Generalized Langevin Equation: An Introductory Review for Biophysicists

An introductory, pedagogical review of the generalized Langevin equation (GLE) within the classical regime is presented. It is intended to be accessible to biophysicists with an interest in molecular dynamics (MD). Section 1 presents why the equation may be of interest within biophysical modeling. A detailed elementary ¯rst principles derivation of the (multidimensional) Kac–Zwanzig model is presented. The literature is reviewed with a focus on biophysical applications and representation by Markovian stochastic di®erential equations. The relationship with the Mori–Zwanzig formalism is discussed. The framework of model reduction and approximation is emphasized. Some open problems are identi¯ed. Keywords: Generalized Langevin equation; Kac–Zwanzig model; Markovian approximation; biophysical modeling; model reduction; Mori–Zwanzig formalism; anomalous di®usion.