  A colloidal particle is a prominent example of a stochastic system, and, if
suspended in a simple viscous liquid, very closely resembles the case of an
ideal random walker. A variety of new phenomena have been observed when such
colloid is suspended in a viscoelastic fluid instead, for example pronounced
nonlinear responses when the viscoelastic bath is driven out of equilibrium.
Here, using a micron-sized particle in a micellar solution, we investigate in
detail, how these nonlinear bath properties leave their fingerprints already in
equilibrium measurements, for the cases where the particle is unconfined or
trapped in a harmonic potential. We find that the coefficients in an effective
linear (generalized) Langevin equation show intriguing inter-dependencies,
which can be shown to arise only in nonlinear baths: For example, the friction
memory can depend on the external potential that acts only on the colloidal
particle (as recently noted in simulations of molecular tracers in water in
Phys. Rev. X 7, 041065 (2017)), it can depend on the mass of the colloid, or,
in an overdamped setting, on its bare diffusivity. These inter-dependencies,
caused by so-called fluctuation renormalizations, are seen in an exact small
time expansion of the friction memory based on microscopic starting points.
Using linear response theory, they can be interpreted in terms of
microrheological modes of force-controlled or velocity-controlled driving. The
mentioned nonlinear markers are observed in our experiments, which are
astonishingly well reproduced by a stochastic Prandtl-Tomlinson model mimicking
the nonlinear viscoelastic bath. The pronounced nonlinearities seen in our
experiments together with the good understanding in a simple theoretical model
make this system a promising candidate for exploration of colloidal motion in
nonlinear stochastic environments.
