{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T21:24:06Z","timestamp":1775683446288,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2017,8,23]],"date-time":"2017-08-23T00:00:00Z","timestamp":1503446400000},"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>A stochastic thermodynamics of Brownian motion is set up in which state functions are expressed in terms of state variables through the same relations as in classical irreversible thermodynamics, with the difference that the state variables are now random fields accounting for the effect of fluctuations. Explicit expressions for the stochastic analog of entropy production and related quantities are derived for a dilute solution of Brownian particles in a fluid of light particles. Their statistical properties are analyzed and, in the light of the insights afforded, the thermodynamics of a single Brownian particle is revisited and the status of the second law of thermodynamics is discussed.<\/jats:p>","DOI":"10.3390\/e19090434","type":"journal-article","created":{"date-parts":[[2017,8,23]],"date-time":"2017-08-23T11:32:27Z","timestamp":1503487947000},"page":"434","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Stochastic Thermodynamics of Brownian Motion"],"prefix":"10.3390","volume":"19","author":[{"given":"Gr\u00e9goire","family":"Nicolis","sequence":"first","affiliation":[{"name":"Center for Nonlinear Phenomena and Complex Systems (CENOLI), Universit\u00e9 Libre de Bruxelles (ULB), Campus Plaine, C.P. 231, B-1050 Brussels, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yannick","family":"De Decker","sequence":"additional","affiliation":[{"name":"Center for Nonlinear Phenomena and Complex Systems (CENOLI), Universit\u00e9 Libre de Bruxelles (ULB), Campus Plaine, C.P. 231, B-1050 Brussels, Belgium"},{"name":"Nonlinear Physical Chemistry Unit, Universit\u00e9 Libre de Bruxelles (ULB), Campus Plaine, C.P. 231, B-1050 Brussels, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,8,23]]},"reference":[{"key":"ref_1","unstructured":"Glansdorff, P., and Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, John Wiley & Sons."},{"key":"ref_2","unstructured":"Nicolis, G., and Prigogine, I. (1977). Self-Organization in Nonequilibrium Systems, John Wiley & Sons."},{"key":"ref_3","unstructured":"De Groot, S.R., and Mazur, P. (2011). Non-Equilibrium Thermodynamics, Dover Publications."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/BF01469698","article-title":"Stability Criteria and Fluctuations around Nonequilibrium States","volume":"56","author":"Luo","year":"1984","journal-title":"Z. Phys. B Condens. Matter"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7011","DOI":"10.1063\/1.450623","article-title":"Stochastic Thermodynamics of Nonequilibrium Steady States in Chemical Reaction systems","volume":"84","author":"Mou","year":"1986","journal-title":"J. Chem. Phys."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"021137","DOI":"10.1103\/PhysRevE.80.021137","article-title":"Extended forms of the second law for general time-dependent stochastic processes","volume":"80","author":"Ge","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"041130","DOI":"10.1103\/PhysRevE.83.041130","article-title":"Irreversible thermodynamics in multiscale stochastic dynamical systems","volume":"83","author":"Qian","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_9","first-page":"155","article-title":"Stochastic thermodynamics: A brief introduction","volume":"184","year":"2013","journal-title":"Phys. Complex Colloids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.physa.2014.04.035","article-title":"Ensemble and trajectory thermodynamics: A brief introduction","volume":"418","author":"Esposito","year":"2015","journal-title":"Physica A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"126001","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_12","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1140\/epjst\/e2015-02438-3","article-title":"Extended local equilibrium approach to stochastic thermodynamics","volume":"224","author":"Nicolis","year":"2015","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"042127","DOI":"10.1103\/PhysRevE.93.042127","article-title":"Stochastic thermodynamics of reactive systems: An extended local equilibrium approach","volume":"93","author":"Derivaux","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Sekimoto, K. (2010). Stochastic Energetics, Springer. Lecture Notes in Physics.","DOI":"10.1007\/978-3-642-05411-2"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"046102","DOI":"10.1103\/PhysRevE.67.046102","article-title":"Stationary and transient work-fluctuation theorems for a dragged Brownian particle","volume":"67","author":"Cohen","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"150601","DOI":"10.1103\/PhysRevLett.98.150601","article-title":"Entropy Production and Time Asymmetry in Nonequilibrium Fluctuations","volume":"98","author":"Andrieux","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"P01002","DOI":"10.1088\/1742-5468\/2008\/01\/P01002","article-title":"Thermodynamic time asymmetry in non-equilibrium fluctuations","volume":"2008","author":"Andrieux","year":"2008","journal-title":"J. Stat. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"021120","DOI":"10.1103\/PhysRevE.82.021120","article-title":"Entropy production in irreversible systems described by a Fokker-Planck equation","volume":"82","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"042140","DOI":"10.1103\/PhysRevE.91.042140","article-title":"Stochastic approach to equilibrium and nonequilibrium thermodynamics","volume":"91","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_20","unstructured":"Landau, L.D., and Lifshitz, E.M. (1987). Fluid Mechanics, Pergamon Press."},{"key":"ref_21","unstructured":"Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions, Dover Publications Inc."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kotz, S., Kozubowski, T., and Podg\u00f3rski, K. (2001). The Laplace Distribution and Generalizations, Springer Science + Business Media.","DOI":"10.1007\/978-1-4612-0173-1"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"050101","DOI":"10.1103\/PhysRevE.76.050101","article-title":"Work and heat probability distribution of an optically driven Brownian particle: Theory and experiments","volume":"76","author":"Imparato","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"051104","DOI":"10.1103\/PhysRevE.82.051104","article-title":"Exact path-integral evaluation of the heat distribution function of a trapped Brownian oscillator","volume":"82","author":"Chatterjee","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"052138","DOI":"10.1103\/PhysRevE.95.052138","article-title":"Heat fluctuations of Brownian oscillators in nonstationary processes: Fluctuation theorem and condensation transition","volume":"95","author":"Crisanti","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/j.crhy.2007.04.010","article-title":"Comparison of far-from-equilibrium work relations","volume":"8","author":"Jarzynski","year":"2007","journal-title":"C. R. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/0031-8914(74)90198-0","article-title":"Brownian motion and fluctuating hydrodynamics","volume":"76","author":"Bedeaux","year":"1974","journal-title":"Physica"},{"key":"ref_28","unstructured":"Mazonka, O., and Jarzynski, C. (arXiv, 1999). Exactly solvable model illustrating far-from-equilibrium predictions, arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/9\/434\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:43:03Z","timestamp":1760208183000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/9\/434"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,23]]},"references-count":28,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["e19090434"],"URL":"https:\/\/doi.org\/10.3390\/e19090434","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,23]]}}}