{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T01:48:12Z","timestamp":1775094492618,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2017,11,29]],"date-time":"2017-11-29T00:00:00Z","timestamp":1511913600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["320823"],"award-info":[{"award-number":["320823"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Langevin dynamics is a versatile stochastic model used in biology, chemistry, engineering, physics and computer science. Traditionally, in thermal equilibrium, one assumes (i) the forces are given as the gradient of a potential and (ii) a fluctuation-dissipation relation holds between stochastic and dissipative forces; these assumptions ensure that the system samples a prescribed invariant Gibbs-Boltzmann distribution for a specified target temperature. In this article, we relax these assumptions, incorporating variable friction and temperature parameters and allowing nonconservative force fields, for which the form of the stationary state is typically not known a priori. We examine theoretical issues such as stability of the steady state and ergodic properties, as well as practical aspects such as the design of numerical methods for stochastic particle models. Applications to nonequilibrium systems with thermal gradients and active particles are discussed.<\/jats:p>","DOI":"10.3390\/e19120647","type":"journal-article","created":{"date-parts":[[2017,11,30]],"date-time":"2017-11-30T03:15:15Z","timestamp":1512011715000},"page":"647","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Langevin Dynamics with Variable Coefficients and Nonconservative Forces: From Stationary States to Numerical Methods"],"prefix":"10.3390","volume":"19","author":[{"given":"Matthias","family":"Sachs","sequence":"first","affiliation":[{"name":"Department of Mathematics, Duke University, Durham, NC 27708, USA"},{"name":"The School of Mathematics and the Maxwell Institute of Mathematical Sciences, James Clerk Maxwell Building, University of Edinburgh, Edinburgh EH9 3FD, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8504-7091","authenticated-orcid":false,"given":"Benedict","family":"Leimkuhler","sequence":"additional","affiliation":[{"name":"The School of Mathematics and the Maxwell Institute of Mathematical Sciences, James Clerk Maxwell Building, University of Edinburgh, Edinburgh EH9 3FD, UK"}]},{"given":"Vincent","family":"Danos","sequence":"additional","affiliation":[{"name":"D\u00e9partement d\u2019informatique, \u00c9cole Normale Sup\u00e9rieure, 45 rue d\u2019Ulm, F-75230 Paris CEDEX 05, France"}]}],"member":"1968","published-online":{"date-parts":[[2017,11,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1209\/epl\/i2001-00103-6","article-title":"Drift without flux: Brownian walker with a space-dependent diffusion coefficient","volume":"54","author":"Lancon","year":"2001","journal-title":"Europhys. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"012116","DOI":"10.1103\/PhysRevE.94.012116","article-title":"Isothermal Langevin dynamics in systems with power-law spatially dependent friction","volume":"94","author":"Regev","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1021\/ct400963d","article-title":"Thermal gradients on graphene to drive nanoflake motion","volume":"10","author":"Becton","year":"2014","journal-title":"J. Chem. Theory Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.matpur.2007.12.002","article-title":"Flocking in noisy environments","volume":"89","author":"Cucker","year":"2008","journal-title":"J. Math. Pures Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"453","DOI":"10.4310\/CMS.2009.v7.n2.a9","article-title":"Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system","volume":"7","author":"Ha","year":"2009","journal-title":"Commun. Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"025003","DOI":"10.1088\/1367-2630\/16\/2\/025003","article-title":"Collective motion in an active suspension of Escherichia coli bacteria","volume":"16","author":"Gachelin","year":"2014","journal-title":"New J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1088","DOI":"10.1073\/pnas.0910390107","article-title":"Coordinate-dependent diffusion in protein folding","volume":"107","author":"Best","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1038\/nature20817","article-title":"Weak synchronization and large-scale collective oscillation in dense bacterial suspensions","volume":"542","author":"Chen","year":"2017","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1209\/0295-5075\/19\/3\/001","article-title":"Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics","volume":"19","author":"Hoogerbrugge","year":"1992","journal-title":"Europhys. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Espa\u00f1ol, P. (2005). Dissipative particle dynamics. Handbook of Materials Modeling, Springer.","DOI":"10.1007\/978-1-4020-3286-8_131"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1142\/S0219493706001670","article-title":"Geometric ergodicity for dissipative particle dynamics","volume":"6","author":"Shardlow","year":"2006","journal-title":"Stoch. Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"103301","DOI":"10.1063\/1.3496895","article-title":"Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises","volume":"51","author":"Ahn","year":"2010","journal-title":"J. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1142\/S0219530513500255","article-title":"Flocking and non-flocking behavior in a stochastic Cucker-Smale System","volume":"12","author":"Ton","year":"2014","journal-title":"Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1137\/15M1030467","article-title":"A Cucker-Smale model with noise and delay","volume":"76","author":"Erban","year":"2016","journal-title":"SIAM J. Appl. Math."},{"key":"ref_15","first-page":"13","article-title":"The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics","volume":"36","author":"Leimkuhler","year":"2016","journal-title":"IMA J. Numer. Anal."},{"key":"ref_16","first-page":"34","article-title":"Rational construction of numerical methods for stochastic molecular dynamics","volume":"2013","author":"Leimkuhler","year":"2013","journal-title":"Appl. Math. Res. Exp."