{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T05:40:07Z","timestamp":1768974007948,"version":"3.49.0"},"reference-count":123,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,15]],"date-time":"2019-10-15T00:00:00Z","timestamp":1571097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the thermodynamic properties of a superconductor\/normal metal\/superconductor Josephson junction in the short limit. Owing to the proximity effect, such a junction constitutes a thermodynamic system where phase difference, supercurrent, temperature and entropy are thermodynamical variables connected by equations of state. These allow conceiving quasi-static processes that we characterize in terms of heat and work exchanged. Finally, we combine such processes to construct a Josephson-based Otto and Stirling cycles. We study the related performance in both engine and refrigerator operating mode.<\/jats:p>","DOI":"10.3390\/e21101005","type":"journal-article","created":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T03:32:54Z","timestamp":1571196774000},"page":"1005","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Thermodynamics of a Phase-Driven Proximity Josephson Junction"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6791-215X","authenticated-orcid":false,"given":"Francesco","family":"Vischi","sequence":"first","affiliation":[{"name":"NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza S. Silvestro 12, I-56127 Pisa, Italy"},{"name":"Dipartimento di Fisica \u201cE. Fermi\u201d, Universit\u00e0 di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matteo","family":"Carrega","sequence":"additional","affiliation":[{"name":"NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza S. Silvestro 12, I-56127 Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2119-1160","authenticated-orcid":false,"given":"Alessandro","family":"Braggio","sequence":"additional","affiliation":[{"name":"NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza S. Silvestro 12, I-56127 Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pauli","family":"Virtanen","sequence":"additional","affiliation":[{"name":"Department of Physics and Nanoscience Center, University of Jyv\u00e4skyl\u00e4, P.O. Box 35 (YFL), FI-40014 Jyv\u00e4skyl\u00e4, Finland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Giazotto","sequence":"additional","affiliation":[{"name":"NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza S. Silvestro 12, I-56127 Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"041003","DOI":"10.1103\/RevModPhys.89.041003","article-title":"Colloquium: Quantum coherence as a resource","volume":"89","author":"Streltsov","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1080\/00107514.2016.1201896","article-title":"Quantum thermodynamics","volume":"57","author":"Vinjanampathy","year":"2016","journal-title":"Contemp. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1038\/nphys3169","article-title":"The role of quantum information in thermodynamics\u2014A topical review","volume":"11","author":"Pekola","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2100","DOI":"10.3390\/e15062100","article-title":"Quantum Thermodynamics: A Dynamical Viewpoint","volume":"15","author":"Kosloff","year":"2013","journal-title":"Entropy"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kosloff, R., and Rezek, Y. (2017). The Quantum Harmonic Otto Cycle. Entropy, 19.","DOI":"10.3390\/e19040136"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1103\/RevModPhys.80.1215","article-title":"Theory of ultracold atomic Fermi gases","volume":"80","author":"Giorgini","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"143001","DOI":"10.1088\/1751-8113\/49\/14\/143001","article-title":"The role of quantum information in thermodynamics\u2014A topical review","volume":"49","author":"Goold","year":"2016","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1800222","DOI":"10.1002\/pssr.201800222","article-title":"Toward Quantum Hall Effect in a Josephson Junction","volume":"13","author":"Guiducci","year":"2019","journal-title":"Phys. Status Solidi (RRL) Rapid Res. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"235419","DOI":"10.1103\/PhysRevB.99.235419","article-title":"Full electrostatic control of quantum interference in an extended trenched Josephson junction","volume":"99","author":"Guiducci","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"197001","DOI":"10.1103\/PhysRevLett.97.197001","article-title":"Cooling Electrons by Magnetic-Field Tuning of Andreev Reflection","volume":"97","author":"Giazotto","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"944","DOI":"10.1038\/nnano.2017.204","article-title":"Towards phase-coherent caloritronics in superconducting circuits","volume":"12","author":"Fornieri","year":"2017","journal-title":"Nat. Nanotechnol."