Inertial stochastic dynamics. I. Long-time-step methods for langevin dynamics

Two algorithms are presented for integrating the Langevin dynamics equation with long numerical time steps while treating the mass terms as finite. Th e development of these methods is motivated by the need for accurate method s for simulating slow processes in polymer systems such as two-site intermo lecular distances in supercoiled DNA, which evolve over the time scale of m illiseconds. Our new approaches refine the common Brownian dynamics (BD) sc heme, which approximates the Langevin equation in the highly damped diffusi ve limit. Our LTID ("long-time-step inertial dynamics") method is based on an eigenmode decomposition of the friction tensor. The less costly integrat or IBD ("inertial Brownian dynamics") modifies the usual BD algorithm by th e addition of a mass-dependent correction term. To validate the methods, we evaluate the accuracy of LTID and IBD and compare their behavior to that o f BD for the simple example of a harmonic oscillator. We find that the LTID method produces the expected correlation structure for Langevin dynamics r egardless of the level of damping. In fact, LTID is the only consistent met hod among the three, with error vanishing as the time step approaches zero. In contrast, BD is accurate only for highly overdamped systems. For cases of moderate overdamping, and for the appropriate choice of time step, IBD i s significantly more accurate than BD. IBD is also less computationally exp ensive than LTID (though both are the same order of complexity as BD), and thus can be applied to simulate systems of size and time scale ranges previ ously accessible to only the usual BD approach. Such simulations are discus sed in our companion paper, for long DNA molecules modeled as wormlike chai ns. (C) 2000 American Institute of Physics. [S0021-9606(00)50717-X].