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1990","DOI":"10.1063\/1.463137","article-title":"Reversible multiple time scale molecular dynamics","volume":"97","author":"Tuckerman","year":"1992","journal-title":"J. Chem. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1088\/1367-2630\/7\/1\/034","article-title":"Position-dependent diffusion coefficients and free energies from Bayesian analysis of equilibrium and replica molecular dynamics simulations","volume":"7","author":"Hummer","year":"2005","journal-title":"New J. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"051111","DOI":"10.1103\/PhysRevE.75.051111","article-title":"Biased diffusion in confined media: Test of the Fick-Jacobs approximation and validity criteria","volume":"75","author":"Burada","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"074108","DOI":"10.1063\/1.3626215","article-title":"Time scale separation leads to position-dependent diffusion along a slow coordinate","volume":"135","author":"Berezhkovskii","year":"2011","journal-title":"J. Chem. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"074103","DOI":"10.1063\/1.4818533","article-title":"Direct evaluation of the position dependent diffusion coefficient and persistence time from the equilibrium density profile in anisotropic fluids","volume":"139","author":"Wilmer","year":"2013","journal-title":"J. Chem. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1017\/S0962492916000039","article-title":"Partial differential equations and stochastic methods in molecular dynamics","volume":"25","author":"Stoltz","year":"2016","journal-title":"Acta Numer."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bellet, L.R. (2006). Ergodic properties of Markov processes. Open Quantum Systems II, Springer.","DOI":"10.1007\/3-540-33966-3_1"},{"key":"ref_24","first-page":"3145","article-title":"Handbook of stochastic methods for physics, chemistry and the natural sciences","volume":"25","author":"Gardiner","year":"1986","journal-title":"Appl. Opt."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Guionnet, A., and Zegarlinksi, B. (2003). Lectures on logarithmic Sobolev inequalities. S\u00e9minaire de Probabilit\u00e9s XXXVI, Springer.","DOI":"10.1007\/978-3-540-36107-7_1"},{"key":"ref_26","unstructured":"H\u00f6rmander, L. (1985). The Analysis of Linear Partial Differential Operators. III, Volume 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"690","DOI":"10.1214\/aop\/1176992166","article-title":"Recurrence and invariant measures for degenerate diffusions","volume":"15","author":"Kliemann","year":"1987","journal-title":"Ann. Probab."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/BF00531822","article-title":"On the functional central limit theorem and the law of the iterated logarithm for Markov processes","volume":"60","author":"Bhattacharya","year":"1982","journal-title":"Zeitschrift F\u00fcr Wahrscheinlichkeitstheorie Und Verwandte Gebiete"},{"key":"ref_29","unstructured":"Villani, C. (2009). Hypocoercivity, American Mathematical Society. Number 949-951."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3807","DOI":"10.1090\/S0002-9947-2015-06012-7","article-title":"Hypocoercivity for linear kinetic equations conserving mass","volume":"367","author":"Dolbeault","year":"2015","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Harris, T.E. (1956). The existence of stationary measures for certain Markov processes. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, University of California Press.","DOI":"10.1525\/9780520350670-011"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"542","DOI":"10.2307\/1427479","article-title":"Stability of Markovian processes I: Criteria for discrete-time chains","volume":"24","author":"Meyn","year":"1992","journal-title":"Adv. Appl. Probab."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/978-3-0348-0021-1_7","article-title":"Yet another look at Harris\u2019 ergodic theorem for Markov chains","volume":"Volume 63","author":"Hairer","year":"2011","journal-title":"Seminar on Stochastic Analysis, Random Fields and Applications VI"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"518","DOI":"10.2307\/1427522","article-title":"Stability of Markovian processes III: Foster\u2013Lyapunov criteria for continuous-time processes","volume":"25","author":"Meyn","year":"1993","journal-title":"Adv. Appl. Probab."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/S0304-4149(02)00150-3","article-title":"Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise","volume":"101","author":"Mattingly","year":"2002","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_36","unstructured":"Meyn, S.P., and Tweedie, R.L. (2012). Markov Chains and Stochastic Stability, Springer Science & Business Media."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Zhang, F. (2006). The Schur Complement and Its Applications, Springer Science & Business Media.","DOI":"10.1007\/b105056"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"852","DOI":"10.1109\/TAC.2007.895842","article-title":"Emergent behavior in flocks","volume":"52","author":"Cucker","year":"2007","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4423","DOI":"10.1063\/1.474784","article-title":"Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation","volume":"107","author":"Groot","year":"1997","journal-title":"J. Chem. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.jcp.2016.07.034","article-title":"Pairwise adaptive thermostats for improved accuracy and stability in dissipative particle dynamics","volume":"324","author":"Leimkuhler","year":"2016","journal-title":"J. Comput. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/12\/647\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:51:49Z","timestamp":1760208709000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/12\/647"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,29]]},"references-count":40,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["e19120647"],"URL":"https:\/\/doi.org\/10.3390\/e19120647","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,11,29]]}}}