Nonextensive Statistical Mechanics

38 3 Generalizing What We Learntwhose inverse function isWe may next wish to consider the following one:whose solution isIts inverse function isand satisfies of coursedydxy = x − 1 . (3.6)= y (y(0) = 1) , (3.7)y = e x . (3.8)y = ln x , (3.9)ln(x A x B ) = ln x A + ln x B . (3.10)Is it possible to unify the three differential equations we considered up to now (i.e.,(3.1), (3.4), and (3.7))? Yes indeed. It is enough to considerdydx= a + by (y(0) = 1) , (3.11)and play with the two parameters a and b. Is it possible to unify the same threedifferential equations with only one parameter? Yes indeed, ...out of linearity!JustconsiderIts solution isIts inverse isdydx = yq (y(0) = 1; q ∈ R) . (3.12)y = [1 + (1 − q)x] 1/(1−q) ≡ e x q (e x 1 = ex ) . (3.13)y = x 1−q − 11 − qand satisfies the following property:≡ ln q x (x > 0; ln 1 x = ln x) , (3.14)ln q (x A x B ) = ln q x A + ln q x B + (1 − q)(ln q x A )(ln q x B ) . (3.15)