Geometric formulation of state-dependent Langevin dynamics using scalar free energy
Abstract
Stochastic dynamics with state-dependent diffusion arise broadly in confined, anisotropic, and hydrodynamically coupled systems. The conventional Langevin equation contains a spurious drift associated with multiplicative noise. Since the restricted free energy is generally non-scalar, the covariance is not explicit. Here, we reformulate state-dependent Langevin dynamics using a scalar free energy and a diffusion metric defined by the inverse diffusion tensor. The conventional spurious drift is then expressed as a Christoffel contribution. Although our formulation is mathematically equivalent to the conventional one through the relation between the restricted and scalar free energies, it makes coordinate covariance explicit and enables the phenomenological construction of thermodynamic potentials directly from system symmetries. We demonstrate its consistency in representative examples of state-dependent diffusion arising from coordinate transformations, geometric confinement, and projection from curved to flat spaces.
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BibTeX
@misc{yasuda2026geometric,
title = {{Geometric formulation of state-dependent Langevin dynamics using scalar free energy}},
author = {Kento Yasuda and Zhongqiang Xiong and Zhanglin Hou and Kenta Ishimoto and Xinpeng Xu and Shigeyuki Komura},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.13617},
eprint = {2606.13617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.13617},
}
RIS
TY - UNPB
TI - Geometric formulation of state-dependent Langevin dynamics using scalar free energy
AU - Kento Yasuda
AU - Zhongqiang Xiong
AU - Zhanglin Hou
AU - Kenta Ishimoto
AU - Xinpeng Xu
AU - Shigeyuki Komura
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2606.13617
UR - https://arxiv.org/abs/2606.13617
AB - Stochastic dynamics with state-dependent diffusion arise broadly in confined, anisotropic, and hydrodynamically coupled systems. The conventional Langevin equation contains a spurious drift associated with multiplicative noise. Since the restricted free energy is generally non-scalar, the covariance is not explicit. Here, we reformulate state-dependent Langevin dynamics using a scalar free energy and a diffusion metric defined by the inverse diffusion tensor. The conventional spurious drift is then expressed as a Christoffel contribution. Although our formulation is mathematically equivalent to the conventional one through the relation between the restricted and scalar free energies, it makes coordinate covariance explicit and enables the phenomenological construction of thermodynamic potentials directly from system symmetries. We demonstrate its consistency in representative examples of state-dependent diffusion arising from coordinate transformations, geometric confinement, and projection from curved to flat spaces.
ER -