ИСТИНА

All observed macroscopic objects consist of a huge number of microscopic parts. The most natural way to describe them mathematically is the apparatus of random processes, which fits well into the architectures of modern high – performance computing systems. We give two classical examples demonstrating the main stages of constructing mathematical models of large systems and their computer implementations. We use stochastic differential equations (SDE) and numerical methods to solve them. The first example relates to the clear problem of the motion of gas from hard spheres, which led Ludwig Boltzmann to his famous equation. He derived it based on the balance of the distribution function describing the gas as a continuous medium in the phase space of coordinates and velocities. In contrast to this Eulerian approach, we consider a gas as a set of particles whose dynamics are described by a system of stochastic differential equations in discontinuous measures, thereby applying Lagrangian formalism. The model is self – sufficient, no parameters that need to be configured are required. The second example shows how to construct a microscopic stochastic model of population dynamics or the spread of infection based on information about the interaction of the agents in question. Traditionally, such phenomena are described by deterministic macroscopic models.