The <Fluctuation-dissipation theorem> for mobility $\widetilde M$ requires
$$
\dot x=-\widetilde M V'(x)
+\sqrt{2\widetilde M k_BT}\,\Lambda_x.
$$
Comparison with the stated noise amplitude gives
$$
\boxed{C^2=2k_BT}.
$$
This choice makes the stationary <Fokker-Planck equation> have the <Boltzmann distribution> proportional to $e^{-V/(k_BT)}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-344/4/b/solution