## 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. The development of these methods is motivated by the need for accurate methods for
simulating slow processes in polymer systems such as two-site intermolecular distances in supercoiled DNA, which evolve
over the time scale of milliseconds. Our new approaches refine the common Brownian dynamics (BD) scheme, which
approximates the Langevin equation in the highly damped diffusive limit. Our LTID ("long-time-step inertial dynamics")
method is based on an eigenmode decomposition of the friction tensor. The less costly integrator IBD ("inertial Brownian
dynamics") modifies the usual BD algorithm by the 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 of BD for the simple example of a
harmonic oscillator. We find that the LTID method produces the expected correlation structure for Langevin dynamics
regardless of the level of damping. In fact, LTID is the only consistent method 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 is significantly more accurate than BD. IBD is also less
computationally expensive 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 previously accessible to only the usual BD approach. Such simulations are
discussed in our companion paper, for long DNA molecules modeled as wormlike chains. (C)2000 American Institute of
Physics.

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