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Using a Mori\u2013Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard\u2013Jones and Coulomb functions.<\/jats:p>","DOI":"10.1007\/s00332-024-10027-5","type":"journal-article","created":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T04:12:42Z","timestamp":1715659962000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials"],"prefix":"10.1007","volume":"34","author":[{"given":"Manh Hong","family":"Duong","sequence":"first","affiliation":[]},{"given":"Hung Dang","family":"Nguyen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,5,14]]},"reference":[{"issue":"96","key":"10027_CR1","first-page":"1","volume":"17","author":"A Athreya","year":"2012","unstructured":"Athreya, A., Kolba, T., Mattingly, J.C.: Propagating Lyapunov functions to prove noise-induced stabilization. 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