Scaling Approach to Obtaining the Tsallis Distributions

Addendum: Jan. 5, 2026 In the note below, we consider p(e/T) as behaving as y power b (where y=e/T) for e>>T. Although this is true, the Tsallis power law form B power p where B is a linear function in y also satisfies the differential equation we obtain so that one does not only need to consider e>>T. In particular, the Kaniadakis distribution also becomes the exponential (Maxwell-Boltzmann) for q=1 where p(e) Kaniadakis = (sqrt(kkyy+1) -ky) power 1/(1-q) and k=1-q. This form also becomes y power b for e>>T, but for intermediate values of e, one does not satisfy the differential equaition, whereas the Tsallis form does. In a previous note (1), we obtained the Maxwell-Boltzmann distribution from a scaling argument based on: Eave = {Integral c1 sqrt(e) de e p(e) } / { Integral c1 sqrt(e) de p(e)}, where p(e) is unnormalized and one works in three dimensions. The scaling feature follows from e→ T e/T so that E = Number * T. Now, one may change T → T+dT and Number should not change, but the integral creating Number involves T and so one may replace T with T+dT and show that the coefficient of dT is 0. Using this approach in (1), one may explicitly find the Maxwell-Boltzmann distribution from a differential equation. Here we try to apply this approach to the Tsallis distribution. In this case, one has: Eave= { Integral c1 sqrt(e) de e p(e) power q } / { Integral c1 sqrt(e) de p(e) } power q, where p(e) is unnormalized and one works in three dimensions. The scaling approach leads to a differential equation (sitting inside an integral (0, infinite) c1 sqrt(e)de ) again. The approximation used in(1), namely that e ranges from 0 to infinite is again used. For q=1, one may show that the differential equation yields the Maxwell-Boltzmann distribution. If one suggests that for e/T large and q not 1, that one should obtain a power series: (e/T) power b, then the differential equation coefficient of dT shows that b = 1/(1-q). From the q=1 exp(-e/T) condition and the power law for large e/T, one may obtain the Tsallis solution (1- (1-q) e/T) power 1/(1-q) using scaling.