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hofmann, R. (2016). SU(2) Yang\u2013Mills Theory: Waves, Particles, and Quantum Thermodynamics. Entropy, 18.","DOI":"10.3390\/e18090310"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1103\/RevModPhys.83.771","article-title":"Colloquium: Quantum fluctuation relations: Foundations and applications","volume":"83","author":"Campisi","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.1103\/RevModPhys.81.1665","article-title":"Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems","volume":"81","author":"Esposito","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"013105","DOI":"10.1063\/1.5010236","article-title":"Fast thermometry with a proximity Josephson junction","volume":"112","author":"Wang","year":"2018","journal-title":"Appl. Phys. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"044068","DOI":"10.1103\/PhysRevApplied.10.044068","article-title":"Nanosecond Thermometry with Josephson Junctions","volume":"10","author":"Zgirski","year":"2018","journal-title":"Phys. Rev. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"014007","DOI":"10.1103\/PhysRevApplied.3.014007","article-title":"Fast Electron Thermometry for Ultrasensitive Calorimetric Detection","volume":"3","author":"Gasparinetti","year":"2015","journal-title":"Phys. Rev. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1103\/RevModPhys.78.217","article-title":"Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications","volume":"78","author":"Giazotto","year":"2006","journal-title":"Rev. Mod. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"044016","DOI":"10.1103\/PhysRevApplied.4.044016","article-title":"Ferromagnetic-Insulator-Based Superconducting Junctions as Sensitive Electron Thermometers","volume":"4","author":"Giazotto","year":"2015","journal-title":"Phys. Rev. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"030802","DOI":"10.1103\/PhysRevLett.117.030802","article-title":"Detection of Zeptojoule Microwave Pulses Using Electrothermal Feedback in Proximity-Induced Josephson Junctions","volume":"117","author":"Govenius","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Lee, G.H., Efetov, D.K., Ranzani, L., Walsh, E.D., Ohki, T.A., Taniguchi, T., Watanabe, K., Kim, P., Englund, D., and Chung Fong, K. (2019). Graphene-based Josephson junction microwave bolometer. arXiv.","DOI":"10.1038\/s41586-020-2752-4"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"162507","DOI":"10.1063\/1.2908922","article-title":"Ultrasensitive proximity Josephson sensor with kinetic inductance readout","volume":"92","author":"Giazotto","year":"2008","journal-title":"Appl. Phys. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"054074","DOI":"10.1103\/PhysRevApplied.11.054074","article-title":"Josephson-Threshold Calorimeter","volume":"11","author":"Guarcello","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Vischi, F., Carrega, M., Braggio, A., Paolucci, F., Bianco, F., Roddaro, S., and Giazotto, F. (2019). Electron cooling with graphene-insulator-superconductor tunnel junctions and applications to fast bolometry. arXiv.","DOI":"10.1103\/PhysRevApplied.13.054006"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"024015","DOI":"10.1103\/PhysRevApplied.10.024015","article-title":"Proximity SQUID Single-Photon Detector via Temperature-to-Voltage Conversion","volume":"10","author":"Solinas","year":"2018","journal-title":"Phys. Rev. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"054027","DOI":"10.1103\/PhysRevApplied.9.054027","article-title":"Josephson Photodetectors via Temperature-to-Phase Conversion","volume":"9","author":"Virtanen","year":"2018","journal-title":"Phys. Rev. