Molecular dynamics

Remarks

The exact flow is a Hamiltonian system that conserves the energy $\sum_i \frac12 m_i \abs{v_i}^2 + U(q)$. The velocity Verlet scheme is a symplectic integrator of second order and is time-reversible, so the computed energy does not drift but oscillates near its initial value for long times. Under the ergodic hypothesis, time averages approximate averages in the microcanonical ensemble; thermostats, such as stochastic Langevin forces, are added to sample the canonical ensemble instead. Example: one particle of mass $m$ in the potential $U(q) = \frac12 k q^2$ is a harmonic oscillator, and velocity Verlet is stable exactly when $\Delta t < 2 / \omega$ with $\omega = \sqrt{k/m}$. Non-example: the explicit Euler scheme $q^{k+1} = q^k + \Delta t \, v^k$, $v^{k+1} = v^k + \Delta t \, F(q^k)/m$ is not symplectic, and for the harmonic oscillator its energy grows by the factor $1 + \omega^2 \Delta t^2$ at every step.