Mpemba effect in a two-dimensional bistable potential

[2603.24148] Mpemba effect in a two-dimensional bistable potential Skip to main content Search Submit Donate Log in Condensed Matter > Statistical Mechanics arXiv:2603.24148 (cond-mat) [Submitted on 25 Mar 2026 ( v1 ), last revised 27 Aug 2026 (this version, v4)] Title: Mpemba effect in a two-dimensional bistable potential Authors: Hisao Hayakawa , Satoshi Takada View PDF HTML (experimental) Abstract: We present an exactly solvable model of the Mpemba effect in an overdamped Langevin system confined to a two-dimensional, radially symmetric bistable potential. The potential is constructed as a piecewise quadratic-logarithmic function that is continuous and differentiable at the matching radii, enabling an exact mapping of the corresponding Fokker-Planck operator to a Schrödinger-type eigenvalue problem. The relaxation spectrum and eigenmodes are obtained analytically in each region in terms of confluent hypergeometric functions, with eigenvalues determined from matching conditions. % Focusing on isotropic equilibrium initial states at inverse temperature $\beta_{\rm ini}$ quenched to a bath at inverse temperature $\beta$, we derive explicit expressions for the mode amplitudes governing long-time relaxation. We demonstrate that the coefficient of the slowest mode exhibits a non-monotonic dependence on $\beta_{\rm ini}$ and identify a sufficient crossing condition for the Kullback-Leibler divergence in terms of the two slowest modes, if the global minimum of the potential is located far away from the origin and the second minimum exists near the origin. For corresponding parameters, we demonstrate that the Mpemba effect can be realized. % Our results provide a rare example of an analytically tractable two-dimensional model exhibiting anomalous relaxation without any confining walls, extending previous one-dimensional constructions with a hard wall and clarifying the role of radial geometry in nonequilibrium relaxation phenomena. Comments: 31 pages, 13 figures Subjects: Statistical Mechanics (cond-mat.stat-mech) Cite as: arXiv:2603.24148 [cond-mat.stat-mech] (or arXiv:2603.24148v4 [cond-mat.stat-mech] for this version) https://doi.org/10.48550/arXiv.2603.24148 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hisao Hayakawa [ view email ] [v1] Wed, 25 Mar 2026 10:13:31 UTC (396 KB) [v2] Thu, 26 Mar 2026 13:16:39 UTC (555 KB) [v3] Wed, 1 Apr 2026 04:13:16 UTC (611 KB) [v4] Thu, 27 Aug 2026 02:38:27 UTC (606 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: cond-mat.stat-mech < prev | next > new | recent | 2026-03 Change to browse by: cond-mat References & Citations NASA ADS 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? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) 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) Major funding support from

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