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1038\/nnano.2014.215","article-title":"Photodetectors based on graphene, other two-dimensional materials and hybrid systems","volume":"9","author":"Koppens","year":"2014","journal-title":"Nat. Nanotechnol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"44240","DOI":"10.1038\/srep44240","article-title":"Near-field terahertz probes with room-temperature nanodetectors for subwavelength resolution imaging","volume":"7","author":"Mitrofanov","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1364\/OPTICA.5.000651","article-title":"Phase-sensitive terahertz imaging using room-temperature near-field nanodetectors","volume":"5","author":"Giordano","year":"2018","journal-title":"Optica"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"164203","DOI":"10.1088\/0953-8984\/27\/16\/164203","article-title":"Ultrasensitive graphene far-infrared power detectors","volume":"27","author":"McKitterick","year":"2015","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"075410","DOI":"10.1103\/PhysRevB.93.075410","article-title":"Electron-phonon cooling in large monolayer graphene devices","volume":"93","author":"McKitterick","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1038\/nnano.2008.199","article-title":"Approaching ballistic transport in suspended graphene","volume":"3","author":"Du","year":"2008","journal-title":"Nat. Nanotechnol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"023056","DOI":"10.1088\/1367-2630\/aa60d4","article-title":"High-efficiency thermal switch based on topological Josephson junctions","volume":"19","author":"Sothmann","year":"2017","journal-title":"New J. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"045418","DOI":"10.1103\/PhysRevB.100.045418","article-title":"Thermal transistor and thermometer based on Coulomb-coupled conductors","volume":"100","author":"Yang","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"092002","DOI":"10.1088\/1367-2630\/aa7641","article-title":"Conduction channels of an InAs-Al nanowire Josephson weak link","volume":"19","author":"Goffman","year":"2017","journal-title":"New J. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"077701","DOI":"10.1103\/PhysRevLett.119.077701","article-title":"Thermal Conductance of a Single-Electron Transistor","volume":"119","author":"Dutta","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1103\/RevModPhys.84.1045","article-title":"Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond","volume":"84","author":"Li","year":"2012","journal-title":"Rev. Mod. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"200601","DOI":"10.1103\/PhysRevLett.116.200601","article-title":"Quantum Thermal Transistor","volume":"116","author":"Joulain","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1038\/s41567-018-0199-4","article-title":"Tunable photonic heat transport in a quantum heat valve","volume":"14","author":"Ronzani","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1038\/nphys4280","article-title":"Heat Coulomb blockade of one ballistic channel","volume":"14","author":"Sivre","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"082601","DOI":"10.1063\/1.4893759","article-title":"Proximity nanovalve with large phase-tunable thermal conductance","volume":"105","author":"Strambini","year":"2014","journal-title":"Appl. Phys. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1038\/nature11702","article-title":"The Josephson heat interferometer","volume":"492","author":"Giazotto","year":"2012","journal-title":"Nature"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"044073","DOI":"10.1103\/PhysRevApplied.11.044073","article-title":"Phase-Tunable Thermal Rectification in the Topological SQUIPT","volume":"11","author":"Bours","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"177208","DOI":"10.1103\/PhysRevLett.99.177208","article-title":"Thermal Logic Gates: Computation with Phonons","volume":"99","author":"Wang","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"024003","DOI":"10.1103\/PhysRevApplied.10.024003","article-title":"Phase-Tunable Thermal Logic: Computation with Heat","volume":"10","author":"Paolucci","year":"2018","journal-title":"Phys. Rev. Appl."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Hwang, S.Y., and Sothmann, B. (2019). Phase-coherent caloritronics with ordinary and topological Josephson junctions. arXiv.","DOI":"10.1140\/epjst\/e2019-900094-y"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"214508","DOI":"10.1103\/PhysRevB.99.214508","article-title":"Phase-dependent heat transport in Josephson junctions with p-wave superconductors and superfluids","volume":"99","author":"Bauer","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1764","DOI":"10.1021\/acs.nanolett.7b04906","article-title":"Phase-Tunable Josephson Thermal Router","volume":"18","author":"Timossi","year":"2018","journal-title":"Nano Lett."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"046501","DOI":"10.1088\/0034-4885\/75\/4\/046501","article-title":"Micrometre-scale refrigerators","volume":"75","author":"Muhonen","year":"2012","journal-title":"Rep. Prog. