The Langevin and generalized Langevin equations

Abstract In Chapter 15, stochastic equations of motion, specifically the Langevin and generalized Langevin equations, are discussed as a means of generating classical ensemble distributions and generating dynamical quantities of systems coupled to harmonic baths. The chapter begins with a derivation of the Caldeira-Leggett Hamiltonian, which is then used to derive the generalized Langevin equation and the Langevin equation as a special case. The physical meaning of the different terms in the generalized Langevin equation are discussed, and a series of increasingly complex analytically solvable examples is presented. Numerical algorithms for integrating the Langevin and overdamped Langevin equations are presented. Following this, the incorporation of stochastic dynamics into multiple time-stepping approaches as a means of eliminating resonances is presented. The chapter then shows how to tailor memory kernels to mimic specific desired bath effects in classical systems. The chapter concludes with discussions of sampling stochastic transition paths and Mori-Zwanzig projection-operator theory for deriving generalized Langevin equations