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

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Abstract

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.

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Notes

  1. At the same time, fractal filaments can be considered as a limiting result of the directed cluster-cluster aggregation in the external field; however, under isotropic conditions, the cluster-cluster aggregation results in formation of isotropic sparse aggregates.

  2. Fractal media are those with a fractional mass dimension (which is a physical analog of the Hausdorff dimension that requires no transition to the limit of infinitesimal diameters of the covering sets).

  3. Fractional partial derivatives are of high importance for developing the generalized hydrodynamics of hereditary nonlocal media (Uchaikin, 2008).

  4. These Rényi measures are transformed to the traditional Boltzmann−Gibbs entropy and the Kullback−Leibler discrimination information under q = 1.

  5. In connection with determining the weight-average of each random variable A, we will note the following: in the Rényi nonextensive statistics, there are three possible ways for averaging over distributions \({{p}_{i}},\)\(p_{i}^{q},\) and \({{P}_{i}}\) (see, Bibliography/ http://tsallis.cat.cbpf.br/biblio.htm). These averaging ways, each of which has its advantages and disadvantages, define the completely different q-thermodynamics, corresponding to these or those statistically anomalous systems. Due to this, a choice of the averaging way in physical applications is principled, since it turns out to be significant in the processing of experimental data (see, Tsallis et al., 1998; Tsallis, 1999; Martinez et al., 2000; Parvan and Biro, 2005; Bashkirov, 2006).

  6. We recall that the Lyapunov function is a function of fixed sign, which vanishes at the equilibrium point of a system. The equilibrium point is an attractor, when the sign of the time derivative of the Lyapunov function is opposite to that of the function itself.

  7. Self-similarity is characteristic only of regular fractals, whose manner of construction is determinate. However, if the algorithm of their generation contains a random element (as, for example, in many processes of the diffusive growing of crystals), the so-called random fractals appear; for this kind of fractal the self-similarity property is true only after averaging over all statistical realizations of the object.

  8. Depending on the task, by “a sphere” is also meant a cube, a square, or simply a straight line.

  9. The Boltzmann−Gibbs entropy is a measure of the amount of information required to define a system in some state i.

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  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    A. V. Kolesnichenko

  2. Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, 119991, Moscow, Russia

    M. Ya. Marov

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  2. M. Ya. Marov
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Correspondence to A. V. Kolesnichenko or M. Ya. Marov.

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Translated by E. Petrova

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Kolesnichenko, A.V., Marov, M.Y. Rényi Thermodynamics as a Mandatory Basis to Model the Evolution of a Protoplanetary Gas−Dust Disk with a Fractal Structure. Sol Syst Res 53, 443–461 (2019). https://doi.org/10.1134/S0038094619060042

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