Critical scaling through Gini index

Soumyaditya Das, Soumyajyoti Biswas

Published 2022-11-02Version 1

In the systems showing critical behavior, various response functions have a singularity at the critical point in the form $M\sim |F-F_c|^{-n}$. 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 $aF_c$ to $bF_c$ ($0<a<b\leq 1$) can be quantified by the Gini index $g$. We show that for $n<1$, the Gini index obeys a scaling of the form $|F-F_c|\sim |g-g_f|^{1/(1-n)}$, where $g_f=g(b=1)=n/(2-n)$. This implies $ {M}\sim |g-g_f|^{-n/(1-n)}$. For $1<n<2$, $|F-F_c| \sim |g-g_f|^{1/(n-1)}$, with $g_f=1$ which implies $ {M}\sim |g-g_f|^{-n/(n-1)}$ and similarly for $n>2$, $|F-F_c| \sim |g-g_f|$, with $g_f=1$, therefore $ {M}\sim |g-g_f|^{-n}$ . Since $g_f$ is either solely a function of the (universal) critical exponent $n$ or a constant, the above relations help in formulating the critical scaling behavior in quantities like $ {M}$ independent of the (non-universal) critical point of the system. We further show for $n>1$, another measure of inequality -- the Kolkata index $k$, coincides with the Gini index value at a point $F_e<F_c$. The coincidence of the two indices, therefore, serve as a crucial indicator of the imminent critical point. We demonstrate the utility of this precursory signal in the fiber bundle model for fracture of disordered solids. This approach can be used in any equilibrium or non-equilibrium systems showing critical behavior.