  We present an extended version of the semi-analytical model, GalICS. Like its
predecessor, eGalICS applies a post-treatment of the baryonic physics on
pre-computed dark-matter merger trees extracted from an N-body simulation. We
review all the mechanisms that affect, at any given time, the formation and
evolution of a galaxy in its host dark-matter halo. We mainly focus on the gas
cycle from the smooth cosmological accretion to feedback processes. To follow
this cycle with high accuracy we introduce some novel prescriptions: i) a
smooth baryonic accretion with two phases: a cold mode and a hot mode built on
the continuous dark-matter accretion. In parallel to this smooth accretion, we
implement the standard photoionisation modelling to reduce the input gas flow
on the smallest structures. ii) a complete monitoring of the hot gas phase. We
compute the evolution of the core density, the mean temperature and the
instantaneous escape fraction of the hot atmosphere by considering that the hot
gas is in hydrostatic equilibrium in the dark-matter potential well, and by
applying a principle of conservation of energy on the treatment of gas
accretion, supernovae and super massive black hole feedback iii) a new
treatment for disc instabilities based on the formation, the migration and the
disruption of giant clumps. The migration of such clumps in gas-rich galaxies
allows to form pseudo-bulges. The different processes in the gas cycle act on
different time scales, and we thus build an adaptive time-step scheme to solve
the evolution equations. The model presented here is compared in detail to the
observations of stellar-mass functions, star formation rates, and luminosity
functions, in a companion paper.
