  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.
