  We study the occurrence of a Bose-Einstein transition in a dilute gas with
repulsive interactions, starting from temperatures above the transition
temperature. The formalism, based on the use of Ursell operators, allows us to
evaluate the one-particle density operator with more flexibility than in
mean-field theories, since it does not necessarily coincide with that of an
ideal gas with adjustable parameters (chemical potential, etc.). In a first
step, a simple approximation is used (Ursell-Dyson approximation), which allow
us to recover results which are similar to those of the usual mean-field
theories. In a second step, a more precise treatment of the correlations and
velocity dependence of the populations in the system is elaborated. This
introduces new physical effects, such as a marked change of the velocity
profile just above the transition: low velocities are more populated than in an
ideal gas. A consequence of this distortion is an increase of the critical
temperature (at constant density) of the Bose gas, in agreement with those of
recent path integral Monte-Carlo calculations for hard spheres.
