(199f) Thermodynamics of Distributions

The Population Balance Equation (PBE) is a mean-field equation that tracks changes in the distribution of a population under a variety of physical mechanisms such as aggregation, breakage, nucleation, etc. Here we present an ensemble theory for generic populations that allows us to treat population balances in the statistical language of formal thermodynamics. The theory reduces to the classical population balance equation when the population is infinitely large, but has the further advantage that it (a) permits the treatment of finite populations and (b) provides the tools to study problems such as gelation and shattering as formal phase transitions. The basis of the theory is the ensemble introduced by Gibbs. We construct the ensemble of all distributions obtained when M individuals are divided into N groups. An important part of the theory is the introduction of the "selection bias" as functional that biases the selection of distributions from the ensemble. We let the system pass to the thermodynamic limit, derive the most probable distribution and formulate a rigorous analogy between the cluster ensemble and thermodynamics to obtain the temperature and pressure of the population. We predict that for certain forms of the selection bias the system undergoes explosive percolation and provide an exactly solvable model that demonstrates this behavior. Finally, we discuss how this new formulation can be applied to solve common problems in particle technology.