  Langevin dynamics has become a popular tool to simulate the Boltzmann
equilibrium distribution. When the repartition of the Langevin equation
involves the exact realization of the Ornstein-Uhlenbeck noise, in addition to
the conventional density evolution, there exists another type of discrete
evolution that may not correspond to a continuous, real dynamical counterpart.
This virtual dynamics case is also able to produce the desired stationary
distribution. Different types of repartition lead to different numerical
schemes, of which the accuracy and efficiency are investigated through studying
the harmonic oscillator potential, an analytical solvable model. By analyzing
the asymptotic distribution and characteristic correlation time that are
derived by either directly solving the discrete equations of motion or using
the related phase space propagators, it is shown that the optimal friction
coefficient resulting in the minimal characteristic correlation time depends on
the time interval chosen in the numerical implementation. When the recommended
"middle" scheme is employed, both analytical and numerical results demonstrate
that for a good numerical performance in efficiency as well as accuracy, one
may choose a friction coefficient in a wide range from around the optimal value
to the high friction limit.
