title: ÐÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ°Ð½Ð¶ÐµÐ²ÐµÐ½Ð° Ð´Ð»Ñ Ñ Ð¸Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐµÐ°ÐºÑÐ¸Ð¹ Ð´Ð»Ñ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ Ð¾ÑÐ½Ð¾Ð²Ð½Ð¾Ð³Ð¾ ÐºÐ¸Ð½ÐµÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ñ Ð¸Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐµÐ°ÐºÑÐ¸Ð¹
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Application of the Langevin equation for chemical reactions for the approximate solution of the main kinetic equation for chemical reactions
Creators: name: ÐÐ¸ ÐÑÐ¹ÑÐ¸
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: 2021
Subjects: ÑÑÐ¾Ñ Ð°ÑÑÐ¸ÑÐ½Ð¾ÑÑÑ
Ð¾ÑÐ½Ð¾Ð²Ð½Ð¾Ðµ ÐºÐ¸Ð½ÐµÑÐ¸ÑÐµÑÐºÐ¾Ðµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð´Ð»Ñ
Ñ Ð¸Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐµÐ°ÐºÑÐ¸Ð¹
Ð¼ÐµÑÐ¾Ð´ Ð­Ð¹Ð»ÐµÑÐ° â ÐÐ°ÑÑÑÐ¼Ñ
ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÐÐ°Ð½Ð¶ÐµÐ²ÐµÐ½Ð° Ð² Ð¿ÑÐ¾ÑÑÐµÐ¹ÑÐµÐ¹
Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸ÑâÐ³Ð¸Ð±ÐµÐ»Ð¸
stochasticity
basic kinetic equation for chemical reactions
Euler -Maruyama method
Langevin equation in the simplest birth â death model

description: Ð¦ÐµÐ»Ñ ÑÐ°Ð±Ð¾ÑÑ â ÐÐ¾ÑÑÑÐ¾ÐµÐ½Ð¸Ðµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ°Ð½Ð¶ÐµÐ²ÐµÐ½Ð° Ð´Ð»Ñ ÑÐ¸Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐµÐ°ÐºÑÐ¸Ð¹ Ð² Ð¿ÑÐ¾ÑÑÐµÐ¹ÑÐµÐ¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ-Ð³Ð¸Ð±ÐµÐ»Ð¸. ÐÐ¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ ÑÑÐ¾ÑÐ°ÑÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð¼ÐµÑÐ¾Ð´Ð° Ð­Ð¹Ð»ÐµÑÐ° â ÐÐ°ÑÑÑÐ¼Ñ Ð¸ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ñ ÑÐ¾ÑÐ½ÑÐ¼ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸ÐµÐ¼ Ð¿ÑÐ¸ ÑÐ°Ð·Ð½ÑÑ Ð·Ð½Ð°ÑÐµÐ½Ð¸ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ° k. Ð¡ÑÑÑÐºÑÑÑÐ° ÑÑÐ°ÑÑÐ¸ ÑÐ»ÐµÐ´ÑÑÑÐ°Ñ, ÑÐ½Ð°ÑÐ°Ð»Ð° Ð¼Ñ Ð¾Ð±ÑÑÐ´Ð¸Ð¼ Ð²Ð°Ð¶Ð½Ð¾ÑÑÑ ÑÑÐ¾ÑÐ°ÑÑÐ¸ÑÐ½Ð¾ÑÑÐ¸ Ð² Ð±Ð¸Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼Ð°Ñ Ð² ÑÐ°Ð·Ð´ÐµÐ»Ðµ 3. ÐÐ°Ð»ÐµÐµ Ð² ÑÐ°Ð·Ð´ÐµÐ»Ðµ 4 Ð¼Ñ Ð¾Ð±ÑÑÐ´Ð¸Ð¼ Ð¼ÐµÑÐ¾Ð´Ñ Ð´ÐµÑÐµÑÐ¼Ð¸Ð½Ð¸ÑÐ¾Ð²Ð°Ð½Ð½Ð¾Ð³Ð¾ Ð¸ ÑÑÐ¾ÑÐ°ÑÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð´Ð»Ñ ÑÐ¸Ð¼Ð¸ÑÐµÑÐºÐ¾Ð¹ ÐºÐ¸Ð½ÐµÑÐ¸ÐºÐ¸. Ð ÑÐ°Ð·Ð´ÐµÐ»Ðµ 6 