Overdamped limits for Langevin dynamics with position-dependent coefficients via $L^2$-hypocoercivity
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917470609080320 |
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| author | Blassel, Noé |
| author_facet | Blassel, Noé |
| contents | This note provides a simple derivation of the overdamped approximation for kinetic (or underdamped) equilibrium Langevin dynamics, in cases where certain coefficients depend on the position variable. The equivalent small-mass limit of these dynamics, known as the Kramers--Smoluchowski approximation, in the case of a state-dependent friction coefficient, has been previously studied by a variety of approaches. Our new approach uses hypocoercivity estimates, which may be of interest in their own right, and lead to a very direct derivation, providing in particular a clear explanation of the ``noise-induced drift'' term in the overdamped equation in the case of a state-dependent friction term. Using the same approach, we also treat effective kinetic dynamical models derived from a coarse-graining approximation of a high-dimensional system, as well as a class of kinetic dynamics with position-dependent mass matrices. All of these models are relevant to applications in computational chemistry. We finally identify a mistake in a related work, and suggest a solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16924 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Overdamped limits for Langevin dynamics with position-dependent coefficients via $L^2$-hypocoercivity Blassel, Noé Probability Statistical Mechanics Mathematical Physics This note provides a simple derivation of the overdamped approximation for kinetic (or underdamped) equilibrium Langevin dynamics, in cases where certain coefficients depend on the position variable. The equivalent small-mass limit of these dynamics, known as the Kramers--Smoluchowski approximation, in the case of a state-dependent friction coefficient, has been previously studied by a variety of approaches. Our new approach uses hypocoercivity estimates, which may be of interest in their own right, and lead to a very direct derivation, providing in particular a clear explanation of the ``noise-induced drift'' term in the overdamped equation in the case of a state-dependent friction term. Using the same approach, we also treat effective kinetic dynamical models derived from a coarse-graining approximation of a high-dimensional system, as well as a class of kinetic dynamics with position-dependent mass matrices. All of these models are relevant to applications in computational chemistry. We finally identify a mistake in a related work, and suggest a solution. |
| title | Overdamped limits for Langevin dynamics with position-dependent coefficients via $L^2$-hypocoercivity |
| topic | Probability Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2602.16924 |