TITLE: Numerical algorithms for a stochastic realization problem

ABSTRACT: The generalized Langevin equation is a linear stochastic integro-differential equation and serves as a coarse-grained model from statistical physics for the effective motion of a system of particles, where dissipative forces are represented by a memory kernel. The solution of this equation, e.g., the velocity of a colloid within some solvent is a Gaussian process in time. For effective numerical simulations, physical chemists are interested in an alternative extended Markov model with a number of auxiliary variables, such that the velocity component of this solution approximates well the measured data, i.e., the autocorrelation function of the velocity at an equidistant grid in time.
The standard workflow for the solution of this problem consists in (i) solving the inverse problem of reconstructing the memory kernel, and (ii) employing a Pade type approach for determining the matrix entries of the Markov model. We present a new approach which is heading directly for a data-driven extended Markov model by formulating the problem as a stochastic realization problem. Our approach has similarities to the usual model reduction ansatz in the pertinent literature. The main differences consist in the use of time domain data and the need of deriving a passive model.
We present numerical examples for the motion of a single colloid in a Lennard-Jones fluid. Data are provided from a molecular dynamics simulation of the all atom system. Then the solvent particles are ignored for the coarse grained model. We show that relatively few (i.e., less than twenty) auxiliary variables are sufficient to determine an extended Markov model which fully captures the dynamics of the colloid. The a posteriori computed memory kernel is in good agreement with those obtained from conventional inverse methods -- in fact, our reconstructions seem to exhibit fewer numerical artefacts due to inherent noise in the empirical autocorrelation data than the traditional ones.