Thermodynamically-consistent semi-classical $\ell$-changing rates
  We compare the results of the semi-classical (SC) and quantum-mechanical (QM)
formalisms for angular-momentum changing transitions in Rydberg atom collisions
given by Vrinceanu & Flannery, J. Phys. B 34, L1 (2001), and Vrinceanu, Onofrio
& Sadeghpour, ApJ 747, 56 (2012), with those of the SC formalism using a
modified Monte Carlo realization. We find that this revised SC formalism agrees
well with the QM results. This provides further evidence that the rates derived
from the QM treatment are appropriate to be used when modelling recombination
through Rydberg cascades, an important process in understanding the state of
material in the early universe. The rates for $\Delta\ell=\pm1$ derived from
the QM formalism diverge when integrated to sufficiently large impact
parameter, $b$. Further to the empirical limits to the $b$ integration
suggested by Pengelly & Seaton, MNRAS 127, 165 (1964), we suggest that the
fundamental issue causing this divergence in the theory is that it does not
fully cater for the finite time taken for such distant collisions to complete.
