Context : Rings or annulus-like features have been observed in most imaged debris discs .
Outside the main ring , while some systems ( e.g. , \beta Pictoris and AU Mic ) exhibit smooth surface brightness profiles ( SB ) that fall off roughly as \sim 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 .

Aims : 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 .

Methods : 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 .

Results : 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 ( 2006 ) 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 > \leq 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 , can not be ruled out by observations for most systems , .

Conclusions : Our “ standard ” profile provides a satisfactory explanation for a large group of debris discs that show smooth outer edges and SB \propto 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 .