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Non-Gaussian statistics, maxwellian derivation and stellar polytropes

In this letter we discuss the Non-gaussian statistics considering two aspects. In the first, we show that the Maxwell's first derivation of the stationary distribution function for a dilute gas can be extended in the context of Kaniadakis statistics. The second one, by investigating the stellar system, we study the Kaniadakis analytical relation between the entropic parameter $κ$ and stellar polytrope index $n$. We compare also the Kaniadakis relation $n=n(κ)$ with $n=n(q)$ proposed in the Tsallis framework.