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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.01553 |
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| _version_ | 1866917727967379456 |
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| author | Ghosh, Asim Manna, S. S. Chakrabarti, Bikas K. |
| author_facet | Ghosh, Asim Manna, S. S. Chakrabarti, Bikas K. |
| contents | We define the $Q$-factor in the percolation problem as the quotient of the size of the
largest cluster and the average size of all clusters. As the occupation probability $p$
is increased, the $Q$-factor for the system size $L$ grows systematically to its maximum
value $Q_{max}(L)$ at a specific value $p_{max}(L)$ and then gradually decays. Our numerical
study of site percolation problems on the square, triangular and the simple cubic lattices
exhibits that the asymptotic values of $p_{max}$ though close, are distinctly different
from the corresponding percolation thresholds of these lattices. We have also shown using
the scaling analysis that at $p_{max}$ the value of $Q_{max}(L)$ diverges as $L^d$ ($d$
denoting the dimension of the lattice) as the system size approaches to their asymptotic
limit. We have further extended this idea to the non-equilibrium systems such as the
sandpile model of self-organized criticality. Here, the $Q(ρ,L)$-factor is the quotient
of the size of the largest avalanche and the cumulative average of the sizes of all the
avalanches; $ρ$ being the drop density of the driving mechanism. This study has been
prompted by some observations in Sociophysics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Q-factor: A measure of competition between the topper and the average in percolation and in SOC Ghosh, Asim Manna, S. S. Chakrabarti, Bikas K. Statistical Mechanics We define the $Q$-factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability $p$ is increased, the $Q$-factor for the system size $L$ grows systematically to its maximum value $Q_{max}(L)$ at a specific value $p_{max}(L)$ and then gradually decays. Our numerical study of site percolation problems on the square, triangular and the simple cubic lattices exhibits that the asymptotic values of $p_{max}$ though close, are distinctly different from the corresponding percolation thresholds of these lattices. We have also shown using the scaling analysis that at $p_{max}$ the value of $Q_{max}(L)$ diverges as $L^d$ ($d$ denoting the dimension of the lattice) as the system size approaches to their asymptotic limit. We have further extended this idea to the non-equilibrium systems such as the sandpile model of self-organized criticality. Here, the $Q(ρ,L)$-factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches; $ρ$ being the drop density of the driving mechanism. This study has been prompted by some observations in Sociophysics. |
| title | Q-factor: A measure of competition between the topper and the average in percolation and in SOC |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2402.01553 |