Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices
In this study, an in-depth analysis of the percolation phenomenon for square matrices with dimensions from <i>L</i> = 50 to 600 for a sample number of 5 × 10<sup>4</sup> was performed using Monte Carlo computer simulations. The percolation threshold value was defined as the n...
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MDPI AG
2023-12-01
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| Series: | Energies |
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| Online Access: | https://www.mdpi.com/1996-1073/16/24/8024 |
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| author | Pawel Zukowski Pawel Okal Konrad Kierczynski Przemyslaw Rogalski Vitalii Bondariev Alexander D. Pogrebnjak |
| author_facet | Pawel Zukowski Pawel Okal Konrad Kierczynski Przemyslaw Rogalski Vitalii Bondariev Alexander D. Pogrebnjak |
| author_sort | Pawel Zukowski |
| collection | DOAJ |
| description | In this study, an in-depth analysis of the percolation phenomenon for square matrices with dimensions from <i>L</i> = 50 to 600 for a sample number of 5 × 10<sup>4</sup> was performed using Monte Carlo computer simulations. The percolation threshold value was defined as the number of conductive nodes remaining in the matrix before drawing the node interrupting the last percolation channel, in connection with the overall count of nodes within the matrix. The distributions of percolation threshold values were found to be normal distributions. The dependencies of the expected value (mean) of the percolation threshold and the standard deviation of the dimensions of the matrix were determined. It was established that the standard deviation decreased with the increase in matrix dimensions, ranging from 0.0262253 for a matrix with <i>L</i> = 50 to 0.0044160 for <i>L</i> = 600, which is almost six-fold lower. The mean value of the percolation threshold was practically constant and amounted to approximately 0.5927. The analysis involved not only the spatial distributions of nodes interrupting the percolation channels but also the overall patterns of node interruption in the matrix. The distributions revealed an edge phenomenon within the matrices, characterized by the maximum concentration of nodes interrupting the final percolation channel occurring at the center of the matrix. As they approached the edge of the matrix, their concentration decreased. It was established that increasing the dimensions of the matrix slowed down the rate of decrease in the number of nodes towards the edge. In doing so, the area in which values close to the maximum occurred was expanded. Based on the approximation of the experimental results, formulas were determined describing the spatial distributions of the nodes interrupting the last percolation channel and the values of the standard deviation from the matrix dimensions. The relationships obtained showed that with increasing matrix dimensions, the edge phenomenon should gradually disappear, and the percolation threshold standard deviation values caused by it will tend towards zero. |
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| id | doaj.art-289b6b72fb9b4d3e895260d8db618a16 |
| institution | Directory Open Access Journal |
| issn | 1996-1073 |
| language | English |
| last_indexed | 2024-03-08T20:48:29Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
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| series | Energies |
| spelling | doaj.art-289b6b72fb9b4d3e895260d8db618a162023-12-22T14:05:51ZengMDPI AGEnergies1996-10732023-12-011624802410.3390/en16248024Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square MatricesPawel Zukowski0Pawel Okal1Konrad Kierczynski2Przemyslaw Rogalski3Vitalii Bondariev4Alexander D. Pogrebnjak5Department of Economics, Vincent Pol University in Lublin, 2, Choiny Str., 20-816 Lublin, PolandDepartment of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, PolandDepartment of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, PolandDepartment of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, PolandDepartment of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, PolandFaculty of Electronics and Information Technology, Sumy State University, 2, Rymskogo-Korsakova Str., 40007 Sumy, UkraineIn this study, an in-depth analysis of the percolation phenomenon for square matrices with dimensions from <i>L</i> = 50 to 600 for a sample number of 5 × 10<sup>4</sup> was performed using Monte Carlo computer simulations. The percolation threshold value was defined as the number of conductive nodes remaining in the matrix before drawing the node interrupting the last percolation channel, in connection with the overall count of nodes within the matrix. The distributions of percolation threshold values were found to be normal distributions. The dependencies of the expected value (mean) of the percolation threshold and the standard deviation of the dimensions of the matrix were determined. It was established that the standard deviation decreased with the increase in matrix dimensions, ranging from 0.0262253 for a matrix with <i>L</i> = 50 to 0.0044160 for <i>L</i> = 600, which is almost six-fold lower. The mean value of the percolation threshold was practically constant and amounted to approximately 0.5927. The analysis involved not only the spatial distributions of nodes interrupting the percolation channels but also the overall patterns of node interruption in the matrix. The distributions revealed an edge phenomenon within the matrices, characterized by the maximum concentration of nodes interrupting the final percolation channel occurring at the center of the matrix. As they approached the edge of the matrix, their concentration decreased. It was established that increasing the dimensions of the matrix slowed down the rate of decrease in the number of nodes towards the edge. In doing so, the area in which values close to the maximum occurred was expanded. Based on the approximation of the experimental results, formulas were determined describing the spatial distributions of the nodes interrupting the last percolation channel and the values of the standard deviation from the matrix dimensions. The relationships obtained showed that with increasing matrix dimensions, the edge phenomenon should gradually disappear, and the percolation threshold standard deviation values caused by it will tend towards zero.https://www.mdpi.com/1996-1073/16/24/8024percolation phenomenonpercolation thresholduncertainty of measurementmetrological approachcomputer simulationMonte Carlo method |
| spellingShingle | Pawel Zukowski Pawel Okal Konrad Kierczynski Przemyslaw Rogalski Vitalii Bondariev Alexander D. Pogrebnjak Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices Energies percolation phenomenon percolation threshold uncertainty of measurement metrological approach computer simulation Monte Carlo method |
| title | Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices |
| title_full | Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices |
| title_fullStr | Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices |
| title_full_unstemmed | Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices |
| title_short | Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices |
| title_sort | monte carlo simulation of percolation phenomena for direct current in large square matrices |
| topic | percolation phenomenon percolation threshold uncertainty of measurement metrological approach computer simulation Monte Carlo method |
| url | https://www.mdpi.com/1996-1073/16/24/8024 |
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