Generalized moment expansion for observables of stochastic processes in dimensions d>1: Application to Mössbauer spectra of proteins

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Abstract
The generalized moment expansion provides an effective algorithm for the approximation of the time dependence of observables that monitor stochastic processes. Up to now this method had been applied mainly to one-variable birth–death processes or to one-dimensional Fokker–Planck systems since in these cases analytical and numerical methods for the evaluation of the generalized moments were available. Here we demonstrate that numerical sparse matrix methods can be used to extend the range of application of the generalized moment expansion to higher dimensions. For this purpose we introduce a simple but general discretization scheme for Fokker–Planck operators of Smoluchowski type which is, for these special operators, superior to common numerical discretization schemes for differential operators. As an application we determine the Mössbauer absorption spectrum of a Brownian particle in certain two- and three-dimensional potentials. This serves as a model for the motion of the heme group in myoglobin.