Stationary state distribution and efficiency analysis of the Langevin equation via real or virtual dynamics

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 minimum 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 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.