Account for the generalized derivative and the collective influence of phases on the homogenization process

The effective transfer coefficients of a heterogeneous medium are obtained based on the formalism of a generalized derivative, which reflects the internal boundaries of a heterogeneous medium. The formula for the generalized derivative is a consequence of applying the variational apparatus to the energy functional for a heterogeneous medium, taking into account the indicator function characterizing the phase at a point. The solution is sought for the averaged Green's function based on the modified operator obtained and the averaging carried out. The solution has the form of the Yukawa potential, which characterizes from the physical point of view the transition layer caused by charge screening. It is a consequence of the integro-differential equation analysis with discontinuities and the introduced hypotheses. This potential is aimed at expressing the solution of the many-body problem in a heterogeneous medium and reflecting the collective influence of the phases on the field propagating through the system. The effective transport coefficients integrally take into account the microstructure of the system (physical properties of phases and characteristic scales) in an explicit form, which is a consequence of the found solution.