title: Ð ÐµÑÐµÐ½Ð¸Ðµ ÐºÐ¸Ð½ÐµÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð»ÑÑÐ¼Ð°Ð½Ð° Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½ÑÑ ÑÑÐ¾Ð»ÐºÐ½Ð¾Ð²Ð¸ÑÐµÐ»ÑÐ½ÑÑ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ¾Ð²
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Solution of the Boltzmann kinetic equation using approximate collisional invariants
Creators: name: ÐÐµÑÑÑÐ¾Ð², ÐÐ¼Ð¸ÑÑÐ¸Ð¹
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2022
Subjects: ÐÐ¾Ð»ÑÑÐ¼Ð°Ð½Ð° ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ
ÑÑÐ¾Ð»ÐºÐ½Ð¾Ð²Ð¸ÑÐµÐ»ÑÐ½ÑÐµ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÑ
Ð¼ÐµÑÐ¾Ð´ Ð§ÐµÐ¿Ð¼ÐµÐ½Ð°-Ð­Ð½ÑÐºÐ¾Ð³Ð°
Ð¸Ð½ÑÐµÐ³ÑÐ°Ð» ÑÑÐ¾Ð»ÐºÐ½Ð¾Ð²ÐµÐ½Ð¸Ð¹
ÐºÐ¸Ð½ÐµÑÐ¸ÑÐµÑÐºÐ°Ñ ÑÐµÐ¾ÑÐ¸Ñ Ð³Ð°Ð·Ð¾Ð²
collision invariants
Chapman-Ensky method
collision integral
kinetic theory of gas

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«Ð ÐµÑÐµÐ½Ð¸Ðµ ÐºÐ¸Ð½ÐµÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð»ÑÑÐ¼Ð°Ð½Ð° Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½ÑÑ ÑÑÐ¾Ð»ÐºÐ½Ð¾Ð²Ð¸ÑÐµÐ»ÑÐ½ÑÑ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ¾Ð²Â». ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° ÑÐµÑÐµÐ½Ð¸Ñ ÐºÐ¸Ð½ÐµÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð»ÑÑÐ¼Ð°Ð½Ð° Ð² ÑÐ»ÑÑÐ°Ðµ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½ÑÑ ÑÑÐ¼Ð¼Ð°ÑÐ¾ÑÐ½ÑÑ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ¾Ð². ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: ÐÐ½Ð°Ð»Ð¸Ð· Ð¿ÑÐ¸Ð¼ÐµÐ½Ð¸Ð¼Ð¾ÑÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð° Ð§ÐµÐ¿Ð¼ÐµÐ½Ð°-Ð­Ð½ÑÐºÐ¾Ð³Ð° Ðº ÑÐµÑÐµÐ½Ð¸Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð»ÑÑÐ¼Ð°Ð½Ð° Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½ÑÑ ÑÑÐ¼Ð¼Ð°ÑÐ¾ÑÐ½ÑÑ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ¾Ð². ÐÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ðµ Ð²Ð¸Ð´Ð° ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ, Ð¸Ð· ÐºÐ¾ÑÐ¾ÑÐ¾Ð³Ð¾ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÐµÑÑÑ Ð¿Ð¾Ð¿ÑÐ°Ð²ÐºÐ° ÑÑÐ½ÐºÑÐ¸Ð¸ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ. ÐÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ðµ Ð¾Ð±ÑÐµÐ³Ð¾ Ð²Ð¸Ð´Ð° Ð¿ÑÐ¾ÑÑÑÐ°Ð½ÑÑÐ²ÐµÐ½Ð½Ð¾-Ð½ÐµÐ¾Ð´Ð½Ð¾ÑÐ¾Ð´Ð½Ð¾Ð¹ ÑÐ°ÑÑÐ¸ Ð¿Ð¾Ð¿ÑÐ°Ð²ÐºÐ¸. ÐÑÐ½Ð¾Ð²Ð½Ð°Ñ Ð¿ÑÐ¾Ð±Ð»ÐµÐ¼Ð° ÑÐ°Ð±Ð¾ÑÑ Ð·Ð°ÐºÐ»ÑÑÐ°ÐµÑÑÑ Ð² ÑÐ¾Ð¼, ÑÑÐ¾ Ð´Ð»Ñ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½ÑÑ ÑÑÐ¼Ð¼Ð°ÑÐ¾ÑÐ½ÑÑ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ¾Ð², Ð² Ð¾ÑÐ»Ð¸ÑÐ¸Ðµ Ð¾Ñ ÑÐ¾ÑÐ½ÑÑ ÑÑÐ¼Ð¼Ð°ÑÐ¾ÑÐ½ÑÑ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ¾Ð², Ð½Ðµ Ð²ÑÐ¿Ð¾Ð»Ð½ÑÐµÑÑÑ Ð°Ð»ÑÑÐµÑÐ½Ð°ÑÐ¸Ð²Ð° Ð¤ÑÐµÐ´Ð³Ð¾Ð»ÑÐ¼Ð°, Ð¸Ð·-Ð·Ð° ÑÐµÐ³Ð¾ ÑÐµÑÐµÐ½Ð¸Ðµ ÑÑÐ°Ð½Ð¾Ð²Ð¸ÑÑÑÑ Ð½Ðµ ÐµÐ´Ð¸Ð½ÑÑÐ²ÐµÐ½Ð½ÑÐ¼. ÐÐ»Ñ ÑÐµÑÐµÐ½Ð¸Ñ ÑÑÐ¾Ð¹ Ð¿ÑÐ¾Ð±Ð»ÐµÐ¼Ñ Ð¿ÑÐµÐ´Ð»Ð°Ð³Ð°ÐµÑÑÑ Ðº ÐºÐ²Ð°Ð·Ð¸ÑÐ°Ð²Ð½Ð¾Ð²ÐµÑÐ½Ð¾Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¸ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ Ð´Ð¾Ð±Ð°Ð²Ð¸ÑÑ Ð¿ÑÐ¸Ð±Ð°Ð²ÐºÑ, Ð±Ð»Ð°Ð³Ð¾Ð´Ð°ÑÑ ÐºÐ¾ÑÐ¾ÑÐ¾Ð¹ Ð°Ð»ÑÑÐµÑÐ½Ð°ÑÐ¸Ð²Ð° Ð¤ÑÐµÐ´Ð³Ð¾Ð»ÑÐ¼Ð° Ð²ÑÐ¿Ð¾Ð»Ð½Ð¸ÑÑÑÑ. Ð ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐ»Ð°ÑÑ Ð¸Ð´ÐµÐ¾Ð»Ð¾Ð³Ð¸Ñ Ð¼ÐµÑÐ¾Ð´Ð° Ð§ÐµÐ¿Ð¼ÐµÐ½Ð°-Ð­Ð½ÑÐºÐ¾Ð³Ð° Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð»ÑÑÐ¼Ð°Ð½Ð°, Ñ ÑÐµÐ¼ Ð¾ÑÐ»Ð¸ÑÐ¸ÐµÐ¼, ÑÑÐ¾ Ð² ÑÐ°Ð±Ð¾ÑÐµ Ð³Ð°Ð· ÑÐ¾ÑÑÐ¾Ð¸Ñ Ð¸Ð· ÑÐ°ÑÑÐ¸Ñ Ð´Ð²ÑÑ ÑÐ°Ð·Ð½ÑÑ ÑÐ¾ÑÑÐ¾Ð². ÐÑÐ»Ð¸ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ñ Ð½ÐµÐ¾Ð±ÑÐ¾Ð´Ð¸Ð¼ÑÐµ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð²ÑÐºÐ»Ð°Ð´ÐºÐ¸, Ð¾Ð±Ð¾ÑÐ½Ð¾Ð²ÑÐ²Ð°ÑÑÐ¸Ðµ Ð²Ð¸Ð´ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ, Ð¸Ð· ÐºÐ¾ÑÐ¾ÑÐ¾Ð³Ð¾ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÐµÑÑÑ Ð¿Ð¾Ð¿ÑÐ°Ð²ÐºÐ° ÑÑÐ½ÐºÑÐ¸Ð¸ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ. ÐÑÐ» Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½ Ð²Ð¸Ð´ Ð¿Ð¾Ð¿ÑÐ°Ð²ÐºÐ¸ ÑÑÐ½ÐºÑÐ¸Ð¸ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ, Ð¸ Ð¼ÐµÑÐ¾Ð´ Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ Ð½ÐµÐ¾Ð±ÑÐ¾Ð´Ð¸Ð¼ÑÑ ÐºÐ¾ÑÑÑÐ¸ÑÐ¸ÐµÐ½ÑÐ¾Ð².
description: Topic of the graduation thesis: "Solution of the Boltzmann kinetic equation using approximate collisional invariants". This work is devoted to solving the Boltzmann kinetic equation in the case of using approximate summation invariants. Tasks that were solved in the course of the study: 1. Analysis of the applicability of the Chapman-Ensky method to the solution of the Boltzmann equation using approximate summation invariants. 2. Determination of the type of equation from which the correction of the distribution function is determined. 3. Determination of the type of amendment. The main problem of the work is that for approximate summation invariants, in contrast to exact summation invariants, the Fredholm alternative is not satisfied, which makes the solution not unique. To solve this problem, it is proposed to add an addition to the quasi-equilibrium distribution function, due to which the Fredholm alternative is fulfilled. The ideology of the Chapman-Ensky method was used in the work to solve the Boltzmann equation, with the difference that in the work the gas consists of particles of two different types. The necessary mathematical calculations were carried out, substantiating the form of the equation, from which the correction of the distribution function is determined. The type of correction of the distribution function was determined, and the method for finding the necessary coefficients.