We estimate the equilibrium size distribution of cholesterol rich micro-domains on a lipid bilayer by solving Smoluchowski equation for coagulation and fragmentation. Towards this aim, we first derive the coagulation kernels based on the diffusion behaviour of domains moving in a two dimensional membrane sheet, as this represents the reality better. We incorporate three different diffusion scenarios of domain diffusion into our coagulation kernel. Subsequently, we investigate the influence of the parameters in our model on the coagulation and fragmentation behaviour. The observed behaviours of the coagulation and fragmentation kernels are also manifested in the equilibrium domain size distribution and its first moment. Finally, considering the liquid domains diffusing in a supported lipid bilayer, we fit the equilibrium domain size distribution to a benchmark solution. © 2016 Elsevier B.V.