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To assess the long-term effect of these stiff forces on dynamics and structure, it is useful to consider the limit where they are treated as constraints, so the system evolves strictly within allowed configurations. Efforts to derive equations involving both constraints, and the stochastic motion appropriate at the scales of soft matter, began around 50 years ago, yet we are still lacking a straightforward way to extract the projected equations and apply them in modern formulations of mesoscale dynamics. Here, we address this gap with two key contributions: (1) a practical summary of the constrained Brownian dynamics equations with \u201csoft\u201d constraints, i.e., constraints imposed by stiff forces, which is illustrated through several representative examples, taking care to highlight the nontrivial effects of the constraints, and (2) a derivation using singular perturbation theory, establishing the validity of these equations over timescales exceeding the relaxation of stiffly constrained degrees of freedom. We further extend our approach to \u201csoft\u2013soft\u201d constraints, where mobility varies on lengthscales comparable with the restraining forces\u2014a scenario typical for particles in fluids experiencing hydrodynamic interactions. We hope our results will be useful for soft matter research, as a robust toolkit for studying tethered or confined systems.<\/jats:p>","DOI":"10.1063\/5.0322900","type":"journal-article","created":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T11:33:51Z","timestamp":1781004831000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Brownian dynamics with soft constraints in soft matter systems"],"prefix":"10.1063","volume":"164","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2427-2065","authenticated-orcid":false,"given":"S.","family":"Marbach","sequence":"first","affiliation":[{"name":"CNRS, Sorbonne Universit\u00e9, Physicochimie des Electrolytes et Nanosyst\u00e8mes Interfaciaux 1 , F-75005 Paris,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-2831-1976","authenticated-orcid":false,"given":"A.","family":"Carter","sequence":"additional","affiliation":[{"name":"CNRS, Sorbonne Universit\u00e9, Physicochimie des Electrolytes et Nanosyst\u00e8mes Interfaciaux 1 , F-75005 Paris,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1556-5963","authenticated-orcid":false,"given":"M.","family":"Holmes-Cerfon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of British Columbia 2 , 1984 Mathematics Rd., Vancouver, British Columbia V6T 1Z2,","place":["Canada"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,6,9]]},"reference":[{"key":"2026060907334939000_c1","doi-asserted-by":"publisher","first-page":"4701","DOI":"10.1021\/ma00016a034","article-title":"Dynamics of reversible networks","volume":"24","year":"1991","journal-title":"Macromolecules"},{"key":"2026060907334939000_c2","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1016\/0043-1648(63)90206-0","article-title":"A theory of dynamic rubber friction","volume":"6","year":"1963","journal-title":"Wear"},{"key":"2026060907334939000_c3","doi-asserted-by":"publisher","first-page":"560","DOI":"10.1126\/science.1181263","article-title":"The free-energy landscape of clusters of attractive hard spheres","volume":"327","year":"2010","journal-title":"Science"},{"key":"2026060907334939000_c4","volume-title":"Colloids and the Depletion Interaction","year":"2024"},{"key":"2026060907334939000_c5","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/s1359-0294(97)80034-9","article-title":"Optical tweezers in colloid and interface science","volume":"2","year":"1997","journal-title":"Curr. 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