Abstract: The momentum correlation functuin is a very important physical quantity in non-equilibrium statistical mechanics.However，where are a few errors for the explain of this function and related contents in our usual textbooks.This paper starts from academic research and discusses those un-rigorous descriptions.The Langevin equation subjected to a white noise is regarded as a model in the present textbooks，but it is difficulty for clearing two characteristic times in the Brownian motion: the correlation time of random force and relaxation time of system.This paper solves this question in terms of the generalized Langevin equation with an Ornstein-Uhlenbeck colored noise. The ergodicity of the momentum variable is studied by using the Khinchin theorem and thus the cost of the momentum correlation function is deepen.

Key words: Brownian motion, momentum correlation function, generalized Langevin equation, ergodicity