On random force correction for large time steps in semi-implicitly discretized overdamped Langevin equations

In this study, we focused on the treatment of random forces in a semi-implicitly discretized overdamped Langevin (OL) equation with large time steps. In the usual implicit approach for a nonstochastic mechanical equation, the product of the time interval and Hessian matrix was added to the friction matrix to construct the coefficient matrix for solution updates, which were performed using Newton iteration. When large time steps were used, the additional term, which could be regarded as an artificial friction term, prevented the amplification of oscillations associated with large eigenvalues of the Hessian matrix. In this case, the damping of the high-frequency terms did not cause any discrepancy because they were outside of our interest. However, in OL equations, the friction coefficient was coupled to the random force; therefore, excessive artificial friction may have obscured the effects caused by the stochastic properties of the fluctuations. Consequently, we modified the random force in the proposed semi-implicit scheme so that the total random force was consistent with the friction including the additional artificial term. By deriving a discrete Fokker-Planck (FP) equation from the discretized OL equation, we showed how our modification improved the distribution of the numerical solutions of discrete stochastic processes. Finally, we confirmed the validity of our approach in numerical simulations of a freely jointed chain.