087","article-title":"Improved equation of state for the classical one-component plasma","volume":"21","author":"Slattery","year":"1980","journal-title":"Phys. Rev. A"},{"key":"ref_29","unstructured":"Abramowitz, M., and Stegun, I.A. (1948). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/1033\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T08:05:09Z","timestamp":1789977909000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/1033"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,20]]},"references-count":29,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["e28091033"],"URL":"https:\/\/doi.org\/10.3390\/e28091033","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,20]]}}}