Rings or annulus-like features have been observed in most imaged debris discs. Outside the main ring, while some systems (e.g., β Pictoris and AU Mic) exhibit smooth surface brightness profiles (SB) that fall off roughly as ∼r–3.5
, others (e.g. HR 4796A and HD 139664) display large drops in luminosity at the ring's outer edge and steeper radial luminosity profiles. We seek to understand this diversity of outer edge profiles under the ``natural'' collisional evolution of the system, without invoking external agents such as planets or gas. We use a multi-annulus statistical code to follow the evolution of a collisional population, ranging in size from dust grains to planetesimals and initially confined within a belt (the ``birth ring''). The crucial effect of radiation pressure on the dynamics and spatial distribution of the smallest grains is taken into account. We explore the dependence of the resulting disc surface brightness profile on various parameters. The disc typically evolves toward a ``standard'' steady state, where the radial surface brightness profile smoothly decreases with radius as r–3.5
outside the birth ring. This confirms and extends the semi-analytical study of Strubbe & Chiang (2006ApJ...648..652S
) and provides a firm basis for interpreting observed discs. Deviations from this typical profile, in the form of a sharp outer edge and a steeper fall-off, occur for two ``extreme'' cases: 1) when the birth ring is so massive that it becomes radially optically thick for the smallest grains. However, the required disc mass is probably too high here to be realistic; 2) when the dynamical excitation of the dust-producing planetesimals is so low (<e> and <i>≤0.01) that the smallest grains, which otherwise dominate the optical depth of the system, are preferentially depleted. This low-excitation case, although possibly not generic, cannot be ruled out by observations for most systems, . Our ``standard'' profile provides a satisfactory explanation for a large group of debris discs that show smooth outer edges and SB∝r–3.5
. Systems with sharper outer edges, barring other confining agents, could still be explained by ``natural'' collisional evolution if their dynamical excitation is very low. We show that such a dynamically-cold case provides a satisfactory fit to the specific HR4796A ring.