  Cooperative effects describe atomic ensembles with exchange of photonic
excitations, such as dipole-dipole interactions. As a particular example,
superradiance arises from spontaneous emission when this exchange leads to
constructive interference of the emitted photons. Here, we introduce an
integrated method for studying cooperative radiation in many-body systems. This
method, which allows to study extended systems with arbitrarily large number of
particles can be formulated by an effective, nonlinear, two-atom master
equation that describes the dynamics using a closed form which treats single-
and many-body terms on an equal footing. We apply this method to a homogeneous
gas of initially inverted two-level atoms, and demonstrate the appearance of
both superradiance and subradiance, identifying a many-body coherence term as
the source of these cooperative effects. We describe the many-body induced
broadening - which is analytically found to scale with the optical depth of the
system - and light shifts, and distinguish spontaneous effects from induced
ones. In addition, we theoretically predict the time-dependence of subradiance,
and the phase change of the radiated field during the cooperative decay.
