Numerical integration of a non-Markovian Langevin equation with a thermal band-passing noise

An accurate and fast approach for numerically solving a non-Markovian Langevin equation with a thermal band-passing noise is proposed. The algorithm combines the closed integration for both damping and noise terms with the Runge-Kutta method for nonlinear force in the Markovian Langevin equation transferred from the original equation. The present algorithm is tested through simulating diffusion of a free particle by using different initial distributions, and then a strong superdiffusion is shown. The mean velocity of a particle in a flashing ratchet driven by the band-passing colored noise is calculated numerically. The dependence of the resulting mean velocity on temperature, asymmetry of the ratchet potential, and inertia of the particle is discussed, and some novel behaviors in comparison with the usual model are observed.