Long-time tails of correlation and memory functions

We review the generalized Langevin equation, which is a transformation and reformulation of equation of motion, from the two viewpoints: the projection operator method developed by Mori and the recurrence relations method developed by Lee. The fluctuating forces acting on the Bloch electrons’ current are clarified the strongly colored quantum fluctuations with the spontaneous interband transitions leading to a long-time tail of 1/t for the envelope of the memory function. The velocity autocorrelation functions in the coupled harmonic oscillator on the Bethe lattice have a long-time tail of 1/tt. The oscillation and the form of decay found in correlation functions affect transport coefficients given by the integrated intensity up to infinity. We also study the force–force correlation functions often used as an approximation to the memory function.