Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment
Abstract
:1. Introduction
- The oscillator frequency is random and there is no external field;
- The oscillator frequency is random and the external field is a regular function;
- The frequency of the oscillator is a regular function and the external force is random.
2. Problem
2.1. Statement of the Problem
- When randomness in a JS generates a complex process , and the second source of the random process is absent , and, accordingly,
- when and randomness in a JS generates the generator , which has a complex character.
2.2. Derivation of Environmental Fields Distribution Equations
3. The Mathematical Expectation of the Trajectory
3.1. The Measure of the Functional Space
3.2. The Stages of the Expected Trajectory Calculation
4. Geometric and Topological Features of a Compactified Space
4.1. Geometry of Two-Dimensional Subspace
4.2. Topology of Two-Dimensional Subspace
5. Statement of the Initial-Boundary Value Problem for the Complex PDE
6. Entropy of a Self-Organizing System
7. Numerical Methods for Solving the Problem
- To calculate Equations (18) and (19), it is also necessary to set two boundary conditions in the form of difference equations on the coordinate axes and , respectively, which can be easily found by approximating Equation (87). Note that these difference equations must be solved taking into account the Dirichlet boundary conditions; , where the index denotes the first and second boundary conditions, respectively.
- The condition is set at the center of the coordinate axes:In addition, the Dirichlet condition is specified on the boundaries of the computational domain, where denotes the boundary.
- As an initial condition, instead of the Dirac delta function, we use the Gaussian distribution:Note that the parameters and included in the function were chosen in such a way to normalize the initial distribution to unit.
- The continuous region for the PDEs system (66) is replaced by a discrete grid, as described in Listing 1.
- Using the PDEs system (66), we can obtain the following system of difference equations on the constructed grid:
- Similarly, as in Listing 1, for the solutions and , boundary conditions are set on the and axes in the form of difference equations, which can be obtained by approximating Equation (88) on the same axes. Note that we solve each equation obtained for the boundary conditions as an internal Dirichlet problem with a zero value of the solution at the boundary.
- As in the case of the probability density (see Listing 1), the solutions and are subject to similar conditions at the center of the coordinate axes:In addition, we will assume that at the boundary of the computational domain, the solutions are subject to the following conditions:
- Finally, as an initial condition for solving the system of Equations (88) for both solutions and , Gaussian distribution with parameters as in Listing 1 is chosen.
7.1. Distributions of the Free Environmental Fields
- When the processes going on in the environment, both elastic and inelastic, are weak;
- When elastic processes are strong and inelastic processes are weak;
- When both elastic and inelastic processes are strong in the environment.
7.2. Distributions of Environmental Fields Taking into Account the Influence of the Oscillator
7.3. Mathematical Expectation of the Oscillator Trajectory
7.4. Calculation of Topological and Geometric Features of the Manifold
7.5. Features of Entropy Calculation
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
SEq | Statistical Equilibrium |
PDE | Partial Differential Equation |
SE | Small Environment |
JS | Joint System |
SDE | Stochastic Differential Equation |
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Gevorkyan, A.S.; Bogdanov, A.V.; Mareev, V.V.; Movsesyan, K.A. Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment. Mathematics 2022, 10, 3868. https://doi.org/10.3390/math10203868
Gevorkyan AS, Bogdanov AV, Mareev VV, Movsesyan KA. Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment. Mathematics. 2022; 10(20):3868. https://doi.org/10.3390/math10203868
Chicago/Turabian StyleGevorkyan, Ashot S., Aleksander V. Bogdanov, Vladimir V. Mareev, and Koryun A. Movsesyan. 2022. "Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment" Mathematics 10, no. 20: 3868. https://doi.org/10.3390/math10203868
APA StyleGevorkyan, A. S., Bogdanov, A. V., Mareev, V. V., & Movsesyan, K. A. (2022). Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment. Mathematics, 10(20), 3868. https://doi.org/10.3390/math10203868