Effective equation of motion of a passive particle immersed in an active fluid
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Abstract
Implicit models of passive, equilibrium systems have been used for many years to
study and understand the physical behavior of such systems. Given the success of
understanding equilibrium systems through such models, recent studies have focused
on coarse-graining non-equilibrium systems. However, non-equilibrium systems are
highly dependent on the system’s dynamic properties, which are necessarily altered
in a coarse-grained model. Furthermore, certain assumptions which are made
in equilibrium coarse-grained modeling are no longer valid beyond equilibrium.
Therefore, coarse-graining of non-equilibrium systems must be done with extreme
care and consciousness of the true system dynamics.
Perhaps the most ubiquitous example of a coarse-grained model is the Brownian
particle model, in which the motions of fluid particles surrounding a much larger,
immersed particle are only implicitly represented to eliminate these degrees of
freedom. In this thesis, we address the question: what happens when the fluid is far
from equilibrium? We explore the dynamics of a passive probe particle immersed in
an active bath and its implications for coarse-grained modeling. We use effective
generalized Langevin equations, which explicitly include memory effects, to examine
the immersed probe dynamics.
In the first part of this thesis, we classify the behaviors that signify the system’s non-
equilibrium nature. Although the probe adopts many active-particle-like behaviors,
the trajectory of the probe does not exhibit obvious non-equilibrium signatures. To
tell that the probe is out of equilibrium requires examination of its behavior in tandem
with that of the active fluid. Alternatively, applying a small perturbation to the probe,
reveals a violation of the first fluctuation dissipation theorem. In the second part of
this thesis, we determine the mechanism behind the active-particle-like behavior of
the probe. This behavior cannot simply be attributed to the convective motion of
the active bath. Instead, the boundary of the probe contributes significantly to these
adopted dynamics by causing active bath particles to accumulate behind the probe
with respect to its instantaneous velocity. This gathering of active bath particles then
pushes the probe, which in turn promotes its active-particle-like behavior. In the
final part of this thesis, we map the dynamics of a probe immersed in an active bath
and subject to an external force onto an equilibrium coarse-grained model. We find
that this system can be mapped onto a physically meaningful coarse-grained model.
However, due to the activity of the bath, the external force in such an equation is
not equal to the physical external force, but rather a renormalized external force.