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

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.