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arXiv · 2606.13617

Geometric formulation of state-dependent Langevin dynamics using scalar free energy

Kento Yasuda·Zhongqiang Xiong·Zhanglin Hou·Kenta Ishimoto·Xinpeng Xu·Shigeyuki Komura
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Abstract

Stochastic dynamics with state-dependent diffusion are widely used for Brownian motion in confined, anisotropic, and hydrodynamically coupled systems. The conventional Langevin formulation includes a spurious drift associated with multiplicative noise, but its free energy generally does not transform as a scalar, meaning that the covariance is not explicit. Here, we formulate a geometrically consistent Langevin equation by introducing a scalar free energy and using the diffusion tensor as a metric on configuration space. The spurious drift is then expressed as a Christoffel contribution of the diffusion metric. While our formulation is equivalent to the conventional one through the relation between the non-scalar and scalar free energies, it makes the coordinate covariance explicit. We demonstrate its consistency in representative examples of state-dependent diffusion arising from coordinate transformations, geometrical confinement, and projection from curved to flat spaces.

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Kento Yasuda, Zhongqiang Xiong, Zhanglin Hou, Kenta Ishimoto, Xinpeng Xu, Shigeyuki Komura. 2026-06-11. Geometric formulation of state-dependent Langevin dynamics using scalar free energy. https://arxiv.org/abs/2606.13617

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