From Generalized Langevin Equations to Brownian Dynamics and Embedded Brownian Dynamics
This work provides theoretical and numerical insights for approximating complex stochastic dynamics, but the results are incremental and domain-specific.
The authors reduce generalized Langevin equations to a coordinate-only stochastic model and show that a fluctuation-dissipation theorem holds. They test the accuracy of Brownian dynamics approximations and propose an embedding method for nonlocal models.
We present the reduction of generalized Langevin equations to a coordinate-only stochastic model, which in its exact form, involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar fluctuation-dissipation theorem still holds at this level. We study the approximation by the typical Brownian dynamics as a first approximation. Our numerical test indicates how the intrinsic frequency of the kernel function influences the accuracy of this approximation. In the case when such an approximate is inadequate, further approximations can be derived by embedding the nonlocal model into an extended dynamics without memory. By imposing noises in the auxiliary variables, we show how the second fluctuation-dissipation theorem is still exactly satisfied.