dc.description.abstract
We develop and apply generalized Langevin equations (GLEs) for modeling non-Markovian dynamics in discrete time-series data. Emphasizing both theoretical and computational aspects, we investigate how memory effects, non-Gaussian noise, and coupling between multi-dimensional reaction coordinates shape the dynamics of stochastic observables.
Through diverse examples from synthetic, simulated, and experimental data, we show that the GLE framework offers a versatile and practical tool for data-driven modeling and prediction of complex dynamics, and establish it as a physically grounded and efficient framework for time-series analysis across molecular, environmental, and financial systems.
We begin by outlining the theoretical basis of the GLE via the projection operator formalism and present numerical techniques for extracting parameters from discrete data, suitable for analyzing systems in and out of equilibrium. In the molecular dynamics setting, we apply these methods to dihedral transitions in butane using molecular dynamics (MD) trajectories to compare various GLE formulations. Incorporating non-Gaussian noise improves predictions of mean first-passage times, surpassing Gaussian-based Markovian embeddings. For multi-dimensional systems, we develop GLE models for the coupled dynamics of the dihedral angles of pentane, revealing strong off-diagonal friction from intramolecular interactions. These models capture cross-displacement statistics and reaction kinetics in MD data.
For real-world applications, convolution filtering decomposes time-series data, and we fit a GLE model to the fast-fluctuating component. This reveals long-range memory in daily weather data, while financial assets exhibit rapid decay, consistent with the efficient-market hypothesis. GLE-based forecasting rivals the performance of state-of-the-art methods like recurrent neural networks while being more computationally efficient and offering physically interpretable insights.
As part of time-series analysis, we explore the microscopic origins of memory friction $\Gamma(t)$. We begin by the computation of static friction coefficients from MD simulations and introduce an integration-based method that accurately estimates friction and separates electrostatic and Lennard-Jones contributions to diffusivity in water. Next, we examine frequency-dependent friction spectra for water and methane in bulk water. Water spectra align with hydrodynamic theory for a diffusing sphere in a viscous medium, enabling feature attribution of $\Gamma(t)$, while methane reveals deviations that can be rationalized by the concept of a viscoelastic hydration shell. We extend these findings to polymeric hydrogels, where a generalized Stokes-Einstein relation considering an interfacial adsorption shell accounts for differences between viscosities from experimental macro- and microrheology data.
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