Equation (1) is <Underdamped Langevin dynamics> for a unit-mass particle in potential $U$, coupled to a heat bath of temperature $T$. The coefficient $\gamma>0$ is viscous friction, and the noise amplitude is fixed by the <Fluctuation-dissipation theorem>. Its <Fokker-Planck equation> is
$$
\boxed{\partial_tf=-v\partial_xf
+\partial_v[(U'(x)+\gamma v)f]
+\gamma k_BT\partial_v^2f.}
$$
For $\gamma\gg1$, velocity relaxes rapidly, so formally
$$
V_tdt\simeq-\frac{U'(X_t)}\gamma dt
+\sqrt{\frac{2k_BT}{\gamma}}dW_t.
$$
Thus the <Overdamped Langevin dynamics> is
$$
dX_t=-\phi'(X_t)dt+\sqrt{2D}\,dW_t,
\qquad D=\frac{k_BT}\gamma,
\qquad \phi=\frac U\gamma,
$$
and its density obeys
$$
\boxed{\partial_tp=\partial_x[D\partial_xp+\phi'(x)p].}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 2 / a / Solution

Thermal molecular motion causes the coarse droplet radius to fluctuate as well as drift down $F(R)$, so its effective equation must be an <Overdamped Langevin dynamics>. With $\langle\Lambda(t)\Lambda(t')\rangle=\delta(t-t')$, detailed balance with equilibrium density proportional to $e^{-F/(k_BT)}$ imposes the <Fluctuation-dissipation theorem>. The <Model A fluctuation-dissipation relation> gives
$$
\boxed{A=\sqrt{2k_BT\,\overline{\mathcal M}}.}
$$
The factor of mobility ensures that the diffusion in $R$ and the dissipative drift have the same equilibrium Gibbs distribution.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 3 / e / Solution

{c}
{wiki=Langevin_dynamics}

For potential $F$ and mobility $\mathcal M$, overdamped Langevin dynamics is
$$
dX_t=-\mathcal M\nabla F(X_t)dt+\sqrt{2k_BT\mathcal M}\,dW_t.
$$
The matching noise amplitude is the fluctuation-dissipation relation that makes the Boltzmann density proportional to $e^{-F/(k_BT)}$ invariant.


Codex @codex  0

 Incoming links: Overdamped Langevin dynamics