  Physical mechanisms of phase separation in living systems can play key
physiological roles and have recently been the focus of intensive studies. The
strongly heterogeneous and disordered nature of such phenomena in the
biological domain poses difficult modeling challenges that require going beyond
mean field approaches based on postulating a free energy landscape. The
alternative pathway we take in this work is to tackle the full statistical
mechanics problem of calculating the partition function in these systems,
starting from microscopic interactions, by means of cavity methods. We
illustrate the procedure first on the simple binary case, and we then apply it
successfully to ternary systems, in which the naive mean field approximations
are proved inadequate. We then demonstrate the agreement with lattice model
simulations, to finally contrast our theory also with experiments of coacervate
formation by associative de-mixing of nucleotides and poly-lysine in aqueous
solution. In this way, different types of evidence are provided to support
cavity methods as ideal tools for quantitative modeling of biomolecular
condensation, giving an optimal balance between the accurate consideration of
spatial aspects of the microscopic dynamics and the fast computational results
rooted in their analytical tractability.
