Mori-Zwanzig formalism: An existence proof for weak solutions of the orthogonal dynamics equation

[2607.25855] Mori-Zwanzig formalism: An existence proof for weak solutions of the orthogonal dynamics equation Skip to main content Search Submit Donate Log in Mathematical Physics arXiv:2607.25855 (math-ph) [Submitted on 28 Jul 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: Mori-Zwanzig formalism: An existence proof for weak solutions of the orthogonal dynamics equation Authors: Christoph Widder View PDF HTML (experimental) Abstract: In classical statistical physics, the Mori-Zwanzig projection operator technique is used to derive generalized Langevin equations for a random variable (observable). Standard derivations implicitly assume the existence of solutions to the so-called orthogonal dynamics equation as well as the validity of the variation of constants formula (Dyson identity). It was pointed out by Givon, Hald and Kupferman that the existence is a subtle issue for infinite-rank projections such as Zwanzig's projection [D. Givon, O. H. Hald, R. Kupferman, Israel Journal of Mathematics, 145 (221-241), 2005]. The authors proved the existence of weak solutions for Zwanzig's projection in the context of stationary Hamiltonian systems. To this date, this is the only existence proof that allows for an infinite-rank projection, whereas the uniqueness and regularity remain open problems. In this article, we generalize the existence proof by Givon et al. to nonstationary non-Hamiltonian systems whose time evolution is given by a quasicontraction semigroup. We establish growth bounds as well as the uniqueness for sufficiently regular solutions (if existent). Finally, we apply our results to Zwanzig's projection using the damped harmonic oscillator as an example. Subjects: Mathematical Physics (math-ph) Cite as: arXiv:2607.25855 [math-ph] (or arXiv:2607.25855v2 [math-ph] for this version) https://doi.org/10.48550/arXiv.2607.25855 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Christoph Widder [ view email ] [v1] Tue, 28 Jul 2026 15:26:17 UTC (27 KB) [v2] Tue, 25 Aug 2026 07:18:24 UTC (27 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math-ph < prev | next > new | recent | 2026-07 Change to browse by: math math.MP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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