  Effective theories describing black hole exteriors resemble open quantum
systems inasmuch as many unmeasurable degrees of freedom beyond the horizon
interact with those we can see. A solvable Caldeira-Leggett type model of a
quantum field that mixes with many unmeasured thermal degrees of freedom on a
shared surface was proposed in arXiv:2106.09854 to provide a benchmark against
which more complete black hole calculations might be compared. We here use this
model to test two types of field-theoretic approximation schemes that also lend
themselves to describing black hole behaviour: Open EFT techniques (as applied
to the fields themselves, rather than Unruh-DeWitt detectors) and mean-field
methods. Mean-field methods are of interest because the effective Hamiltonians
to which they lead can be nonlocal; a possible source for the nonlocality that
is sometimes entertained as being possible for black holes in the near-horizon
regime. Open EFTs compute the evolution of the field state, allowing discussion
of thermalization and decoherence even when these occur at such late times that
perturbative methods fail (as they often do). Applying both of these methods to
a solvable system identifies their domains of validity and shows how their
predictions relate to more garden-variety perturbative tools.
