Stochastic processes with finite correlation time: modeling and application to the generalized Langevin equation

The kangaroo process (KP) is characterized by various forms of the covariance and can serve as a useful model of random noises. We discuss properties of that process for the exponential, stretched exponential and algebraic (power-law) covariances. Then we apply the KP as a model of noise in the generalized Langevin equation and simulate solutions by a Monte Carlo method. Some results appear to be incompatible with requirements of the fluctuation-dissipation theorem because probability distributions change when the process is inserted into the equation. We demonstrate how one can construct a model of noise free of that difficulty. This form of the KP is especially suitable for physical applications.Comment: 22 pages (RevTeX) and 4 figure

Time correlation functions for the one-dimensional Lorentz gas

The velocity autocorrelation function and related quantities are investigated for the one-dimensional deterministic Lorentz gas, consisting of randomly distributed fixed scatterers and light particles moving back and forth between two of these at a constant given speed. An expansion for the velocity autocorrelation function given by Grassberger, which is useful for short times, is reconstructed. The long time behavior is investigated by Fourier transform techniques. For large time t the velocity autocorrelation function decays as exp(-ct't1/2) and in addition oscillates with a period increasing as t1/2. A velocity average over a Maxwellian changes this long time behavior to exp(-c't2/3), while the oscillations are removed. The Green's function is also investigated. Its spatial and temporal Fourier transform, the incoherent scattering function, exhibits strongly non-Lorentzian behavior