Langevin Equation for General Harmonic Heat Bath. I

A generalization of Mori's projection operator is used to obtain an exact equation for the force which is exerted on a particular particle in an arbitrarily disordered harmonic lattice. The equation consists of two terms, one of which is a "random force" and the other is the convolution between the correlation function of random force and the momentum of the particular particle. The random force may be identified with the force which would be exerted on a particular particle if its mass M is changed to a different mass m*. Thus the equation represents a generalization of that of Deutch and Silbey, obtained for the case m*=oo. Secondly it is shown that the same equation can be derived purely mechanically, without having recourse to any statistical tool such as Mori's operator, and that the correlation functions of the random force and the momentum of the particle under consideration are completely determined by the mechanical property of the· reference system, in which the mass of the particle is changed. For the case in which the reference system is periodic, our results reduce to what have been obtained by Rubin. Thirdly it is shown that the choice m*=oo is optimum in the sense that in this case the correlation function of the random force takes its extremal value. Generalization to the many-particle and quantum-mechanical cases and possibility of different choice of the operator are discussed.