Adopted inertial velocity Langevin bath

Assume the Axiom of Choice The Axiom of Choice, the filtered Brownian probability-space convention of Ornstein–Uhlenbeck integral equation, and the adopted Newtonian spacetime and frame assumptions of Inertial frames, stationary observers and the adopted Galilean transformations between frames. Fix one inertial frame $\mathrm{\Gamma }=(o,u,(e_1,e_2,e_3))$ of that postulate: it is a fixed global adapted affine chart, with $o\in A$, $u\in V$, $\tau (u)=1$, and a fixed positively oriented $g$-orthonormal basis $(e_1,e_2,e_3)$ of the spatial displacement space $K=\ker \tau$. Its chart map assigns each event $p\in A$ the unique coordinates $(t_{\mathrm{\Gamma }}(p),{\mathbf{x}}_{\mathrm{\Gamma }}(p))$ satisfying $p=o+t_{\mathrm{\Gamma }}(p)u+{\sum }_{i=1}^3{x}_{\mathrm{\Gamma },i}(p)e_i$. Let $X=(X_t{)}_{t\ge 0}$ be the scalar position-coordinate process in this chart, measured in metres, whose value is the signed first spatial coordinate along the fixed axis $e_1$. The model tracks only $X$; it does not specify a full particle worldline or assert a law for either of the other spatial coordinates. Let $X_0$ be an ${\mathcal{F}}_0$-measurable real-valued random variable. On a measurable event ${\mathrm{\Omega }}_{\ast }\in \mathcal{F}$ with $P({\mathrm{\Omega }}_{\ast })=1$, assume $t\mapsto X_t(\omega )$ is continuously differentiable and define the velocity coordinate $V_t(\omega ):=d{X}_t(\omega )\mathrm{/}dt$ there, measured in metres per second. Thus $X_t=X_0+{\int }_0^t{V}_{s}ds$ on ${\mathrm{\Omega }}_{\ast }$; values on the null complement do not affect the stochastic conclusions.

Take a primitive particle mass $m>0$ in kilograms, a primitive constant bath temperature $T_b>0$ in kelvins, and the primitive SI Boltzmann constant $k_B>0$ in joules per kelvin. The positive damping rate $\gamma$ is a primitive model parameter in seconds${}^{-1}$; the positive noise amplitude $\sigma$ is derived as the square root of $2\gamma k_B{T}_b\mathrm{/}m$ and has units metres seconds${}^{-3\mathrm{/}2}$. Let $B_t$ be standard scalar Brownian motion with $B_0=0$, variance $t$ in seconds, and future increments independent of the present filtration. Prepare an ${\mathcal{F}}_0$-measurable $V_0\sim N(0,k_B{T}_b\mathrm{/}m)$ independently of future Brownian increments. The force $F:[0,\mathrm{\infty })\to \mathbb{R}$ is a prescribed deterministic locally continuous input in newtons. Adopt

$V_t=V_0-\gamma {\int }_0^t{V}_{s}ds+\frac{1}{m}{\int }_0^{t}F(s)ds+\sigma B_t,{\sigma }^2=\frac{2\gamma k_B{T}_b}{m}\mathrm{.}$

The displayed integral equation is the definition of the stochastic differential notation $d{V}_t=-\gamma V_{t}dt+F(t)dt\mathrm{/}m+\sigma d{B}_t$. Brownian increments have units seconds${}^{1\mathrm{/}2}$, so every term in the equation for $V_t$ has units metres seconds${}^{-1}$. When $F=0$, the preparation is the stationary OU Gaussian law because its variance is ${\sigma }^2\mathrm{/}(2\gamma )=k_B{T}_b\mathrm{/}m$; the cited mathematical supplier establishes the specified OU model properties. The thermal noise normalization, Gaussian preparation, constant damping, and deterministic force coupling are adopted assumptions. No measurement, material calibration, nonlinear response, or microscopic derivation of this bath is asserted.