Abstract
The generalized Langevin equation is derived for a particle of arbitrary mass in an assembly of harmonic oscillators with general interaction matrix and with an external force acting on the particle. The reduction of the equation to the ordinary Langevin equation is studied in various limits. The reduction to the Langevin equation is achieved apart from the distribution of the bath particles, and the results thus obtained are valid whether the bath is in equilibrium or not. It is found that if the Ford-Kac-Mazur interaction is assumed, the particle achieves Brownian motion regardless of its mass ratio to the bath particle. The weak coupling limit is effected by scaling the equation with the mass ratio. In the weak coupling limit, the explicit formula for the friction coefficient is obtained assuming a general interaction matrix. It is demonstrated that the generalized Langevin equation is a very convenient starting point for the study of Brownian motion.
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