Cluster growth poised on the edge of break-up, II: From reaction kinetics to thermodynamics

The cluster size distributions of power-law form n(s)∝s−τ with small exponents 0<τ<1 are ubiquitous in many naturally occurring growth processes, where one may expect that aggregation driven cluster growth is poised on the edge of cluster break-up. We propose here a statistical thermodynamics description of such a growth process governed by size dependent aggregation and break-up rates of form sα with 0<α<2. By using the maximum entropy method the energy levels and statistical ensemble corresponding the kinetic model are deduced and α is identified as the inverse of thermodynamic temperature, conjugated in the standard way to the total energy E of the system. In addition, the macroscopic free energy F, the entropy S and the heat capacity C are derived. The thermodynamic behavior of the system strongly suggests that at α≈1 there is a phase transition in growth.