{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T02:58:51Z","timestamp":1781578731812,"version":"3.54.5"},"reference-count":38,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2017,11,7]],"date-time":"2017-11-07T00:00:00Z","timestamp":1510012800000},"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>This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known, the system then obeys an equation, which in the bulk and in the suitable limit tends to the Kramers\u2013Langevin equation of physical kinetics. I consider time-dependent system-bath coupling and show that this leads to an additional harmonic force acting on the system. When the coupling is switched on and switched off rapidly, the force has delta-function support at the initial and final time. I further show that the work and heat functionals as recently defined in stochastic thermodynamics at strong coupling contain additional terms depending on the time derivative of the system-bath coupling. I discuss these terms and show that while they can be very large if the system-bath coupling changes quickly, they only give a finite contribution to the work that enters in Jarzynski\u2019s equality. I also discuss that these corrections to standard work and heat functionals provide an explanation for non-standard terms in the change of the von Neumann entropy of a quantum bath interacting with a quantum system found in an earlier contribution (Aurell and Eichhorn, 2015).<\/jats:p>","DOI":"10.3390\/e19110595","type":"journal-article","created":{"date-parts":[[2017,11,7]],"date-time":"2017-11-07T11:46:01Z","timestamp":1510055161000},"page":"595","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["On Work and Heat in Time-Dependent Strong Coupling"],"prefix":"10.3390","volume":"19","author":[{"given":"Erik","family":"Aurell","sequence":"first","affiliation":[{"name":"KTH\u2014Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden"},{"name":"Departments of Computer Science and Applied Physics, Aalto University, FIN-00076 Aalto, Finland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1103\/PhysRevLett.78.2690","article-title":"Nonequilibrium Equality for Free Energy Differences","volume":"78","author":"Jarzynski","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"040602","DOI":"10.1103\/PhysRevLett.95.040602","article-title":"Entropy Production along a Stochastic Trajectory and an Integral Fluctuation Theorem","volume":"95","author":"Seifert","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sekimoto, K. (2010). Stochastic Energetics, Springer.","DOI":"10.1007\/978-3-642-05411-2"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1146\/annurev.physchem.58.032806.104555","article-title":"Fluctuation Theorems","volume":"59","author":"Sevick","year":"2008","journal-title":"Annu. Rev. Phys. Chem."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1146\/annurev-conmatphys-062910-140506","article-title":"Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale","volume":"2","author":"Jarzynski","year":"2011","journal-title":"Annu. Rev. Condens. Matter Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"P126001","DOI":"10.1088\/0034-4885\/75\/12\/126001","article-title":"Stochastic thermodynamics, fluctuation theorems and molecular machines","volume":"75","author":"Seifert","year":"2012","journal-title":"Rep. Prog. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"P07006","DOI":"10.1088\/1742-5468\/2004\/07\/P07006","article-title":"A note on the Jarzynski equality","volume":"2004","author":"Cohen","year":"2004","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"P09005","DOI":"10.1088\/1742-5468\/2004\/09\/P09005","article-title":"Nonequilibrium work theorem for a system strongly coupled to a thermal environment","volume":"2004","author":"Jarzynski","year":"2004","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"235440","DOI":"10.1103\/PhysRevB.92.235440","article-title":"Nature of heat in strongly coupled open quantum systems","volume":"92","author":"Esposito","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"020601","DOI":"10.1103\/PhysRevLett.116.020601","article-title":"First and Second Law of Thermodynamics at Strong Coupling","volume":"116","author":"Seifert","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","first-page":"011008","article-title":"Stochastic and Macroscopic Thermodynamics of Strongly Coupled Systems","volume":"7","author":"Jarzynski","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"062123","DOI":"10.1103\/PhysRevE.95.062123","article-title":"Entropy production and time asymmetry in the presence of strong interactions","volume":"95","author":"Miller","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"022143","DOI":"10.1103\/PhysRevE.94.022143","article-title":"Open system trajectories specify fluctuating work but not heat","volume":"94","author":"Talkner","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"062101","DOI":"10.1103\/PhysRevE.95.062101","article-title":"Stochastic thermodynamics in the strong coupling regime: An unambiguous approach based on coarse-graining","volume":"95","author":"Strasberg","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"065007","DOI":"10.1088\/1367-2630\/17\/6\/065007","article-title":"On the von Neumann entropy of a bath linearly coupled to a driven quantum system","volume":"17","author":"Aurell","year":"2015","journal-title":"New J. