Another derivation of generalized Langevin equations

TL;DR

This paper provides a comprehensive, self-contained derivation of generalized Langevin equations using classical mechanics, clarifying key concepts and extending applicability to various physical systems.

Contribution

It offers a detailed, self-contained derivation of generalized Langevin equations, clarifying conceptual issues and broadening applicability to diverse physical systems.

Findings

Clarifies the meaning of divergence of Poisson brackets

Explains the role of nonlinear damping coefficients

Derivation applies to a wide range of classical systems

Abstract

The formal derivation of Langevin equations (and, equivalently Fokker-Planck equations) with projection operator techniques of Mori, Zwanzig, Kawasaki and others apparently not has widely found its way into textbooks. It has been reproduced dozens of times on the fly with many references to the literature and without adding much substantially new. Here we follow the tradition, but strive to produce a self-contained text. Furthermore, we address questions that naturally arise in the derivation. Among other things the meaning of the divergence of the Poisson brackets is explained, and the role of nonlinear damping coefficients is clarified. The derivation relies on classical mechanics, and encompasses everything one can construct from point particles and potentials: solids, liquids, liquid crystals, conductors, polymers, systems with spin-like degrees of freedom ... Einstein relations and Onsager reciprocity relations come for free.