Starting from the orthogonal dynamics of any given set of variables with respect to the projection variable used to derive the Mori–Zwanzig equation, a set of coupled Volterra equations is obtained that relate the projected time correlation functions between all the variables of interest. This set of equations can be solved using standard numerical inversion methods for Volterra equations, leading to a very convenient yet efficient strategy to obtain any projected time correlation function or contribution to the memory kernel entering a generalized Langevin equation. Using this strategy, the memory kernel related to the diffusion of tagged particles in a bulk Lennard–Jones fluid is investigated up to the long-term regime to show that the repulsive–attractive cross-contribution to memory effects represents a small but non-zero contribution to the self-diffusion coefficient.
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Duhamel’s principle gives the solution to an inhomogeneous initial value problem (t) − Du(t) = F(t), with u(0) = u0 for a time-independent operator D and a function F as . Identifying u = M+, and from Eq. (21) straightforwardly shows that Eq. (27) derives from Duhamel’s principle, which is equivalent to applying Dyson’s identity [Eq. (6)] to M(0). Similarly, Eq. (28) derives from Duhamel’s principle (u = N−, and ) in Agreement with the corresponding Dyson identity [Eq. (30)].
