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Langevin dynamics has become a popular tool to simulate the Boltzmannequilibrium distribution. When the repartition of the Langevin equationinvolves the exact realization of the Ornstein-Uhlenbeck noise, in addition tothe conventional density evolution, there exists another type of discreteevolution that may not correspond to a continuous, real dynamical counterpart.This virtual dynamics case is also able to produce the desired stationarydistribution. Different types of repartition lead to different numericalschemes, of which the accuracy and efficiency are investigated through studyingthe harmonic oscillator potential, an analytical solvable model. By analyzingthe asymptotic distribution and characteristic correlation time that arederived by either directly solving the discrete equations of motion or usingthe related phase space propagators, it is shown that the optimal frictioncoefficient resulting in the minimal characteristic correlation time depends onthe time interval chosen in the numerical implementation. When the recommended"middle" scheme is employed, both analytical and numerical results demonstratethat for a good numerical performance in efficiency as well as accuracy, onemay choose a friction coefficient in a wide range from around the optimal valueto the high friction limit.

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