By using the Zubarev nonequilibrium statistical operator method, and the Liouville equation with fractional derivatives, a generalized diffusion equation with fractional derivatives is obtained within the Renyi statistics. Averaging in generalized diffusion coefficient is performed with a power distribution with the Renyi parameter q.

A new approach is proposed to calculate the thermodynamic potential, which consists in reducing the relevant non-Gaussian functional integral to its Gaussian form with a renormalized "density-density" correlator. It is shown that the knowledge of the effective potential of electron-electron interaction is sufficient to calculate the thermodynamic potential in this approach.

An effective potential of electron-electron interaction and a two-particle "density-density" correlator are calculated for semi-infinite jellium. Their asymptotics at large distances between electrons are studied in a plane parallel to the surface.

Abstract. General expression for the thermodynamic potential of the model of semiinfinite jellium is obtained. By using this expression, the surface energy for infinite barrier model is calculated. The behavior of the surface energy and of chemical potential as functions of the Wigner-Seitz radius and the influence of the Coulomb interaction between electrons on the calculated values is studied. It is shown that taking into account the Coulomb interaction between electrons leads to growth of the surface energy. The surface energy is positivein the entire area of the Wigner-Seitz radius. It is shown that taking into account the Coulomb interaction between electrons leads to a decrease of the chemical potential.

The generalization of the Zubarev nonequilibrium statistical operator (NSO) method for the case of Renyi statistics is proposed when the relevant statistical operator (or distribution function) is obtained based on the principle of maximum for the Renyi entropy. The nonequilibrium statistical operator and corresponding generalized transport equations for the reduced-description parameters are obtained. A consistent description of kinetic and hydrodynamic processes in the system of interacting particles is considered as an example.