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1996","DOI":"10.1063\/1.115651","article-title":"Efficient Peltier refrigeration by a pair of normal metal\/insulator\/superconductor junctions","volume":"68","author":"Leivo","year":"1996","journal-title":"Appl. Phys. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1016\/j.crhy.2016.08.010","article-title":"High-performance electronic cooling with superconducting tunnel junctions","volume":"17","author":"Courtois","year":"2016","journal-title":"C. R. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"054011","DOI":"10.1103\/PhysRevApplied.6.054011","article-title":"Cascade Electronic Refrigerator Using Superconducting Tunnel Junctions","volume":"6","author":"Nguyen","year":"2016","journal-title":"Phys. Rev. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"224521","DOI":"10.1103\/PhysRevB.93.224521","article-title":"Microwave quantum refrigeration based on the Josephson effect","volume":"93","author":"Solinas","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"165502","DOI":"10.1103\/PhysRevLett.102.165502","article-title":"Phonon Cooling of Nanomechanical Beams with Tunnel Junctions","volume":"102","author":"Koppinen","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"245304","DOI":"10.1103\/PhysRevB.99.245304","article-title":"Nonlinear chiral refrigerators","volume":"99","author":"Sothmann","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"184503","DOI":"10.1103\/PhysRevB.94.184503","article-title":"Otto refrigerator based on a superconducting qubit: Classical and quantum performance","volume":"94","author":"Karimi","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"054014","DOI":"10.1103\/PhysRevApplied.6.054014","article-title":"Self-Oscillating Josephson Quantum Heat Engine","volume":"6","author":"Marchegiani","year":"2016","journal-title":"Phys. Rev. Appl."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"3238","DOI":"10.1038\/s41598-019-40202-8","article-title":"Thermodynamic cycles in Josephson junctions","volume":"9","author":"Vischi","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Haack, G., and Giazotto, F. (2019). Efficient and tunable Aharonov-Bohm quantum heat engine. arXiv.","DOI":"10.1103\/PhysRevB.100.235442"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"054034","DOI":"10.1103\/PhysRevApplied.11.054034","article-title":"Superconducting Quantum Refrigerator: Breaking and Rejoining Cooper Pairs with Magnetic Field Cycles","volume":"11","author":"Manikandan","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"062111","DOI":"10.1103\/PhysRevA.99.062111","article-title":"Optimal work-to-work conversion of a nonlinear quantum Brownian duet","volume":"99","author":"Carrega","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.05.008","article-title":"Fundamental aspects of steady-state conversion of heat to work at the nanoscale","volume":"694","author":"Benenti","year":"2017","journal-title":"Phys. Rep."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"100540","DOI":"10.1016\/j.progsurf.2019.100540","article-title":"Thermal, electric and spin transport in superconductor\/ferromagnetic-insulator structures","volume":"94","author":"Silaev","year":"2019","journal-title":"Prog. Surf. Sci."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"48005","DOI":"10.1209\/0295-5075\/124\/48005","article-title":"On-chip cooling by heating with superconducting tunnel junctions","volume":"124","author":"Marchegiani","year":"2018","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1060","DOI":"10.1016\/j.crhy.2016.08.005","article-title":"Nonlinear phenomena in quantum thermoelectrics and heat","volume":"17","year":"2016","journal-title":"C. R. Phys."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"075429","DOI":"10.1103\/PhysRevB.99.075429","article-title":"Nonlocal thermoelectricity in a Cooper-pair splitter","volume":"99","author":"Hussein","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Marchegiani, G., Braggio, A., and Giazotto, F. (2019). Nonlinear thermoelectricity with particle-hole symmetry in transport. arXiv.","DOI":"10.1103\/PhysRevLett.124.106801"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1023\/A:1004635226825","article-title":"Andreev Reflection and Proximity effect","volume":"118","author":"Pannetier","year":"2000","journal-title":"J. Low Temp. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1021885617107","article-title":"The Spectral Conductance of a Proximity Superconductor and the Reentrance Effect","volume":"116","author":"Courtois","year":"1999","journal-title":"J. Low Temp. Phys."