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

<Invariant distribution of an Itô diffusion>, specialized to <Underdamped Langevin dynamics>, has density
$$
\pi(x,p)=Z^{-1}\exp\left\{-U(x)-\frac{\lVert p\rVert^2}{2\eta}\right\}.
$$
Thus $X$ has density proportional to $e^{-U(x)}$, and conditionally and marginally $P\sim N(0,\eta I_d)$. The Hamiltonian transport between $x$ and $p$ preserves this density, while the <Ornstein-Uhlenbeck process> in momentum has exactly that Gaussian invariant law.

The <Euler-Maruyama method> with step size $\delta$ and independent $\xi_k\sim N(0,I_d)$ is
$$
\begin{aligned}
P_{k+1}
&=P_k-\gamma\delta P_k-\eta\delta\nabla U(X_k)
+\sqrt{2\gamma\eta\delta}\,\xi_k,\\
X_{k+1}&=X_k+\delta P_k.
\end{aligned}
$$

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 4 / c / Solution

{c}
{wiki}

Underdamped Langevin dynamics augments position by momentum and combines Hamiltonian transport with friction and matching Gaussian noise. For potential $U$, momentum variance $\eta$, and friction $\gamma$, its invariant density is proportional to $\exp[-U(x)-|p|^2/(2\eta)]$.


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