  We review various methods to investigate the statics and the dynamics of
collective composition fluctuations in dense polymer mixtures within
fluctuating-field approaches. The central idea of fluctuating-field theories is
to rewrite the partition function 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 like the average composition and the structure factor can be expressed
exactly as averages of these fields. We discuss different analytical and
numerical approaches to studying such a theory: The self-consistent field
approach solves the integrals over the fluctuating fields in saddle-point
approximation. Generalized random phase approximations allow 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 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 approximation
is accurate. Finally, we discuss recent approaches to formulate realistic
dynamical 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.