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1103\/RevModPhys.51.101","article-title":"Superconducting weak links","volume":"51","author":"Likharev","year":"1979","journal-title":"Rev. Mod. Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/0921-4526(88)90016-6","article-title":"On the thermodynamics of a superconducting ring interrupted by a weak superconducting junction","volume":"154","year":"1988","journal-title":"Phys. B Condens. Matter"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1016\/0921-4526(89)90003-3","article-title":"On the thermodynamics of a superconducting ring interrupted by a weak superconducting junction: II (continuation)","volume":"159","author":"Ouboter","year":"1989","journal-title":"Phys. B Condens. Matter"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/S0378-4371(96)00424-4","article-title":"Superconducting correlations and the thermodynamics of Josephson junctions","volume":"237","author":"Dekker","year":"1997","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"R8697","DOI":"10.1103\/PhysRevB.55.R8697","article-title":"Josephson-junction thermodynamics and the superconducting phase transitionin a SQUID device","volume":"55","author":"Dekker","year":"1997","journal-title":"Phys. Rev. B"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/S0921-4526(97)00345-1","article-title":"Periodic oscillations of the critical temperature versus the applied magnetic flux in a four-terminal SQUID","volume":"239","author":"Vleeming","year":"1997","journal-title":"Phys. B Condens. Matter"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1103\/PhysRevB.5.861","article-title":"Thermodynamics of the Proximity Effect: Specific-Heat Jumps in Lamellar Lead-Tin Eutectic Alloys","volume":"5","author":"Lechevet","year":"1972","journal-title":"Phys. Rev. B"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1103\/PhysRevB.14.78","article-title":"Specific-heat jump of superconducting lamellas with pair-breaking boundary conditions","volume":"14","author":"Manuel","year":"1976","journal-title":"Phys. Rev. B"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"5729","DOI":"10.1103\/PhysRevB.25.5729","article-title":"Heat capacity and density of states of a normal-metal\u2014Superconductor bilayer","volume":"25","author":"Zaitlin","year":"1982","journal-title":"Phys. Rev. B"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"11268","DOI":"10.1103\/PhysRevB.38.11268","article-title":"Free-energy calculations in a self-consistent model of the proximity effect","volume":"38","author":"Kobes","year":"1988","journal-title":"Phys. Rev. B"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"9365","DOI":"10.1103\/PhysRevB.58.9365","article-title":"Free energy of an inhomogeneous superconductor: A wave-function approach","volume":"58","author":"Kosztin","year":"1998","journal-title":"Phys. Rev. B"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF00120865","article-title":"Bogoliubov equations, boundary conditions, and the free energy of inhomogeneous superconductors: A new method","volume":"20","author":"Eilenberger","year":"1975","journal-title":"J. Low Temp. Phys."},{"key":"ref_82","first-page":"96","article-title":"Contribution to the microscopic theory of the Josephson effect in superconducting bridges","volume":"21","author":"Kulik","year":"1975","journal-title":"JETP Lett."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1103\/RevModPhys.76.411","article-title":"The current-phase relation in Josephson junctions","volume":"76","author":"Golubov","year":"2004","journal-title":"Rev. Mod. Phys."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"245311","DOI":"10.1103\/PhysRevB.96.245311","article-title":"Quasiparticle entropy in superconductor\/normal metal\/ superconductor proximity junctions in the diffusive limit","volume":"96","author":"Virtanen","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"064514","DOI":"10.1103\/PhysRevB.76.064514","article-title":"Density of states and supercurrent in diffusive SNS junctions: Roles of nonideal interfaces and spin-flip scattering","volume":"76","author":"Hammer","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1103\/PhysRev.111.132","article-title":"Superconducting Transition in Aluminum","volume":"111","author":"Cochran","year":"1958","journal-title":"Phys. Rev."