Pedagogical derivation of the generalized Langevin equation for an oscillator bath system via the Mori–Zwanzig formalism

Abstract

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. This article gives a concrete and pedagogic 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. The effect of the irrelevant variables appears as two terms in the GLE: (a) a memory kernel: a time-dependent "drag" term that accounts for the history-dependent influence of the irrelevant dynamics, and, (b) random noise: a stochastic term that represents the fluctuating, noisy forces that arise from the irrelevant degrees of freedom. It is shown that the time dependence of the noise term is given by the free oscillations of the bath coordinates.