Microscopic theory of brownian motion: Mori friction kernel and langevin-equation derivation

A derivation of the phenomenological Langevin equation for the momentum of a brownian particle from the generalized Langevin equation of Mori is presented. This derivation requires a detailed examination of the Mori friction kernel (or memory function). It is demonstrated, on the basis of prior work of Mazur and Oppenheim, that the Mori kernel does not admit of a well behaved expansion in the ratio of bath- and brownian-particle masses. In addition, the Mori kernel is found to decay on the slow time scale of the brownian-particle momentum. Both features, which contradict standard assumption, are traced to the influence of coupling to nonlinear powers of the momentum and preclude a Langevin-equation derivation solely on the basis of time-scale separation arguments. The Langevin-equation is recovered, however, when the small magnitude of slowly decaying contributions is taken into account.