  This paper is an attempt to extend the recent understanding of the Page curve
for evaporating black holes to more general systems coupled to a heat bath.
Although calculating the von Neumann entropy by the replica trick is usually a
challenge, we have identified two solvable cases. For the initial section of
the Page curve, we sum up the perturbation series in the system-bath coupling
$\kappa$; the most interesting contribution is of order $2s$, where $s$ is the
number of replicas. For the saturated regime, we consider the effect of an
external impulse on the entropy at a later time and relate it to OTOCs. A
significant simplification occurs in the maximal chaos case such that the
effect may be interpreted in terms of an intermediate object, analogous to the
branching surface of a replica wormhole.
