Stability Method and the Fokker-Planck Equation

In (1), we introduced a stability method which yields, using a single differential equation, the Maxwell-Boltzmann, Tsallis and Kaniadakis distributions. The equation is: F(p(ei)) + p(ei) dF/dp(ei) = a1F(p(ei) + a2 ((1)), where F(p(ei))= -ei/T for p(ei) unnormalized. It is well known that there exists a Fokker-Planck differential equation which may also yield the Maxwell-Boltzmann equation under certain conditions (2). The Fokker-Planck equation uses -ei/T = -V(x)/T, where V(x) is potential and is of the form: dp/dt partial = -d/dx { u(x) p(x) } + d/dx d/dx {D(x,t) p(x,t)}. For an equilibrium solution, one seeks dp/dt partial=0 and p(x,t)=p(x). In such a case, one may factor out a d/dx leaving an equation of the form: u(x)p(x) + D dp/dx =constant if one sets D(x,t)=D as in (2). This solution from (2) matches the Maxwell-Boltzmann case. Here we try to show how the differential equation ((1)) matches the Fokker-Planck equilibrium equation for the Maxwell-Boltzmann, Tsallis and Kaniadakis ca