A Lanczos-based method for computing Ornstein-Uhlenbeck representations for the generalized langevin equation
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Abstract
The generalized Langevin equation describes the velocity of a macromolecule via the convolution of the history of its velocity and a memory kernel. It employs the coarse graining approach of only simulating the macromolecule's velocity explicitly while collisions with other particles are modeled via a random force process. If the memory is a Prony series, it can be represented by an Ornstein-Uhlenbeck process by introducing auxiliary variables.
We are interested in the inverse problem, which consists in determining a good approximation of the memory kernel given the covariance function of the velocity. We present a new method for solving this problem. Instead of first approximating the memory kernel in order to determine a suitable Prony series and an Ornstein-Uhlenbeck process approximation, this method directly yields an Ornstein-Uhlenbeck process which approximates the given velocity covariance function. It is based on a variant of the (multi-dimensional) Lanczos algorithm followed by several post-processing steps and can also be applied to the multi-dimensional generalized Langevin equation.
We test this method using data from molecular dynamics simulations. For the one-dimensional case we introduce an extension of the algorithm for the setting where the macromolecule is subject to an additional constant external force.