Cluster size distributions in equilibrium

Equilibrium size distributions for finite system containingM monomers in a volumeV undergoing coagulation (polymerization)/fragmentation, are determined for coagulation rateK _ij and fragmentation rateF _ij , corresponding to the processA _i +A _j A _i+j (hereA _k denotes a cluster containingk monomeric units). It is shown that microscopic detailed balance impliesF _ij /K _ij =a _i a _j /a _i+j , where anda _k are arbitrary. The mean and most probable size distributionc _k = ^–1 a _k e ^–k coincide in the thermodynamic limitM, V, =M/V fixed. If , 2<<3, then the theory predicts a (solgel) phase transition at a well defined value ofq = ^–1, with critical exponents = and = –2. The gel fraction exponent assumes its classical value unity, probably due to the neglect of spatial fluctuations. Finally it is indicated how the theory can in principle be extended to account for isomerism and cyclization.