Underdamped Langevin dynamics

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 isHere $W$ 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 distributionThe Hamiltonian flow preserves the total energy and phase-space volume; the momentum Ornstein-Uhlenbeck process preserves a centered normal distribution with coordinate variance $m{k}_{B}T$. These two facts verify the invariant density. In dimensionless units with $k_{B}T=1$ 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.

A free particle following Langevin dynamics initialized at zero position and velocity has relaxation time $\tau =m/\zeta$ and thermal diffusivity $D=k_{B}T/\zeta$. Its position is the convolution of the random force with $[1-e^{-t/\tau }]/\zeta$. The fluctuation-dissipation relation for a Langevin particle gives force covariance amplitude $2\zeta k_{B}T$, and integrating the squared kernel yields the displayed variance. At short times it grows as $2D{t}^3/(3{\tau }^2)$ because the initial velocity is cold; at long times it grows as $2Dt$. A Maxwellian initial velocity produces a different short-time transient.