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1103\/PhysRev.122.1101","article-title":"Study of Superconductors by Electron Tunneling","volume":"122","author":"Giaever","year":"1961","journal-title":"Phys. Rev."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1109\/PROC.1966.4776","article-title":"Josephson-type superconducting tunnel junctions as generators of microwave and submillimeter wave radiation","volume":"54","author":"Langenberg","year":"1966","journal-title":"IEEE Proc."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"113110","DOI":"10.1063\/1.4930934","article-title":"Normal metal tunnel junction-based superconducting quantum interference proximity transistor","volume":"107","author":"Meissner","year":"2015","journal-title":"Appl. Phys. Lett."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1038\/nphys1537","article-title":"Superconducting quantum interference proximity transistor","volume":"6","author":"Giazotto","year":"2010","journal-title":"Nat. Phys."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"024005","DOI":"10.1103\/PhysRevApplied.2.024005","article-title":"Highly Sensitive Superconducting Quantum-Interference Proximity Transistor","volume":"2","author":"Ronzani","year":"2014","journal-title":"Phys. Rev. Appl."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1166\/jctn.2005.201","article-title":"Properties of Mesoscopic hybrid Superconducting Systems","volume":"2","author":"Taddei","year":"2005","journal-title":"J. Comp. Theor. Nanosci."},{"key":"ref_93","unstructured":"Kulik, I., and Ellialtiogammalu, R. (2012). Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, Springer."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"1251","DOI":"10.1006\/spmi.1999.0710","article-title":"Quasiclassical Green\u2019s function approach to mesoscopic superconductivity","volume":"25","author":"Belzig","year":"1999","journal-title":"Superlattices Microstruct."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"1420","DOI":"10.1103\/PhysRevLett.73.1420","article-title":"Circuit Theory of Andreev Conductance","volume":"73","author":"Nazarov","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"1221","DOI":"10.1006\/spmi.1999.0738","article-title":"Novel circuit theory of Andreev reflection","volume":"25","author":"Nazarov","year":"1999","journal-title":"Superlattices Microstruct."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1038\/nphys4070","article-title":"Signatures of interaction-induced helical gaps in nanowire quantum point contacts","volume":"13","author":"Heedt","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1023\/A:1022628927203","article-title":"Density of States in Superconductor-Normal Metal-Superconductor Junctions","volume":"110","author":"Zhou","year":"1998","journal-title":"J. Low. Temp. Phys."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1007\/s10909-008-9826-2","article-title":"The Vortex State and Josephson Critical Current of a Diffusive SNS Junction","volume":"153","author":"Bergeret","year":"2008","journal-title":"J. Low Temp. Phys."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"054504","DOI":"10.1103\/PhysRevB.95.054504","article-title":"Coherent transport properties of a three-terminal hybrid superconducting interferometer","volume":"95","author":"Vischi","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"184513","DOI":"10.1103\/PhysRevB.66.184513","article-title":"Supercurrent-carrying density of states in diffusive mesoscopic Josephson weak links","volume":"66","author":"Wilhelm","year":"2002","journal-title":"Phys. Rev. B"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"214502","DOI":"10.1103\/PhysRevB.84.214502","article-title":"Hybrid superconducting quantum magnetometer","volume":"84","author":"Giazotto","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"197002","DOI":"10.1103\/PhysRevLett.100.197002","article-title":"Phase Controlled Superconducting Proximity Effect Probed by Tunneling Spectroscopy","volume":"100","author":"Joyez","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_104","unstructured":"Grosso, G., and Parravicini, G. (2000). Solid State Physics, Elsevier Science."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"012503","DOI":"10.1103\/PhysRevB.78.012503","article-title":"Phase-dependent electronic specific heat of mesoscopic Josephson junctions","volume":"78","author":"Rabani","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"093904","DOI":"10.1063\/1.3121210","article-title":"Influence of interface transmissivity and inelastic scattering on the electronic entropy and specific heat of diffusive superconductor-normal metal-superconductor Josephson junctions","volume":"105","author":"Rabani","year":"2009","journal-title":"J. Appl. Phys."