= Underdamped Langevin dynamics
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Underdamped Langevin dynamics retains both <position> $q$ and <momentum> $p$. For <mass> $m>0$, <temperature> $T>0$, friction rate $\gamma>0$ and smooth <potential energy> $U$, one common normalization is
$$
dq_t=\frac{p_t}{m}dt,\qquad dp_t=-\nabla U(q_t)dt-\gamma p_tdt+\sqrt{2\gamma mk_BT}\,dW_t.
$$
Here $W$ is vector <Brownian motion>. The noise amplitude is tied to friction by the <fluctuation-dissipation relation for a Langevin particle>. If the density is normalizable and the dynamics is nonexplosive, its <invariant distribution> is the <Boltzmann distribution>
$$
\pi(q,p)\propto\exp\!\left[-\frac{U(q)+|p|^2/(2m)}{k_BT}\right].
$$
The <Hamiltonian flow> preserves the total energy and phase-space volume; the momentum <Ornstein-Uhlenbeck process> preserves a centered <normal distribution> with coordinate <variance> $mk_BT$. These two facts verify the invariant density. In dimensionless units with $k_BT=1$ and momentum variance $\eta$, it is $\exp[-U(q)-|p|^2/(2\eta)]$, with the corresponding choice $m=\eta$.
Retaining inertia resolves the short-time <ballistic-to-diffusive Brownian displacement>. The <overdamped Langevin dynamics> approximation removes momentum relaxation when only much longer times are relevant.
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