Dual Lanczos vector space. II. Lanczos transformations and contracted descriptions of relaxation processes

Abstract

We consider the relationship of the dual Lanczos transformation method, introduced in paper I, to the projection operator methods of Mori and Zwanzig for treating relaxation processes. It is shown that Mori’s memory function formalism and ‘‘generalized’’ Langevin description of relaxation processes represents a special case in the reversible limit of the ‘‘global’’ dual Lanczos transformation method. Unlike Mori’s memory function formalism, the present results are not limited to the description of autocorrelation functions for reversible systems. Our results apply to auto- and cross-correlation functions, as well as, the ensemble average of observables for both reversible and irreversible systems. In fact, we are able to construct a continued fraction representation of both auto- and cross-correlation functions. Introducing a ‘‘bath’’ dual Lanczos transformation scheme, we show that the first bath dual Lanczos projection gives rise to the Zwanzig projection operator method for constructing an equation of motion for a reduced distribution function. Additional bath dual Lanczos projections allow us to go beyond the usual results obtained in applications of the Zwanzig method.