Conditional instrument-record diffusion coefficient

Assume full AC, Specified effective GKSL bath and instrument assumption, and the finite connected Gibbs-jump model with its selected energy-resolved instrument. Prepare the system with diagonal Gibbs law $\pi$. Assign a real pointer velocity $v_a$ in metres per second to each recorded energy label, with ${\sum }_a{\pi }_a{v}_a=0$, and define the model record displacement $Y_T={\int }_0^T{v}_{a(s)}ds$. The postulate's physical applicability is limited to $0\le t\le t_{\mathrm{r}\mathrm{e}\mathrm{s}}$. For formulas at arbitrary $T\ge 0$ and their limit, continue the same finite-rate chain as an idealized mathematical model; this gives no physical-applicability claim for $T>t_{\mathrm{r}\mathrm{e}\mathrm{s}}$. Thus the displayed formula is a conditional physical-model prediction for $0\le T\le t_{\mathrm{r}\mathrm{e}\mathrm{s}}$ and a mathematical model identity for all $T\ge 0$ in the continuation:

${\mathbb{E}}_{\pi }[Y_T]=0,{\mathrm{Var}}_{\pi }(Y_T)=2{\int }_0^T(T-s)C_v(s)ds,$

where $C_v(s)={\sum }_a{\pi }_a{v}_a(e^{sW}v{)}_a$. The limit

$D_{\mathrm{r}\mathrm{e}\mathrm{c}}:={\lim }_{T\to \mathrm{\infty }}\frac{{\mathrm{Var}}_{\pi }(Y_T)}{2T}={\int }_0^{\mathrm{\infty }}{C}_v(s)ds=⟨(-W{\mathrm{\mid }}_{{\pi }^{\perp }}{)}^{-1}v,v{⟩}_{\pi }\ge 0$

exists and has units square metres per second. This long-time limit is a derived coefficient of the idealized mathematical continuation. It is not a measurement of a system's position, a calibrated material coefficient, or evidence that the microscopic bath obeys the effective model.

The stochastic record is the selected instrument output. It is not a continuous position trajectory and does not follow uniquely from the one-time GKSL channel family.