  Stochastic Langevin molecular dynamics for nuclei is derived from the
Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics)
in a Kac-Zwanzig setting, with the initial data for the electrons
stochastically perturbed from the ground state and the ratio, $M$, of nuclei
and electron mass tending to infinity. The Ehrenfest nuclei dynamics is
approximated by the Langevin dynamics with accuracy $o(M^{-1/2})$ on bounded
time intervals and by $o(1)$ on unbounded time intervals, which makes the small
$\mathcal{O}(M^{-1/2})$ friction and $o(M^{-1/2})$ diffusion terms visible. The
initial electron probability distribution is a Gibbs density at low temperture,
derived by a stability and consistency argument: starting with any equilibrium
measure of the Ehrenfest Hamiltonian system, the initial electron distribution
is sampled from the equilibrium measure conditioned on the nuclei positions,
which after long time leads to the nuclei positions in a Gibbs distribution
(i.e. asymptotic stability); by consistency the original equilibrium measure is
then a Gibbs measure.The diffusion and friction coefficients in the Langevin
equation satisfy the Einstein's fluctuation-dissipation relation.
