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Using the projection operator technique, the exact homogeneous generalized master equation (HGME) for the relevant part of a distribution function (statistical operator) is derived. The exact (mass) operator governing the evolution of the relevant part of a distribution function and comprising arbitrary initial correlations is found. Neither the Bogolyubov principle of weakening of initial correlations with time nor any other approximation such as random phase approximation has been used to obtain the HGME. These approximations are usually used to derive the approximate homogeneous equation for the relevant part of a distribution function from the conventional exact generalized master equation (GME), which has a source containing the irrelevant part (initial correlations). The HGME does not have a source and contains only the linear, relative to the relevant part of a distribution function, terms of the GME modified by the dynamics of initial correlations. The obtained equation is valid on any timescale, for any initial moment of time and any initial correlations. In particular, it describes the short-time behaviour and allows for treating the influence of initial correlations consistently. As an example, we have considered a dilute gas of classical particles. By selecting the appropriate projection operator, we have derived the homogeneous equation for a one-particle distribution function retaining initial correlations in the linear approximation on the small density parameter and for the space homogeneous case. This equation allows for considering all stages of the time evolution. It converts into the conventional Boltzmann equation on the appropriate timescale if the contribution of all initial correlations vanishes on this timescale.

The problem of finding the exact spacetime particle's propagator in the presence of a potential step (interface between different materials) is revisited. In contrast to the conventional Feynman path-integral approach, integration over all energy values of the particle's spectral density matrix (discontinuity of the energy-dependent Green's function across the real energy axis) is suggested for obtaining the exact spacetime propagator. The energy-dependent Green's functions are found in the framework of the multiple scattering theory (MST). The problem of finding the step-localized energy-dependent potentials responsible for the particle's reflection from and transmission through a potential step, which are needed for MST application, is solved. The obtained exact result for the particle's propagator is expressed in terms of integrals of elementary functions and has a significantly simpler form than that reported earlier. The obtained expressions allow easy evaluation of all limiting cases, including the case of the infinitely large potential step, as well as simple numerical visualization. The square of the absolute value of the propagator, which represents the relative transition probability density between two spacetime points, is plotted and discussed in detail for the cases of particle reflection and transmission.

A method for calculating the two-particle electron Green function (alloy conductivity) based on the cluster expansion for the scattering T-matrix is developed. Taking into account scattering processes on all pairs of atoms, an analytical expression for the conductivity of alloys with short-range and long-range order is obtained. The coherent potential approximation is selected as a zero approximation. It is shown that the change in the electronic spectrum due to ordering leads to an essential change in alloy conductivity.