ISSN 0278-6419 (*printed)
ISSN 1934-8428 (electronic version)
ISSN 1934-8428 (electronic version)
Recieved: 02/23/2026
Accepted: 03/03/2026
Keywords: Boltzmann equation, Kolmogorov–Fokker–Planck equation, Navier–Stokes equation, equations of stochastic gas dynamics and quasi gas dynamics, random processes, SDE with respect to Bernoulli and Wiener measures, particle methods, Monte-Carlo methods
Available in the on-line version with: 15.09.2026
Bogomolov S.V., Esikova N. B. Stochastic lagrangian approach to the equations of gas dynamics. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2026. N 3, p.3-19 https://doi.org/10.55959/MSU/0137–0782–15–2026–50–3–3–19.
This paper demonstrates the process of deriving macroscopic equations of gas dynamics from a stochastic microscopic model describing the behavior of gases from hard spheres. As the Knudsen number decreases, the resulting equations acquire a form similar to quasi-gasdynamic (QGD) equations with regularizing components that differ from the classical Navier–Stokes system.
A classic example of a simple discrete problem of the motion of a gas consisting of absolutely elastic spheres is considered. This example illustrates the key stages of constructing mathematical models for large systems using stochastic differential equations (SDE) and their computer implementation.
Ludwig Boltzmann derived his fundamental equation based on the balance of the distribution function describing the gas as a continuous medium in the phase space of coordinates and velocities, that is, using the Eulerian approach. We consider a gas as a set of individual particles, the dynamics of which is described by the SDE system, which means the Lagrange formalism.
The proposed model is self-sufficient and does not require setting additional equations of state or configurable parameters.