Ð¿Ð¾Ð´ÑÐ¾Ð±Ð½Ð¾ Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÑÐµÐ¼ Ð¼ÐµÑÐ¾Ð´ Ð­Ð¹Ð»ÐµÑÐ° â ÐÐ°ÑÑÑÐ¼Ñ. Ð ÑÐ°Ð·Ð´ÐµÐ»Ðµ 5 ,7 Ð´Ð°ÐµÐ¼ Ð¾Ð±ÑÑÐ¶Ð´ÐµÐ½Ð¸Ñ ÑÐ¸Ð¼Ð¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ°Ð½Ð¶ÐµÐ²ÐµÐ½Ð° Ð¸ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ°Ð½Ð¶ÐµÐ²ÐµÐ½Ð° Ð² Ð¿ÑÐ¾ÑÑÐµÐ¹ÑÐµÐ¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ-Ð³Ð¸Ð±ÐµÐ»Ð¸. Ð ÑÐ°Ð·Ð´ÐµÐ»Ðµ 7 Ð¼Ñ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð¼ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ¸Ð¼Ð¸ÑÐµÑÐºÐ¾Ðµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÐÐ°Ð½Ð¶ÐµÐ²ÐµÐ½Ð° Ð¸ ÑÑÐ°Ð²Ð½Ð¸Ð²Ð°ÐµÐ¼ ÑÐ¾ÑÐ½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ. ÐÐ°ÐºÐ¾Ð½ÐµÑ, Ð¼Ñ Ð¿Ð¾Ð´Ð²Ð¾Ð´Ð¸Ð¼ Ð¸ÑÐ¾Ð³Ð¸ Ð² Ð Ð°Ð·Ð´ÐµÐ»Ðµ 8. ÐÑÐ²Ð¾Ð´: Ð¿ÑÐ¸ Ð±Ð¾Ð»ÑÑÐ¸Ñ Ð·Ð½Ð°ÑÐµÐ½Ð¸ÑÑ "k" ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ, Ð¿Ð¾Ð»ÑÑÐ°ÐµÐ¼ÑÐµ Ð¿ÑÐ¸ Ð¿Ð¾Ð¼Ð¾ÑÐ¸ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ°Ð½Ð¶ÐµÐ²ÐµÐ½Ð°, Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶Ð°ÑÑ ÑÐ¾ÑÐ½ÑÐµ Ñ Ð²ÑÑÐ¾ÐºÐ¾Ð¹ ÑÑÐµÐ¿ÐµÐ½ÑÑ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸. ÐÑÐ¸ ÑÑÐ¾Ð¼ ÑÐ´Ð°ÐµÑÑÑ Ð·Ð½Ð°ÑÐ¸ÑÐµÐ»ÑÐ½Ð¾ ÑÐ¾ÐºÑÐ°ÑÐ¸ÑÑ Ð²ÑÑÐ¸ÑÐ»Ð¸ÑÐµÐ»ÑÐ½ÑÐµ Ð·Ð°ÑÑÐ°ÑÑ.
description: The aim of the work is to construct the Langevin equation for chemical reactions in the simplest birth-death model. Simulation of a stochastic equation using the Euler - Maruyama method and comparison with the exact distribution for different values of the parameter k. The structure of this article is as follows. We first discuss the importance of stochasticity in biological systems in section 3. Further, in section 4, we will discuss the methods of deterministic and stochastic modeling for chemical kinetics. In Section 6, we present in detail the Euler - Maruyama method. In section 5, 7, we discuss the chemical Langevin equation and the Langevin equation in the simplest birth-death model. In section 7, we carry out a numerical study of the chemical Langevin equation and compare the exact solution. Finally, we summarize in Section 8. Conclusion: for large values of "k", the distributions obtained using the Langevin equation approximate the exact ones with a high degree of accuracy. At the same time, it is possible to significantly reduce computing costs.