Generalized Langevin equations for systems with local interactions

Abstract

We present a new method to approximate the Mori-Zwanzig (MZ) memory integral in generalized Langevin equations (GLEs) describing the evolution of smooth 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 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 observable. We also develop a new stochastic process representation of the MZ fluctuation term for systems in statistical equilibrium. Numerical applications are presented for the Fermi-Pasta-Ulam model, and for random wave propagation in homogeneous media.

arXiv comment27 pages, 6 figures

References in corpus (15)

• Mode-Coupling Theory (MCT) Lecture Notes
• Data-driven parameterization of the generalized Langevin equation
• Current fluctuations for anharmonic chains in thermal equilibrium
• A Dynamic Subgrid Scale Model for Large Eddy Simulations Based on the Mori-Zwanzig Formalism
• Non-Markovian Closure Models for Large Eddy Simulations using the Mori-Zwanzig Formalism
• Data-driven closures for stochastic dynamical systems
• A comparative study of two stochastic mode reduction methods
• The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
• Optimal Prediction for Hamiltonian partial differential equations
• On the estimation of the Mori-Zwanzig memory integral
• Invariant measures of the 2D Euler and Vlasov equations
• Renormalized reduced models for singular PDEs
• Memory effects in the Fermi-Pasta-Ulam Model
• Duality and Conditional Expectations in the Nakajima-Mori-Zwanzig Formulation
• A series expansion for the time autocorrelation of dynamical variables

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• Data-driven molecular modeling with the generalized Langevin equation
• Generalized second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications
• Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations
• Connection among stochastic Hamilton-Jacobi-Bellman equation, path-integral, and Koopman operator on nonlinear stochastic optimal control
• Statistics for stochastic differential equations and approximations of resolvent