An Exact derivation of the Generalized Langevin Equation for a Harmonic Oscillator Coupled to N "Bath" Oscillators via the Mori-Zwanzig Projection Operator Method

An overarching goal of statistical physics is to derive macroscopic irreversible equations starting with microscopic reversible equations of motion. The Mori-Zwanzig Projection Operator Method achieves this by projecting out the motions of irrelevant variables and studying the equations of the relevant variables. We give a concrete derivation of the Langevin equation of the motion of an oscillator that is coupled to N other "bath" oscillators using the Mori–Zwanzig projection operator technique, starting from the deterministic Newtonian equations of motion and projecting out the motion of the bath oscillators. The resulting equation of motion for the relevant variables takes the form of a Non-Markovian, Generalized Langevin Equation (GLE), which is an exact description of the coarse-grained dynamics. It is shown that the time dependence of the noise term is given by the free oscillations of the bath coordinates.