Rényi Thermodynamics as a Mandatory Basis to Model the Evolution of a Protoplanetary Gas−Dust Disk with a Fractal Structure

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For the purpose of mathematical simulations of the formation processes for planetesimals in the Solar protoplanetary disk, statistical thermodynamics for nonextensive fractal systems was developed and its properties were determined on the basis of the Rényi parametric entropy taking account of fractal conceptions on the properties of disperse dust aggregates in the disk medium. It has been found that there is a close relationship between the Rényi thermodynamics of nonextensive systems on the one hand and the technique for obtaining fractal and multifractal dimensions based on geometry and stochastics on the other. It has been shown that temporal evolution of a closed system to the equilibrium state depends on a sign of the deformation parameter that is a measure of nonextensivity of a fractal system. Different scenarios for constructing fractal dimensions of various orders for fractals and multifractals are discussed and their properties are analyzed. The approach developed allows the evolution of cosmologic and cosmogonic objects, from galaxies and gas−dust astrophysical disks to cosmic dust, to be modeled on the basis of the generalized thermodynamics with fractional derivatives and the thermodynamics for fractal media. A specific feature of these objects is remoteness and globality of force interactions between the system elements, a hierarchical pattern (usually, multifractality) of the geometric and phase spaces, a significant range of spatial−temporal correlations, and the prevalence of asymptotic power-law statistical distributions.