We consider memory effects in two examples of colloidal dispersions
far from equilibrium: A driven colloid that is pulled by an external
force through a Lennard-Jones fluid, and a colloid immersed in a bath
of active Brownian particles. These systems are mapped to an implicit
solvent model containing colloids only, and the effective dynamical
equation takes the form of a Generalized Langevin Equation (GLE). The
nonequilibrium driving force have a profound effect on the amplitude
and shape of memory kernels. The first fluctuation dissipation
relation, which relates the non-equilibrium response of the system to
a perturbation with the stationary correlation functions, breaks down
far from equilibrium. Nevertheless, the second fluctuation
dissipation theorem that relates the noise fluctuations and the memory
kernel is always strictly fulfilled.
We present a mathematical argument why this generally holds for memory
kernels that are reconstructed from a deterministic Volterra equation
for correlation functions, even for non-stationary, non-Hamiltonian,
non-equilibrium systems. We also discuss the relation to the
corresponding result from the Mori-Zwanzig formalism for Hamiltonian
systems.