Relaxation and Memory Times in a Non-Markovian Generalized Langevin–Landau–Lifshitz Equation

The memory time τK is identified with the collision time τc by calculating the memory time using an Ornstein–Uhlenbeck model. The effective momentum-transfer cross-section is used to derive the relaxation time τr for various relativistic models with and without quantum effects and the radiation reaction. Models based on Møller scattering are also considered. The distinction between collision and relaxation times is established, as is the role they play in a non-Markovian relativistic generalized Langevin equation based on the Landau–Lifshitz equation. The memory time τK does not generally coincide with the relaxation time, but we show that in unphysical cases, they coincide. Relaxation times are calculated progressively by considering the differential cross-sections from various models ranging across the relativistic proposals of Rutherford; Huschilt and Baylis; Aguiar, Barroso, and Barone; and a modified Mott model. We also improve a Landau–Lifshitz model, but the effects due to the radiation reaction are negligible. Finally, the Møller model yields the best result, as it accounts for quantum effects in electron–electron collisions. Given the nondegenerate energy levels and fields considered here, a non-Markovian generalized Langevin equation based on the Landau–Lifshitz equation is derived.