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Breuer, H.P., and Petruccione, F. (2002). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1007\/3-540-44874-8_4"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"045030","DOI":"10.1088\/1367-2630\/17\/4\/045030","article-title":"Functional integral approach to time-dependent heat exchange in open quantum systems: General method and applications","volume":"17","author":"Carrega","year":"2015","journal-title":"New J. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"240403","DOI":"10.1103\/PhysRevLett.116.240403","article-title":"Energy Exchange in Driven Open Quantum Systems at Strong Coupling","volume":"116","author":"Carrega","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","unstructured":"Funo, K., and Quan, H.T. (arXiv, 2017). On the thermodynamic implications of path integral formalism of quantum mechanics, arXiv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1143\/PTPS.130.17","article-title":"Langevin equation and thermodynamics","volume":"180","author":"Sekimoto","year":"1998","journal-title":"Prog. Theor. Phys. Suppl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"051121","DOI":"10.1103\/PhysRevE.79.051121","article-title":"Thermodynamics of a subensemble of a canonical ensemble","volume":"79","author":"Gelin","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_23","unstructured":"Kurchan, J. (arXiv, 2000). A Quantum Fluctuation Theorem, arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1103\/RevModPhys.83.771","article-title":"TBD","volume":"83","author":"Campisi","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/0003-4916(63)90068-X","article-title":"The Theory of a General Quantum System Interacting with a Linear Dissipative System","volume":"24","author":"Feynman","year":"1963","journal-title":"Ann. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/0378-4371(83)90013-4","article-title":"Path Integral Approach to Quantum Brownian Motion","volume":"121","author":"Caldeira","year":"1983","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/0370-1573(88)90023-3","article-title":"Quantum Brownian Motion: The Functional Intergral Approach","volume":"168","author":"Grabert","year":"1988","journal-title":"Phys. Rep."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"022107","DOI":"10.1103\/PhysRevA.61.022107","article-title":"Exact diagonalization of two quantum models for the damped harmonic oscillator","volume":"61","author":"Caldeira","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1038\/nature10872","article-title":"Experimental verification of Landauer\u2019s principle linking information and thermodynamics","volume":"483","author":"Arakelyan","year":"2012","journal-title":"Nature"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/s10955-012-0478-x","article-title":"Refined Second Law of Thermodynamics for fast random processes","volume":"147","author":"Aurell","year":"2012","journal-title":"J. Stat. Phys."},{"key":"ref_31","first-page":"031038","article-title":"Quantum-Classical Correspondence Principle for Work Distributions","volume":"5","author":"Jarzynski","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/BF01008729","article-title":"Nonlinear generalized Langevin equations","volume":"9","author":"Zwanzig","year":"1973","journal-title":"J. Stat. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/BF01305831","article-title":"Microscopic preparation and macroscopic motion of a Brownian particle","volume":"39","author":"Bez","year":"1980","journal-title":"Z. Phys. B Condens. Matter"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0378-4371(94)90146-5","article-title":"Translational symmetry and microscopic preparation in oscillator models of quantum dissipation","volume":"212","author":"Sols","year":"1994","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_35","unstructured":"H\u00e4nggi, P. (1997). Generalized Langevin Equations: A Useful Tool for the Perplexed Modeller of Nonequilibrium Fluctuations?, Springer."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"026105","DOI":"10.1063\/1.1853631","article-title":"Fundamental aspects of quantum Brownian motion","volume":"15","author":"Ingold","year":"2005","journal-title":"Chaos"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"160401","DOI":"10.1103\/PhysRevLett.111.160401","article-title":"Past Quantum States of a Monitored System","volume":"111","author":"Gammelmark","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"062116","DOI":"10.1103\/PhysRevA.91.062116","article-title":"Forward-backward analysis of the photon-number evolution in a cavity","volume":"91","author":"Rybarczyk","year":"2015","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/11\/595\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:48:32Z","timestamp":1760208512000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/11\/595"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,7]]},"references-count":38,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2017,11]]}},"alternative-id":["e19110595"],"URL":"https:\/\/doi.org\/10.3390\/e19110595","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,11,7]]}}}