Generalized Langevin equation revisited: mechanical random force and self-consistent structure

The generalized Langevin equation (GLE) is considered with emphasis put on a mechanical random force , whose time evolution is not natural due to the presence of a projection operator in a propagator. We first derive the GLE for a set of dynamical variables of interest not by formal manipulation of the operator identity as is usually done, but by expressing the random force in terms of natural trajectory in the phase space. Based on this new derivation we observe two structural characteristics, (1) a linear functional interdependence between and and (2) a self-consistent structure (SCS) inherent in the GLE. The SCS gives an iterative procedure to determine and its time correlation function, i.e. a memory function K ( t ), once we have numerical data for supplied by computer experiments. This method generates and a memory function K ( t ), nicely for an exactly solvable many-body system after several iterations. Some merits of using the SCS are further discussed in connection with numerical methods to calculate the autocorrelation function of .