Berne and Pecora[@berne1976a], especially their Appendix 5.A and Section 5.9 leading to their eq 5.9.6, also prove the other important consequence of the Langevin model and Doob’s theorem, namely that the Langevin model determines the exact value of $\langle (\Delta x(t))^{2} \rangle$. On the time and distance scales accessible to quasielastic scattering, the Langevin model requires $$\langle (\Delta x(t))^{2} \rangle = 2 D t.
     \label{eq:meansquare}$$ Here $k_{B}$ is Boltzmann’s constant and $T$ is the absolute temperature. $D = k_{B}T/f$ is the diffusion constant, a quantity *that does not depend on time*. Time independence of $D$ is required by the calculation, because $D$ results from a time integral over ($0 \leq t \leq \infty$).

Equations \[eq:g1swrong\] and \[eq:meansquare\] come as a package; they are equally consequences of the Langevin model. Correspondingly, Berne and Pecora show for diffusing monodisperse Brownian particles that the Langevin model requires that the field correlation function is a simple exponential $$g^{(1s)}(q,t) = \exp(- q^{2} D t).
      \label{eq:g1ssimple}$$ For unclear reasons – the literature error noted in the introduction – Berne and Pecora’s entirely correct Chapter 5 is being misread as proving that eq \[eq:g1swrong\] is always correct, even when the time relaxation of $g^{(1s)}(q,t)$ is not a 