The phase behavior is investigated for systems composed
of a large
number of macromolecular components N, with N ≥ 2. Liquid–liquid phase separation is modeled
using a virial expansion up to the second order of the concentrations
of the components. Formal analytical expressions for the spinodal
manifolds in N dimensions are derived, which simplify
their calculation (by transforming the original problem into inequalities
that can be evaluated numerically using linear programming techniques).
In addition, a new expression is obtained to calculate the critical
manifold and composition of the coexisting phases. The present analytical
procedure complements previous attempts to handle spinodal decomposition
for many components using a statistical approach based on random matrix
theory. The results are relevant for predicting the effects of polydispersity
on phase behavior in fields like polymer or food science and liquid–liquid
phase separation in the cytosol of living cells.