A simple and effective Verlet-type algorithm for simulating Langevin   dynamics
  We present a revision to the well known Stormer-Verlet algorithm for simulating second order differential equations. The revision addresses the inclusion of linear friction with associated stochastic noise, and we analytically demonstrate that the new algorithm correctly reproduces diffusive behavior of a particle in a flat potential. For a harmonic oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping, frequency, and time step for both underdamped and over damped behavior within the usual the stability limit of the Verlet algorithm. Given the structure and simplicity of the method we conclude this approach can trivially be adapted for contemporary applications, including molecular dynamics with extensions such as molecular constraints. 
Niels Gr{\o}nbech-Jensen and Oded Farago
cond-mat.stat-mech cond-mat.soft physics.comp-ph