  We analyse the near-adiabatic dynamics in a ramp through the critical point
(CP) of the classical transverse field Ising chain. This is motivated,
conceptually, by the fact that this CP -- unlike its quantum counterpart --
experiences no thermal or quantum fluctuations, and technically by the
tractability of its effective model. For a `half-ramp' from ferromagnet to CP,
the longitudinal and transverse magnetization scale as $\tau^{-1/3}$ and
$\tau^{-2/3}$, respectively, with $1/\tau$ the ramp rate, in accord with
Kibble-Zurek theory. For ferro- to paramagnetic ramps across the CP, however,
they stay closer, $\tau^{-1/2}$ and $\tau^{-1}$, to adiabaticity. This
adiabaticity enhancement compared to the half ramp is understood by casting the
dynamics in the paramagnet in the form of a non-hermitian Dirac Hamiltonian,
with the CP playing the role of an exceptional point, opening an additional
decay channel.
