Separation of time scales in weakly interacting diffusions Article Swipe
Zachary P. Adams
,
Maximilian Engel
,
Rishabh S. Gvalani
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2502.12881
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2502.12881
We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a "droplet state" which is $metastable$, i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to $0$. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is $O(1)$ as the inverse temperature $β\to \infty$. Meanwhile the rate of leakage away from its centre of mass is $O(e^{-β})$. Futhermore, the quasi-stationary distribution is localised on a length scale of order $O(β^{-\frac12})$. We thus provide a partial answer to a question posed by Carrillo, Craig, and Yao (Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits. In: Active Particles, Volume 2, 2019) in the microscopic setting.
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Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2502.12881
- https://arxiv.org/pdf/2502.12881
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
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All OpenAlex metadata
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https://openalex.org/W4407759908Canonical identifier for this work in OpenAlex
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https://doi.org/10.48550/arxiv.2502.12881Digital Object Identifier
- Title
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Separation of time scales in weakly interacting diffusionsWork title
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preprintOpenAlex work type
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enPrimary language
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2025Year of publication
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2025-02-18Full publication date if available
- Authors
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Zachary P. Adams, Maximilian Engel, Rishabh S. GvalaniList of authors in order
- Landing page
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https://arxiv.org/abs/2502.12881Publisher landing page
- PDF URL
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https://arxiv.org/pdf/2502.12881Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2502.12881Direct OA link when available
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Statistical physics, Separation (statistics), Mathematics, Physics, StatisticsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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10Other works algorithmically related by OpenAlex
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