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1103\/PhysRev.108.1175","article-title":"Theory of Superconductivity","volume":"108","author":"Bardeen","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"9545","DOI":"10.1103\/PhysRevB.59.9545","article-title":"Gradient expansion for the free energy of a clean superconductor","volume":"59","author":"Kos","year":"1999","journal-title":"Phys. Rev. B"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"100501","DOI":"10.1103\/PhysRevB.77.100501","article-title":"Quantitative study of quasiparticle traps using the single-Cooper-pair transistor","volume":"77","author":"Court","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_110","unstructured":"de Gennes, P.G. (1999). Superconductivity of Metals and Alloys, Advanced Book Program, Perseus Books."},{"key":"ref_111","unstructured":"Abrikosov, A., Gorkov, L., and Dzyaloshinski, I. (1975). Methods of Quantum Field Theory in Statistical Physics, Dover Publications."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1119\/1.4945266","article-title":"Not all counterclockwise thermodynamic cycles are refrigerators","volume":"84","author":"Dickerson","year":"2016","journal-title":"Am. J. Phys."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1119\/1.5005928","article-title":"Comment on \u201cNot all counterclockwise thermodynamic cycles are refrigerators\u201d [Am. J. Phys. 84, 413\u2013418 (2016)]","volume":"85","author":"Bizarro","year":"2017","journal-title":"Am. J. Phys."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"055101","DOI":"10.1088\/1361-6404\/aa776d","article-title":"Coefficient of performance of Stirling refrigerators","volume":"38","author":"Mungan","year":"2017","journal-title":"Eur. J. Phys."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1088\/0031-9120\/29\/3\/013","article-title":"Demonstration experiments with a Stirling engine","volume":"29","author":"Deacon","year":"1994","journal-title":"Phys. Educ."},{"key":"ref_116","first-page":"2","article-title":"The natural heat engine","volume":"14","author":"Wheatley","year":"1986","journal-title":"Los Alamos Sci."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"134502","DOI":"10.1103\/PhysRevB.80.134502","article-title":"Nonequilibrium electrons in tunnel structures under high-voltage injection","volume":"80","author":"Kopnin","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"257002","DOI":"10.1103\/PhysRevLett.100.257002","article-title":"Quasiparticle Relaxation in Optically Excited High-Q Superconducting Resonators","volume":"100","author":"Barends","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"3695","DOI":"10.1063\/1.356060","article-title":"Broadband ultrafast superconducting NbN detector for electromagnetic radiation","volume":"75","author":"Gousev","year":"1994","journal-title":"J. Appl. Phys."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"6041","DOI":"10.1103\/PhysRevB.61.6041","article-title":"Electron-phonon interaction in disordered conductors: Static and vibrating scattering potentials","volume":"61","author":"Sergeev","year":"2000","journal-title":"Phys. Rev. B"},{"key":"ref_121","doi-asserted-by":"crossref","unstructured":"Kopnin, N.B. (2001). Theory of Nonequilibrium Superconductivity, Clarendon Press.","DOI":"10.1093\/acprof:oso\/9780198507888.001.0001"},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"017003","DOI":"10.1103\/PhysRevLett.102.017003","article-title":"Recombination-Limited Energy Relaxation in a Bardeen-Cooper-Schrieffer Superconductor","volume":"102","author":"Timofeev","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"147001","DOI":"10.1103\/PhysRevLett.111.147001","article-title":"Excitation of Single Quasiparticles in a Small Superconducting Al Island Connected to Normal-Metal Leads by Tunnel Junctions","volume":"111","author":"Maisi","year":"2013","journal-title":"Phys. Rev. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/10\/1005\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:26:34Z","timestamp":1760189194000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/10\/1005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,15]]},"references-count":123,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,10]]}},"alternative-id":["e21101005"],"URL":"https:\/\/doi.org\/10.3390\/e21101005","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,15]]}}}