Abstract
The chapter discusses the approach to the Brownian limit starting with the microscopic Liouville equation. The Deutch-Oppenheimer approach is used to derive the Langevin equation in the Brownian limit. The Brownian limit arises when the conditions discussed above are valid, which essentially means that all dynamical processes have ceased in the bath system in the presence of a Brownian particle. The general forms of the Langevin equation for any function of the dynamical variables characterizing the state of the large particles are exact. It is useful to consider the relationship between the many-body Langevin equation derived from a microscopic starting point and the macroscopic approach to the derivation. The Brownian limit is valid when all dynamical processes have ceased in the bath system in the presence of a Brownian particle before the Brownian particle changes its dynamical state significantly. The intermediate scattering function and the longitudinal current autocorrelation function show the time-scale separation required for the Fokker–Planck, Smoluchowski, or Langevin equations to be applicable.
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