ИСТИНА

It is well known that a human beeing is more complecated than the universe not only because of that the body, brain, blood, organs consist of a huge number of microscopic particles but mostly of that the complex structure gives birth to a huge number of processes which we call life. Manifestations of life processes are a macroscopic result of that interactions. Science has accumulated a vast amount of knowledge on both micro and macro levels. The both levels are important in gaining an understanding which means control. The quantitative analysis of these big data for understanding the laws of the functioning of complex systems can be carried out only on the basis of mathematical modeling. The latter has to include the proper description of micro and macro levels. Moreover, they have to be properly connected. Micro models are hard to observe but easy to imagine. They are the origin of what is going on at macro level. The complexity of micro models is overcome by the probability tools. At the example of a simple and clear, but far from trivial, model of hard sphere gas, we will try to show the main stages in constructing the mathematical formalization of such a connection. This methodology is developed by many scientists, in particular, in the biological and sociological contexts. We are considering a set of about 1025 solid balls that just fly and collide. A mathematical description of the evolution of such a system inevitably leads to the necessity of using the apparatus of the theory of random processes. To identify the mathematical and computational features of the problem under study it is important to write it in a dimensionless form. This procedure leads to the appearance of the Knudsen number, the physical meaning of which is the ratio of the mean free path of molecules to the characteristic size of the problem. The hierarchy of micro-macro models is constructed in accordance with the change in this parameter from values ​​of the order of unity (micro) to magnitudes of the order of 0.1 (meso) and further to 0.01 (macro). Accurate movement along this path leads to more accurate, in comparison with traditional, mathematical models, which affects their greater computational fitness - nature pays for a careful attitude towards it. In particular, macroscopic equations are obtained softer for calculations than the classical Navier-Stokes equations. This hierarchy of mathematical statements generates a corresponding chain of computational methods. Microscopic problems are most often solved using Monte Carlo methods, although there are groups that are committed to nonrandom methods for solving the Boltzmann equation. Recently, much attention has been paid to meso models based on modeling the Brownian motion or solving the deterministic Fokker - Planck - Kolmogorov equations. To solve the problems of a continuous medium, different approaches are used: difference methods, finite element methods, and particle methods. The latter, in our opinion, are particularly promising for the entire hierarchy, uniting different statements with a single computational ideology. References [1] L.Boltzmann. Weitere Studien ueber das Waerme gleichgenicht unfer Gasmolaekuler. Sitzungsberichte der Akademie der Wissenschaften. 66 (1872), 275-370. [2] A. V. Skorokhod, Stochastic Equations for Complex Systems. Moscow: Nauka, 1983; (Dordrecht: Kluwer Academic, 1987). [3] A. A. Arsen’ev. On the approximation of the solution of the Boltzmann equation by solutions of the Itô stochastic differential equations. USSR Comput. Math. Math. Phys. 1987, 27 (2), 51–59. [4] S. V. Bogomolov, N. B. Esikova, A. E. Kuvshinnikov. Micro-Macro Kolmogorov–Fokker–Planck Models for a Rigid-Sphere Gas. Mathematical Models and Computer Simulations, 2016, 8(5), 533–547.