  To investigate the impact of non-linear interactions on dynamic coarse
graining, we study a simplified model system, featuring a tracer particle in a
complex environment. Using a projection operator formalism and computer
simulations, we systematically derive generalized Langevin equations describing
the dynamics of this particle. We compare different kinds of linear and
non-linear coarse-graining procedures to understand how non-linearities enter
reconstructed generalized Langevin equations and how they influence the
coarse-grained dynamics. For non-linear external potentials, we show
analytically and numerically that the non-Gaussian parameter and the incoherent
intermediate scattering function will not be correctly reproduced by the
generalized Langevin equation if a linear projection is applied. This, however,
can be overcome by using non-linear projection operators. We also study
anharmonic coupling between the tracer and the environment and demonstrate that
the reconstructed memory kernel develops an additional trap-dependent
contribution. Our study highlights some open challenges and possible solutions
in dynamic coarse graining.
