Variation of Gini and Kolkata indices with saving propensity in the Kinetic Exchange model of wealth distribution: An analytical study

We study analytically the change in wealth ($x$) distribution $P(x)$ against saving propensity $\lambda$ a closed economy, using Kinetic theory. estimate Gini ($g$) and Kolkata ($k)$ indices by deriving (using $P(x)$) Lorenz function $L(f)$, giving cumulative fraction $L$ of possessed $f$ people ordered ascending order wealth. First, exact result for when $\lambda = 0$ we derive from there index values $g$ $k$. then proceed with an approximate gamma form non-zero $\lambda$. Then results $k$ at 0.25$ as \rightarrow 1$. note that 1$ becomes Dirac $\delta$-function. Using this assuming larger centered around most probable (a $\lambda$). utilize to evaluate along known analytical expression $g$, $k(\lambda)$. These different are compared numerical (Monte Carlo) Chakraborti-Chakrabarti model. Next relation between From expressions $k$, thermodynamic mapping show former corresponds entropy latter inverse temperature.