From the Boltzmann equation with non-local correlations to a standard non-linear Fokker-Planck equation

In this work, we study the formal connections between the non-linear Fokker-Planck Equation associated with the non-additive entropy and the Boltzmann Equation with the non-additive correlation functional. The collisional term following the $q$-algebra is adopted. In the derivation of the non-additive Fokker-Planck Equation, two constraints are imposed on the final result: i) that the entropic index $q$ is a characteristic parameter of the non-additive systems with a value that does not change with time, and ii) that for $q\to 1$ a smooth transition for the standard Fokker-Planck Equation is obtained.