  Fluctuation-dissipation relations or "theorems" (FDTs) are fundamental for
statistical physics and can be rigorously derived for equilibrium systems.
Their applicability to non-equilibrium systems is, however, debated. Here, we
simulate an active microrheology experiment, in which a spherical colloid is
pulled with a constant external force through a fluid, creating
near-equilibrium and far-from-equilibrium systems. We characterize the
structural and dynamical properties of these systems, and reconstruct an
effective generalized Langevin equation (GLE) for the colloid dynamics.
Specifically, we test the validity of two FDTs: The first FDT relates the
non-equilibrium response of a system to equilibrium correlation functions, and
the second FDT relates the memory friction kernel in the GLE to the stochastic
force. We find that the validity of the first FDT depends strongly on the
strength of the external driving: it is fulfilled close to equilibrium and
breaks down far from it. In contrast, we observe that the second FDT is always
fulfilled. We provide a mathematical argument why this generally holds for
memory kernels reconstructed from a deterministic Volterra equation for
correlation functions, even for non-stationary non-equilibrium systems.
  Motivated by the Mori-Zwanzig formalism, we therefore suggest to impose an
orthogonality constraint on the stochastic force, which is in fact equivalent
to the validity of this Volterra equation. Such GLEs automatically satisfy the
second FDT and are unique, which is desirable when using GLEs for
coarse-grained modeling.
