  A theoretical framework is developed to clarify the relation between the
profiles of density and temperature in the Scrape Off Layer (SOL) with the
fluctuations (filaments) that generate them. The framework is based on the
dynamics of independent filaments and on their statistical behaviour and can be
used to rigorously understand the mechanisms that lead to the non-exponential
nature of the radial SOL profiles as well as the increase of the relative
fluctuation amplitude in the far SOL. Several models for the dynamics of the
filaments, which can be applied to the framework, are derived and discussed for
the purpose of identifying how different assumptions lead to the emergence of
features in the profiles. It is found that multiple alternative models can
explain the observations, thus motivating more stringent and focused
experimental analysis. In particular, radially accelerating filaments, less
efficient parallel exhaust and also a statistical distribution of the velocity
of the filaments can all contribute to induce flatter profiles in the far SOL.
A quite general result is the resiliency of the non-exponential nature of the
profiles. At the same time, several of the models discussed can also capture
the increase of the relative fluctuation amplitude observed in the far SOL. It
is also shown that several scenarios are compatible with the broadening of the
SOL, which could be caused by charge exchange interactions with neutral
particles or by a significant radial acceleration of the filaments.
