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Resistance distribution in the hopping percolation model
Yakov M Strelniker1, Shlomo Havlin, Richard Berkovits
1Minerva Center, Jack and Pearl Resnick Institute of Advanced Technology, and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel.
Summary
We analyzed effective resistance in disordered resistor networks. The distribution of resistance follows a log-normal function, depending on disorder strength and network size.
Area of Science:
- Physics
- Statistical Mechanics
- Network Science
Background:
- Random resistor networks are crucial for understanding conductivity in disordered materials.
- Percolation models describe the formation of conducting paths in such systems.
- Disorder in bond conductivity significantly impacts network properties.
Purpose of the Study:
- To investigate the distribution function of effective resistance in 2D and 3D random resistor networks.
- To analyze the behavior of resistance distribution under varying degrees of disorder.
- To determine the functional form and scaling of the resistance distribution.
Main Methods:
- Simulating random resistor networks with bond conductivity following an exponential distribution.
- Analyzing the distribution function P(rho) of effective resistance (rho).
- Examining networks in both strong-disorder (L/kappa(nu) > 1) and extreme-disorder (L/kappa(nu) < 1) regimes.
Main Results:
- The distribution function P(rho) depends only on the ratio L/kappa(nu), a measure of disorder relative to network size.
- P(rho) can be accurately approximated by a log-normal distribution.
- The dispersion of the log-normal distribution is proportional to kappa(nu)/L.
Conclusions:
- The effective resistance distribution in disordered resistor networks is robust across different disorder regimes.
- A log-normal distribution provides a universal description for resistance distribution, dependent on disorder and size.
- The findings offer insights into the statistical properties of conductivity in complex disordered systems.