Linear Ornstein-Uhlenbeck bath model

Choose a Galilean inertial chart $\mathcal{F}$ as defined in Galilean inertial charts and the center-of-mass rest frame for the specified fixed closed trap-and-bath body, assumed to have positive total mass and the center-of-mass map required there, and supply its componentwise differentiable center-of-mass trajectory on the whole modeled time interval. Assume that its center-of-mass velocity is one constant vector on that interval, so the single constant boost of that supplier selects $\mathcal{F}$ with zero center-of-mass velocity throughout. Take $\mathcal{F}$ to be this common rest chart. The trap-and-bath background is held fixed in this idealization; center-of-mass recoil or accelerating-frame dynamics are not part of the scalar process. In this chart let ${\mathbf{r}}_c\in {\mathbb{R}}^3$ be the fixed trap-center position and let $\mathbf{e}\in {\mathbb{R}}^3$ be a fixed unit vector along the modeled trap axis. For a particle position $\mathbf{r}\in {\mathbb{R}}^3$, define the signed displacement $x=⟨\mathbf{r}-{\mathbf{r}}_c,\mathbf{e}⟩$ in metres; the process $X_t$ in Ornstein–Uhlenbeck integral equation is this coordinate of the particle position at time $t$. Thus $U(x)=k{x}^2\mathrm{/}2$ is the harmonic potential relative to the trap center, not an absolute spatial coordinate. Using the SI conventions and exact $k_{\mathrm{B}}$ of SI quantities and unit conventions, take stiffness $k>0$ in joules per square metre, constant mobility $\mu >0$ in square metres per joule-second, and bath temperature $T_b>0$ kelvin. Adopt the one-dimensional overdamped model $d{X}_t=-\gamma X_{t}dt+\sigma d{B}_t$, interpreted through the integral equation of Ornstein–Uhlenbeck integral equation, with $\gamma =\mu k$ in seconds${}^{-1}$ and ${\sigma }^2=2\mu k_{\mathrm{B}}{T}_b$ in square metres per second. Here $B$ is standard Brownian motion with its stated filtration and increment hypotheses, and the bath noise is independent of the initial preparation. This Einstein noise relation and the linear Markov model are physical assumptions. They do not follow from equilibrium thermodynamics alone and do not report a measured spectrum or calibration.