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Condensed Matter > Statistical Mechanics
Title: Critical scaling through Gini index
(Submitted on 2 Nov 2022 (v1), revised 9 Jul 2023 (this version, v3), latest version 23 Sep 2023 (v4))
Abstract: In the systems showing critical behavior, various response functions have a singularity at the critical point in the form $M\propto |F-F_c|^{-n}$. The value of $ {M}$, therefore, changes drastically as the driving field $F$ is tuned towards its critical value $F_c$. The inequality in the values of $M$, within a range $F=aF_c$ to $F=bF_c$, can be quantified through the Gini index ($g$). When the range is extended till the critical point $b \to 1$, then $g\to g_f$, where $g_f$ is either a function of the critical exponent $n$ (if $0<n<1$) or is 1. The response function now can be written in a form free from the non-universal critical point $F_c$, as $M\propto |g-g_f|^{-n^*}$, where $n^*$ and $g_f$ are only dependent on $n$, the universal critical exponent. The finite size and other scalings change accordingly. This is useful in determining the critical point and exponent values for systems where it is not known. Also, another measure of inequality, the Kolkata index ($k$) crosses $g$ at a point just prior to the critical point. Therefore, measuring $g$ and $k$ for a system where the critical point is not known, can produce a precursory signal for the imminent criticality. This could be useful in many systems, including fracture. The generality and numerical validity of the calculations are shown for the Monte Carlo simulations of the two dimensional Ising model, site percolation on square lattice and the fiber bundle model of fracture. The results are, however, applicable to any equilibrium or non-equilibrium critical phenomenon.
Submission history
From: Soumyajyoti Biswas [view email][v1] Wed, 2 Nov 2022 17:04:15 GMT (278kb)
[v2] Thu, 16 Mar 2023 16:14:47 GMT (277kb)
[v3] Sun, 9 Jul 2023 04:35:37 GMT (854kb)
[v4] Sat, 23 Sep 2023 11:22:34 GMT (855kb)
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