We review various methods to investigate the statics and the dynamics of
collective composition fluctuations in dense polymer mixtures within fluctuating-field2 M. Müller · F. Schmid
approaches. The central idea of fluctuating-field theories is to rewrite the partition func￾tion of the interacting multi-chain systems in terms of integrals over auxiliary, often
complex, fields, which are introduced by means of appropriate Hubbard–Stratonovich
transformations. Thermodynamic averages such as the average composition and the
structure factor can be expressed exactly as averages of these fields. We discuss differ￾ent analytical and numerical approaches to studying such a theory: The self-consistent
field approach solves the integrals over the fluctuating fields in saddle-point approxi￾mation. Generalized random phase approximations allow one to incorporate Gaussian
fluctuations around the saddle point. Field theoretical polymer simulations are used to
study the statistical mechanics of the full system with Complex Langevin or Monte Carlo
methods. Unfortunately, they are hampered by the presence of a sign problem. In a dense
system, the latter can be avoided without losing essential physics by invoking a saddle
point approximation for the complex field that couples to the total density. This leads to
the external potential theory. We investigate the conditions under which this approxima￾tion is accurate. Finally, we discuss recent approaches to formulate realistic time evolution
equations for such models. The methods are illustrated by two examples: A study of
the fluctuation-induced formation of a polymeric microemulsion in a polymer-copolymer
mixture and a study of early-stage spinodal decomposition in a binary blend.