Rényi Thermodynamics as a Mandatory Supporting Basis for Adequate Modeling of the Evolution of a Protoplanetary Gas-Dust Disk with a Fractal Structure

In the context of the problem of mathematical modeling of the processes of formation of protoplanetesimals in the Solar protoplanetary disk, taking into account fractal concepts about the properties of dispersed dust aggregates in the disk medium, in the paper, on the basis of the parametric Rényi entropy, statistical thermodynamics for nonextensive fractal systems has been constructed and its properties have been determined. It has been established that between the Rényi thermodynamics of nonextensive systems, on the one hand, and the technique of obtaining fractal and multifractal dimensions, based on geometry and stochastics, on the other hand, there is a close relationship. It has been shown that the time evolution of a closed thermodynamic systems to the equilibrium state depends on the sign of the deformation parameter, which is a measure of the nonextensiveness of the fractal system. Various options for constructing fractal dimensions of different orders for fractals and multifractals were discussed and their features have been analyzed. The developed approach allows, from a unified position based on generalized hydrodynamics with fractional derivatives and thermodynamics for fractal media, to model the evolution of cosmological and cosmogonic objects from galaxies and gas-dust astrophysical disks to cosmic dust, a specific feature of which is the remoteness and globality of force interactions between elements of the system, the hierarchy (usually multifractality) of geometric and phase spaces, a large range of spatio-temporal correlations, as well as the presence of asymptotically power statistical distributions.