 Solution

Take an overdamped particle with viscous drag coefficient $\gamma$, so its deterministic drift in the harmonic potential $U=k{x}^2/2$ is $\dot{x}=-kx/\gamma$. The one-dimensional Fokker-Planck equation has probability currentAt thermal equilibrium, $J=0$ and the Boltzmann distribution is $p(x)=Z^{-1}{e}^{-k{x}^2/(2k_{B}T)}$, with $p^{\prime }/p=-kx/(k_{B}T)$. Inserting this into the zero-current condition givesThe result is the Einstein diffusion relation between diffusion and mobility. It assumes a bath at one temperature, linear drag and thermal equilibrium. The same Gaussian probability density gives $⟨x^2⟩=k_{B}T/k$, providing an independent calibration of trap stiffness.