Generalized Langevin equations for systems with local interactions
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We present a new method to approximate the Mori-Zwanzig (MZ) memory integral in generalized Langevin equations (GLEs) describing the evolution of observables in high-dimensional nonlinear systems with local interactions. Building upon the Faber operator series we recently developed for the orthogonal dynamics propagator, and an exact combinatorial algorithm that allows us to compute memory kernels to any desired accuracy from first principles, we demonstrate that the proposed method is effective in computing auto-correlation functions, intermediate scattering functions and other important statistical properties of the observables. We also develop a new stochastic process representation that combines MZ memory kernels and Karhunen-Loève (KL) series expansions to build reduced-order models of observables in statistical equilibrium. Numerical applications are presented for the Fermi-Pasta-Ulam model, and for random wave propagation in homegeneous media.
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