The classical theory of multicomponent nucleation, including binary nucleation, ternary nucleation, etc., makes similar key assumptions as in CNT for single-component nucleation. Clusters are modeled as spherical liquid droplets consisting of an ideal multicomponent solution. The surface tension is assumed to equal that of a flat surface of liquid having the same composition, in equilibrium with its multicomponent vapor. The Gibbs free energy of cluster formation is a function of the cluster’s size and composition and of the gas-phase partial pressures of each component. The critical size and composition are found at a saddle point on the multidimensional free energy surface, where the free energy of cluster formation is a maximum in one direction and a minimum in all orthogonal directions. Several improvements have been proposed within the framework of classical theory. These include models of the cluster growth trajectory near the critical point; studies that account for composition-dependent surface tension; and models that consider the existence of surface-active layers that cause the chemical composition near the cluster surface to be different than in the core.

Optimisation of two-variable functions is motivated via the example of a firm producing two goods, where the concepts of complementary and substitute goods are discussed. The general idea of a critical point is discussed and it is explained that there can be different types of critical point: maxima, minima and saddle points. It is explained how the second partial derivatives may be used to determine what type of critical point one has. This is shown explicitly for the case in which the two-variable function is quadratic, and then stated in general. Examples involving profit-maximisation are given.

We introduce the notion of a mathematical game. We give examples and classify them into various types, such as two-person games vs. n-person games (where n > 2), and zero-sum vs. constant-sum vs. variable-sum games. We carefully delineate the assumptions under which we operate in game theory. We illustrate how two-person games can be described by payoff matrices or by game trees. Using examples, including an analysis of the Battle of the Bismarck Sea from World War II, we develop the notions of a strategy, dominant strategy, and Nash equilibrium point of a game. Specializing to constant-sum games, we show the equivalence between Nash equilibrium and saddle point of a payoff matrix. We then consider games where the payoff matrix has no saddle point and develop the notion of a mixed strategy, after a quick review of some basic probability notions. Finally, we introduce the minimax theorem, which states that all constant-sum games have an optimal solution, and give a novel proof of the theorem in case the payoff matrix is 2 x 2.

This chapter introduces optimal recovery games on Banach spaces, presents their natural lift to mixed strategies, and then characterizes their saddle points interms of Gaussian measures, cylinder measures, and fields. The canonical Gaussian field is shown to be a universal field in the sense that its conditioningwith respect to linear measurements producesoptimal strategies. When those measurements form a nested hierarchy, hierarchies of optimal approximations form a martingale obtained by conditioningthe Gaussian field on the filtration formed by those measurements.