  We critically review the literature on the Debye absorption peak of liquid
water and the excess response found on the high frequency side of the Debye
peak. We find a lack of agreement on the microscopic phenomena underlying both
of these features. To better understand the molecular origin of Debye peak we
ran large scale molecular dynamics simulations and performed several different
distance-dependent decompositions of the low frequency dielectric spectra,
finding that it involves processes that take place on scales of 1.5-2.0 nm. We
also calculated the $k$-dependence of the Debye relaxation, finding it to be
highly dispersive. These findings are inconsistent with models that relate
Debye relaxation to local processes such as the rotation/translation of
molecules after H-bond breaking. We introduce the spectrumfitter Python package
for fitting dielectric spectra and analyze different ways of fitting the high
frequency excess, such as including one or two additional Debye peaks. We
propose using the generalized Lydanne-Sachs-Teller (gLST) equation as a way of
testing the physicality of model dielectric functions. Our attempts at fitting
the experimental spectrum using the gLST relation as a constraint indicate that
the traditional way of fitting the excess response with secondary and tertiary
Debye relaxations is problematic. All of our work is consistent with the recent
theory of Popov et al. (2016) that Debye relaxation is due to the migration of
Bjerrum-like defects in the hydrogen bond network. Under this theory, the
mechanism of Debye relaxation in liquid water is similar to the mechanism in
ice, but the heterogeneity and power-law dynamics of the H-bond network in
water results in excess response on the high frequency side of the peak.
