= Solution
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
$$
\boxed{\partial_tp=\partial_x[D\partial_xp+\phi'(x)p].}
$$
Back to article page
