Long term convergence rate of Smoluchowski-Kramers approximation by Stein's method

We consider the following second-order stochastic differential equation on $\mathbb{R}^{2d}$: \begin{equation*} dX_t^m=Y_t^mdt, \quad mdY_t^m=b(X_t^m)dt+\sigma(X_t^m)dB_t-Y^m_tdt, \end{equation*} where $X^m_t$ and $Y^m_t$ represent the position and velocity of a particle at time $t$, $m>0$ denotes its mass, $b:\mathbb{R}^d \rightarrow \mathbb{R}^d$ is the drift field, $\sigma:\mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$ is the diffusion coefficient, and $\{B_t\}_{t \ge 0}$ is a $d$-dimensional standard Brownian motion. The Smoluchowski--Kramers approximation states that as $m \rightarrow 0$, this system converges to the limiting equation: \begin{equation*} dX_t=b(X_t)dt+\sigma(X_t)dB_t. \end{equation*} Utilizing Stein's method, we prove that the $1$-Wasserstein distance between the invariant distribution of $X_t^m$ and that of its small-mass limit $X_t$ is of order $O(\sqrt{m}|\ln m|).$ Particularly, in the one-dimensional case, the convergence rate can be improved to $O(\sqrt{m}).$